Ising n-fold integrals as diagonals of rational functions and integrality of series expansions
A. Bostan, S. Boukraa, G. Christol, S. Hassani, J.-M. Maillard
aa r X i v : . [ m a t h - ph ] M a r Ising n -fold integrals as diagonals of rationalfunctions and integrality of series expansions March 17th, 2013A. Bostan ¶ , S. Boukraa || , G. Christol ‡ , S. Hassani § , J-M.Maillard £ ¶ INRIA, Bˆatiment Alan Turing, 1 rue Honor´e d’Estienne d’Orves, Campus del’´Ecole Polytechnique, 91120 Palaiseau, France || LPTHIRM and D´epartement d’A´eronautique, Universit´e de Blida, Algeria ‡ Institut de Math´ematiques de Jussieu, UPMC, Tour 25, 4`eme ´etage, 4 PlaceJussieu, 75252 Paris Cedex 05, France § Centre de Recherche Nucl´eaire d’Alger, 2 Bd. Frantz Fanon, B.P. 399, 16000Alger, Algeria £ LPTMC, UMR 7600 CNRS, Universit´e de Paris 6, Tour 23, 5`eme ´etage, case121, 4 Place Jussieu, 75252 Paris Cedex 05, France
Abstract.
We show that the n -fold integrals χ ( n ) of the magnetic susceptibilityof the Ising model, as well as various other n -fold integrals of the “Ising class”,or n -fold integrals from enumerative combinatorics, like lattice Green functions,correspond to a distinguished class of functions generalising algebraic functions:they are actually diagonals of rational functions . As a consequence, the powerseries expansions of the, analytic at x = 0, solutions of these linear differentialequations “Derived From Geometry” are globally bounded , which means that, afterjust one rescaling of the expansion variable, they can be cast into series expansionswith integer coefficients . We also give several results showing that the uniqueanalytical solution of Calabi-Yau ODEs, and, more generally, Picard-Fuchs linearODEs with solutions of maximal weights, are always diagonal of rational functions.Besides, in a more enumerative combinatorics context, generating functions whosecoefficients are expressed in terms of nested sums of products of binomial termscan also be shown to be diagonals of rational functions . We finally address thequestion of the relations between the notion of integrality (series with integercoefficients, or, more generally, globally bounded series) and the modularity ofODEs. This paper is the short version of the larger (100 pages) version [19], availableon http: // arxiv. org/ abs/ 1211. 6031 , where all the detailed proofs are given andwhere a larger set of examples is displayed.
PACS : 05.50.+q, 05.10.-a, 02.30.Hq, 02.30.Gp, 02.40.Xx
AMS Classification scheme numbers : 34M55, 47E05, 81Qxx, 32G34, 34Lxx,34Mxx, 14Kxx
Key-words : Diagonals of rational functions, Hadamard products, series withinteger coefficients, globally bounded series, differential equations Derived FromGeometry, Hauptmoduls, modular forms, Calabi-Yau ODEs, modularity, modular iagonals of rational functions iagonals of rational functions
1. Introduction
The series expansions of many magnetic susceptibilities (or many other quantities,like the spontaneous magnetisation) of the Ising model on various lattices in arbitrarydimensions are actually series with integer coefficients [1, 2, 3]. This is a consequenceof the fact that, in a van der Waerden type expansion of the susceptibility, allthe contributing graphs are the ones with exactly two odd-degree vertices and thenumber of such graphs is an integer. When series expansions in theoretical physics,or mathematical physics, do not have such an obvious counting interpretation, thepuzzling emergence of series with integer coefficients is a strong indication that somefundamental structure, symmetry, concept have been overlooked, and that a deeperunderstanding of the problem remains to be discovered ‡ . Algebraic functions areknown to produce series with integer coefficients . Eisenstein’s theorem [5] statesthat the Taylor series of a (branch of an) algebraic function can be recast into aseries with integer coefficients, up to a rescaling by a constant (Eisenstein constant).An intriguing result due to Fatou [6] (see pp. 368–373) states that a power serieswith integer coefficients and radius of convergence (at least) one, is either rational,or transcendental. This result also appears in P´olya and Szeg¨o’s famous Aufgabenbook [7] (see Problem VIII-167). P´olya [8] conjectured a stronger result, namelythat a power series with integer coefficients which converges in the open unit disk iseither rational, or admits the unit circle as a natural boundary (i.e. it has no analyticcontinuation beyond the unit disk). This was eventually proved ¶ by Carlson [10].Along this natural boundary line, it is worth recalling [11, 12, 13, 15, 16] that the seriesexpansions of the full magnetic susceptibility of the 2D Ising model [17] correspond toa power series with integer coefficients † . For them, the unit circle certainly arises asa natural boundary [18] (with respect to the modulus variable k ), but, unfortunately,this cannot be justified by Carlson’s theorem †† .A series with natural boundaries cannot be D-finite ♯ , i.e. solution of a lineardifferential equation with polynomial coefficients [22, 23]. For simplicity, let us restrictto series with integer coefficients (or series that have integer coefficients up to avariable rescaling), that are series expansions of D-finite functions. Wu, McCoy,Tracy and Barouch [24] have shown that the previous full magnetic susceptibilityof the 2D Ising model can be expressed (up to a normalisation factor (1 − s ) / /s ,see [13, 25]) as an infinite sum of n -fold integrals, denoted by ˜ χ ( n ) , which are actuallyD-finite § . We found out that the corresponding (minimal order) differential operatorsare Fuchsian [11, 13], and, in fact, “special” Fuchsian operators: the critical exponents ‡ The emergence of positive integer coefficients corresponds to the existence of some underlyingmeasure [4]. ¶ The P´olya-Carlson result can be used to prove that some integer sequences, such as the sequenceof prime numbers ( p n ) [9], do not satisfy any linear recurrence relation with polynomial coefficients. † In some variable w [11, 12, 13, 15]. In the modulus variable k , one needs to perform a simplerescaling by a factor 2 or 4 according to the type of (high, or low temperature) expansions. †† The radius of convergence is 1 with respect to the modulus variable k , in which the series does nothave integer coefficients, being globally bounded only (this means that it can be recast into a serieswith integer coefficients by one rescaling of the variable k ). If one considers the series expansion withrespect to another variable (such as w ) in which the series does have integer coefficients, then theradius of convergence is not 1. ♯ D-finite series are sometimes called holonomic . A priori, for multivariate functions, these notionsdiffer [19]. The equivalence of these notions is proved by deep results of Bernˇste˘ın [20] andKashiwara [21]. § For Ising models on higher dimensional lattices [1, 2, 3] no such decomposition of susceptibilities,as an infinite sum of D-finite functions, should be expected at first sight. iagonals of rational functions all their singularities are rational numbers , and their Wronskians are N -th rootsof rational functions [26]. Furthermore, it has been shown later that these ˜ χ ( n ) ’s are,in fact, solutions of globally nilpotent operators [27], or G -operators [28, 29]. It isworth noting that the series expansions, at the origin, of the ˜ χ ( n ) ’s, in a well-suitedvariable [13, 25] w , actually have integer coefficients , even if this result does not havean immediate proof † for all integers n (in contrast with the full susceptibility). Fromthe first truncated series expansions of ˜ χ ( n ) , the coefficients for generic n can beinferred [27]˜ χ ( n ) ( w ) = 2 n · w n · (cid:16) n · w + 2 · (4 n + 13 n + 1) · w + 83 · ( n + 4) (4 n + 23 n + 3) · w (1)+ 13 · (32 n + 624 n + 4006 n + 8643 n + 1404) · w + 415 · ( n + 8) · (32 n + 784 n + 6238 n + 16271 n + 3180) · w + · · · (cid:17) . Note that the coefficients of ˜ χ ( n ) ( w ) / n , which depend on n , are integer coefficients when n is any integer , this integrality property of the coefficients for any integer n being not straightforward (see [19]). These coefficients are valid up to w for n ≥ w for n ≥ w for n ≥ w for n ≥
9, and w for n ≥
11 (in particularit should be noted that ˜ χ ( n ) is an even function of w only for even n ). Furtherstudies on these ˜ χ ( n ) ’s showed the fundamental role played by the theory of ellipticfunctions ¶ (elliptic integrals, modular forms ) and, much more unexpectedly, Calabi-Yau ODEs [30, 31]. These recent structure results thus suggest to see the occurrenceof series with integer coefficients as a consequence of modularity [32] (modular forms,mirror maps [30, 31, 32, 33], etc) in the Ising model.Along this line, many other examples of series with integer coefficients emergedin mathematical physics (differential geometry, lattice statistical physics, enumerativecombinatorics, replicable functions ♯ . . . ). One must, of course, also recall Ap´ery’sresults [39]. Appendix A gives a list of modular forms , and their associated series withinteger coefficients, corresponding to various lattice Green functions [40, 41, 42, 43],that are, often, expressed in terms of HeunG functions ‡ which can be writtenas hypergeometric functions with two alternative pullbacks (see also sections (6.1)and (6.2) below). Let us underline, in Appendix A, the Green function for thediamond lattice [43], the
Green function for the face-centred cubic lattice (see equation(19) in [43]), and more examples corresponding to the spanning tree generatingfunctions [44] (and Mahler measures). This integrality is also seen in the nome andin other quantities like the
Yukawa coupling [30].In this paper we restrict on series with integer coefficients , or, more generally, globally bounded [45] series of one complex variable , but it is clear that this integralityproperty does also occur in physics with several complex variables : they can, for † We are interested in this paper in the emergence of integers as coefficients of D-finite series. Ingeneral, this emergence is not obvious : it cannot be simply explained at the level of the linearrecurrence satisfied by the coefficients, as illustrated by the case of Ap´ery’s calculations (see alsosection (6.2) and Appendix D). ¶ Which is not a surprise for Yang-Baxter integrability specialists. ♯ The concept of replicable functions is closely related to modular functions [34], (see the replicabilityof Hauptmoduls), Calabi-Yau threefolds, and more generally the concept of modularity [32, 35, 36,37, 38]. ‡ Generically HeunG functions are far from being modular forms. iagonals of rational functions § ) n -fold integrals ˜ χ ( n ) for the anisotropicIsing model [46] (or for the Ising model on the checkerboard lattice), or on the exampleof the lattice Ising models with a magnetic field ‡ (see for instance, Bessis et al. [4]).We take, here, a learn-by-example approach: on such quite technical questionsone often gets a much deeper understanding from highly non-trivial examples thanfrom sometimes too general, or slightly obfuscated, mathematical demonstrations.The main result of the paper will be to show that the ˜ χ ( n ) ’s are globally bounded series, as a consequence of the fact that they are actually diagonals of rational functionsfor any value of the integer n . We will generalise these ideas, and show that anextremely large class of problems of mathematical physics can be interpreted in termsof diagonals of rational functions : n -fold integrals with algebraic integrand of a certaintype that we will characterise, Calabi-Yau ODEs, MUM linear ODEs [48], series whosecoefficients are nested sums of products of binomials , etc.Another purpose of this paper is to “disentangle” the notion of series with integercoefficients ( integrality ) and the notion of modularity [32, 35, 36, 37, 38, 49, 50]. In this“down-to-earth” paper we essentially restrict to Picard-Fuchs ODEs and to a “Calabi-Yau” framework, therefore modularity ♯ will just mean that the series solutions ofPicard-Fuchs ODEs, as well as the corresponding nome series, and the Yukawa series,have integer coefficients.The paper is organised as follows. Section (2) introduces the main concepts weneed in this very paper, namely the concept of diagonals of rational or algebraicfunctions , and the concept of globally bounded series , recalling that diagonals ofrational or algebraic functions are necessarily globally bounded series. Section (3)shows the main result of the paper, namely that the n -fold Ising integrals ˜ χ ( n ) are diagonals of rational functions for any value of the integer n , the corresponding seriesbeing, thus, globally bounded. Section (4) shows that series with (nested sums ofproducts of) binomials coefficients are diagonals of rational functions. Section (5)discusses, in the most general framework, the conjecture that D-finite globally boundedseries could be necessarily diagonals of rational functions. Section (6) provides a setof modular forms examples (in particular lattice Green functions see Appendix A).Beyond modular forms, using new determinantal identities on the Yukawa couplings,and focusing on Hadamard products of modular forms, section (7) analyses thedifference between integrality and modularity, showing that the two concepts areactually quite different. Section (8) addresses, more specifically, the Calabi-Yaumodularity, and the difference between integrality and modularity, underlining thatthe integrality of the nome series is crucial for modularity, the integrality of theYukawa series being not sufficient. The conclusion, section (9), emphasises thedifference between the “special properties” of geometrical nature and the ones of arithmetic nature, emerging in theoretical physics. Several large appendices providedetailed examples illustrating pedagogically the previous sections. In particularAppendix A provides many modular forms examples associated with lattice Greenfunctions , and Appendix E provides new representations of Yukawa couplings as ratios § For several complex variables the ODEs of the paper are replaced by Picard-Fuchs systems. ‡ Along this line, original alternative representations of the partition function of the Ising model ina magnetic field are also worth recalling [47]. ♯ Modularity is a wider concept than this “Calabi-Yau” modularity (see modular up to a Tate twist,modular Galois representations [51]). Modular forms provide the simplest examples (see Appendix A)of modularity (see also Serre’s modularity conjecture, and the Taniyama-Shimura conjecture). For afirst introduction to these ideas see [52]. iagonals of rational functions of determinants .
2. Series integrality, diagonal of rational functions
Let us recall some concepts that will be fundamental in this paper, first the notion of globally bounded series , and, then, the concept of diagonal of a function ¶ , and some ofits most important properties. The main reason to introduce this concept of diagonal of function, not very familiar to physicists, is that it enables to consider diagonal ofrational functions , this class of functions filling the gap between algebraic functionsand G -series: they can be seen as generalisations of algebraic functions . Thus thisclass of functions can play a key role to decipher the complexity of functions occurringin theoretical physics . Let us first recall the definition of being globally bounded [45] for a series. Consider aseries expansion with rational coefficients, with non-zero radius of convergence † . Theseries is said to be globally bounded if there exists an integer N such that the seriescan be recast into a series with integer coefficients with just one rescaling x → N x .A necessary condition for being globally bounded is that only a finite number ofprimes occur as factors of the denominators of the rational number series coefficients.There is also a condition on the growth of these denominators, that must be boundedexponentially [45], in such a way that the series has a non-zero p -adic radius ofconvergence for all primes p . When this is the case, it is easy to see that theseseries can be recast, with just one rescaling, into series with integer coefficients ♯ . Assume that F ( z , . . . , z n ) = P ( z , . . . , z n ) /Q ( z , . . . , z n ) is a rational function,where P and Q are polynomials of z , · · · , z n with rational coefficients such that Q (0 , . . . , = 0. This assumption implies that F can be expanded at the origin as aTaylor series with rational number coefficients F (cid:16) z , z , . . . , z n (cid:17) = ∞ X m = 0 · · · ∞ X m n = 0 F m , ..., m n · z m · · · z m n n . (2)The diagonal of F is defined as the series of one variable Diag (cid:16) F (cid:16) z , z , . . . , z n (cid:17)(cid:17) = ∞ X m = 0 F m, m, ..., m · z m . (3)More generally, one can define, in a similar way, the diagonal of any multivariatepower series F , with rational number coefficients, or with coefficients in a finite field ‡ . ¶ The functions are in fact defined by series of several complex variables: they have to be Taylor, orLaurent, series (no Puiseux series). † A series like the Euler-series P ∞ n =0 n ! · x n which has integer coefficients is excluded. ♯ For a first set of series with integer coefficients, see Appendix A, where a set of such series withinteger coefficients corresponding to modular forms is displayed. See also (6.1) and (6.2) below. ‡ The definition even extends to multivariate Laurent power series, see e.g. [53]. iagonals of rational functions The concept of diagonal of a function has a lot of interesting properties (see forinstance [54]). Let us recall, through examples, some of the most important ones.The study of diagonals goes back, at least, to P´olya [55], in a combinatorialcontext, and to Cameron and Martin [56] in an analytical context related to Hadamardproducts [57]. P´olya showed that the diagonal of a rational function in two variables is always an algebraic function . The most basic example is F = 1 / (1 − z − z ), forwhich Diag ( F ) = Diag ∞ X m =0 ∞ X m =0 (cid:18) m + m m (cid:19) · z m z m ! = ∞ X m =0 (cid:18) mm (cid:19) · z m = 1 √ − z . (4)The proof of P´olya’s result is based on the simple observation that the diagonal Diag ( F ) is equal to the coefficient of z in the expansion of F ( z , z/z ). Therefore,by Cauchy’s integral theorem, Diag ( F ) is given by the contour integral Diag ( F ) = 12 πi I γ F ( z , z/z ) dz z , (5)where the contour γ is a small circle around the origin. Therefore, by Cauchy’sresidue theorem, Diag ( F ) is the sum of the residues of the rational function G = F ( z , z/z ) /z at all its singularities s ( z ) with zero limit at z = 0. Since the residuesof a rational function of two variables are algebraic functions, Diag ( F ) is itself analgebraic function.For instance, when F = 1 / (1 − z − z ), then G = F ( z , z/z ) /z has twopoles at s = (1 ± √ − z ). The only one approaching zero when z → s = (1 − √ − z ). If p ( s ) /q ( s ) has a simple pole at s , then its residue at s is p ( s ) /q ′ ( s ). Therefore Diag ( F ) = 12 πi I γ dz z − z − z = 11 − s = 1 √ − z . (6) When passing from two to more variables , diagonalisation may still be interpretedusing contour integration of a multiple complex integral over a so-called vanishingcycle [58]. However, the result is not an algebraic function anymore. A simple exampleis F = 1 / (1 − z − z − z z − z z ), for which Diag ( F ) = 1 + 4 z + 36 z + 400 z + 4900 z + 63504 z + · · · (7)is equal to the complete elliptic integral of the first kind Diag ( F ) = X m ≥ (cid:18) mm (cid:19) · z m = F (cid:16) [ 12 ,
12 ] , [1]; 16 z (cid:17) , (8)which is a transcendental function. A less obvious example (see [59] for a relatedexample with a combinatorial flavor) is Diag (cid:18) − z − z − z − z z − z z − z z − z z z (cid:19) (9)= 11 − z · F (cid:16) [ 13 ,
23 ] , [1]; 54 z (1 − z ) (cid:17) . iagonals of rational functions Diag ( F ) of any rationalfunction F is D-finite , in the sense that it satisfies a linear differential equationwith polynomial coefficients ¶ . Moreover, the diagonal of any algebraic power serieswith rational coefficients is a G -function coming from geometry , i.e. it satisfies thePicard-Fuchs type differential equation associated with some one-parameter familyof algebraic varieties. Diagonals of algebraic power series thus appear to be a distinguished class of G -functions ♯ . It will be seen below (see (2.5)) that algebraicfunctions with n variables can be seen as diagonals of rational functions with 2 n variables. Thus diagonals of rational functions also appear to be a distinguished class of G -functions. It is worth noting that this distinguished class is stable by the Hadamardproduct: the Hadamard product of two diagonals of rational functions is the diagonalof rational function .An immediate, but important property of diagonals of rational functions, withrational number coefficients, is that they are globally bounded , which means that theyhave integer coefficients up to a simple change of variable z → N z , where N ∈ Z . Furstenberg [65] showed that the diagonal of any multivariate rational power series with coefficients in a field of positive characteristic is algebraic . Deligne [58, 53]extended this result to diagonals of algebraic functions. For instance, when F =1 / (1 − z − z − z z − z z ), one gets modulo 7 Diag ( F ) mod 7 = 1 + 4 z + z + z + 4 z + 2 z + 4 z + · · · = 1 √ z + z + z mod 7 . (10)More generally, in this example, for any prime p , one has Diag ( F ) = P ( z ) / (1 − p ) mod p (11)where the polynomial P ( z ) is nothing, but [66, 67, 68] P ( z ) = F (cid:16) [ 12 ,
12 ] , [1]; 16 z (cid:17) − p mod p = ( p − / X n =0 (cid:18) p − / n (cid:19) · (16 z ) n . (12)Note, however, that the Furstenberg-Deligne result [65, 58], that we illustrate,here, with F = 1 / (1 − z − z − z z − z z ), goes far beyond the case of hypergeometricfunctions for which simple closed formulae can be displayed. Let us also recall the notion of
Hadamard product [57, 69] of two series, that we willdenote by a star.If f ( x ) = ∞ X n =0 a n · x n , g ( x ) = ∞ X n =0 b n · x n , then: f ( x ) ⋆ g ( x ) = ∞ X n =0 a n · b n · x n . (13) ¶ A more general result was proved by Lipshitz [63]: the diagonal of any D-finite series is D-finite ,see also [64]. ♯ Such diagonals are solutions of G -operators. They are functions that are always algebraic moduloany prime p . They fill the gap between algebraic functions and G -series: they can be seen as generalisations of algebraic functions . iagonals of rational functions Diag (cid:16) f ( x ) · f ( x ) · · · f n ( x n ) (cid:17) = f ( x ) ⋆ f ( x ) ⋆ · · · ⋆ f n ( x ) . (14)In other words, the diagonal of a product of functions with separate variables is equalto the Hadamard product of these functions in a common variable. In particular,the Hadamard product of n rational (or algebraic, or even D-finite) power series isD-finite § .The Hadamard product of two series with integer coefficients is straightforwardlya series with integer coefficients. Furthermore, the Hadamard product of two operators ,annihilating two series, defined as the (minimal order, monic) linear differentialoperator annihilating the Hadamard product of these two series, is a product compatiblewith a large number of structures and concepts ‡ that naturally occur in latticestatistical mechanics. We have a similar compatibility property between the diagonaland the Hurwitz product [19, 70]. It was shown by Furstenberg [65] that any algebraic series in one variable canbe written as the diagonal of a rational function of two variables . The basis ofFurstenberg’s result is the fact that if f ( x ) is a power series without constant term,and is a root of a polynomial P ( x, y ) such that P y (0 , = 0, then f ( x ) = Diag (cid:18) y · P y ( xy, y ) P ( xy, y ) (cid:19) where: P y = ∂P∂y . (15)When P y (0 ,
0) = 0, formula (15) is not true anymore. However, Furstenberg’s resultstill holds [19].Note that this representation as diagonal of a rational function is, by no meansunique, as can be seen on the algebraic function † f = x √ − x = x + 12 x + 38 x + 516 x + 35128 x + 63256 x + · · · (16)which is the diagonal of (2 x y − cx + cy ) / ( x + y + 2) for any rational number c .Furstenberg’s proof does not necessarily produce the simplest rational function (see [19]).Furstenberg’s result has been generalised to power series expansions of algebraicfunctions in an arbitrary number of variables n : any algebraic power series ¶ withrational coefficients is the diagonal of a rational function with 2 n variables (see Denefand Lipshitz [71]).
3. Selected n -fold integrals are diagonals of rational functions Among many multiple integrals that are important in various domains of mathematicalphysics, and before considering other n -fold integrals of the “Ising class †† ”, let us first § The Hadamard product of rational power series is still rational, but the Hadamard product ofalgebraic series is in general transcendental. ‡ For instance, the Hadamard product of two globally nilpotent [27] operators is also globallynilpotent . † Here, f is annihilated by P ( x, y ) = (1 − x ) y − x , which is precisely such that P y (0 ,
0) = 0. ¶ In the one-variable case, Puiseux series could be considered but only after ramifying the variable. †† Using the terminology introduced by Bailey et al. [14], see also [15]. iagonals of rational functions n -particle contribution to the magnetic susceptibility of the Ising modelwhich we denote ˜ χ ( n ) ( w ). They are given by ( n − χ ( n ) ( w ) = (2 w ) n n ! (cid:16) n − Y j =1 Z π d Φ j π (cid:17) · Y · X − X · X n − · G , (17)where, defining Φ by P n − i =0 Φ i = 0, we set X = n − Y i =0 x i , x i = 2 wA i + p A i − w , Y = n − Y i =0 y i , y i = 1 p A i − w ,G = Y ≤ i 1) times the residue formula, one finds˜ χ ( n ) ( w ) = Diag (cid:16) (2 z · · · z n − ) n n ! · F ( z · · · z n − , z , . . . , z n − ) (cid:17) . (27)To check that this is actually true, we introduce an auxiliary set, namely T n the subsetof Laurent series Q [ z , . . . , z n − , z − , . . . , z − n − ][[ w ]], consisting of series f ( w, z , . . . , z n − ) = ∞ X m =0 P m · w m , where P m belongs to Q [ z , . . . , z n − , z − , . . . , z − n − ] and is such that the degree of P m , in each of the z − k , is at most m .Then to prove that F ( z z . . . z n − , z , . . . , z n − ) has a Taylor expansion, we onlyhave to verify that F ( w, z , . . . , z n − ) belongs to this auxiliary set T n . Checking thisis a straightforward step-by-step computation on auxiliary functions: A k = 1 − w · (cid:18) z k + 1 z k (cid:19) , for: 1 ≤ i ≤ n − ,A k = 1 − w · (cid:18) z · · · z n − + z · · · z n − (cid:19) , for: i = 0 . Hence A k belongs to this auxiliary set T n . So to be sure that the inverse or thesquare root of some function in this auxiliary set T n is also in this auxiliary set T n we only have to check that its first Taylor coefficient is actually 1, or w , w , or w n , n integer. It is straightforward to see that: A k − w = 1 − w · (cid:18) z k + 1 z k (cid:19) + w · (cid:18) z k − z k (cid:19) , (28)hence: q A k − w = 1 + · · · ,y k = 1 p A k − w = 1 + · · · , Y = 1 + · · · ,x k = 2 wA k + p A k − w = w + · · · , x k x j = w + · · · ,X = w n + · · · , X − X = 1 + · · · ,G = Y ≤ k 4. Calabi-Yau ODEs solutions and series with binomials seen as diagonals Calabi-Yau ODEs have been defined in [75] as order-four linear differential ODEsthat satisfy the following conditions: they are maximal unipotent monodromy [76, 77](MUM), they satisfy a “Calabi-Yau condition” which amounts to imposing that theexterior squares of these order-four operators are of order five (instead of the ordersix one expects in the generic case), the series solution, analytic at x = 0, is globallybounded (can be reduced to integer coefficients), the series of their nome and Yukawacoupling are globally bounded. In the literature, one finds also a cyclotomic conditionon the monodromy at the point at ∞ , x = ∞ , and/or the conifold † character of oneof the singularities [79].Let us recall that a linear ODE has MUM ( maximal unipotent monodromy [30, 78])if all the exponents at (for instance) x = 0 are zero. In a hypergeometric frameworkthe MUM condition amounts to restricting to hypergeometric functions of the type n +1 F n ([ a , a , · · · a n ] , [1 , , · · · , x ), since the indicial exponents at x = 0 are thesolutions of ρ ( ρ + b − · · · ( ρ + b n − 1) = ρ n +1 = 0, where the b j are the lowerparameters which are here all equal to 1.Let us consider a MUM order-four linear differential operator. The four solutions y , y , y , y of this order-four linear differential operator read: y , y = y · ln( x ) + ˜ y , y = y · ln( x ) y · ln( x ) + ˜ y ,y = y · ln( x ) y · ln( x ) y · ln( x ) + ˜ y , where y , ˜ y , ˜ y , ˜ y are analytical at x = 0 (with also ˜ y (0) = ˜ y (0) = ˜ y (0) = 0).The nome of this linear differential operator reads: q ( x ) = exp (cid:16) y y (cid:17) = x · exp (cid:16) ˜ y y (cid:17) . (36)Calabi-Yau ODEs have been defined as being MUM, thus having one solutionanalytical at x = 0. As far as Calabi-Yau ODEs are concerned, the fact that thissolution analytical at x = 0 has an integral representation, and, furthermore, anintegral representation of the form (34) together with (35), is far from clear, even ifone may have a “Geometry-prejudice” that this solution, analytical at x = 0, can beinterpreted as a “Period” and “Derived From Geometry” [28, 29, 80].Large tables of Calabi-Yau ODEs have been obtained by Almkvist et al. [78,81, 82]. It is worth noting that the coefficients A n of the series corresponding to † The local exponents are 0 , , , 2. For the cyclotomic condition on the monodromy at ∞ , seeProposition 3 in [78]. iagonals of rational functions x = 0, are, most of the time, nested sums of productof binomials , less frequently nested sums of product of binomials and of harmonicnumbers ¶ H n , and, in rare cases, no “closed formula” is known for these coefficients.Let us show, in the case of A n coefficients being nested sums of product ofbinomials , that the solution of the Calabi-Yau ODE, analytical at x = 0, whichis by construction a series with integer coefficients, is actually a diagonal of rationalfunction , and furthermore, that this rational function can actually be easily built. For pedagogical reasons we will just consider, here, a very simple example § of a series S ( x ), with integer coefficients, given by a sum of product of binomials S ( x ) = ∞ X n =0 n X k =0 (cid:18) nk (cid:19) · x n = HeunG ( − / , / , , , , − x ) (37)= 1 + 2 x + 10 x + 56 x + 346 x + 2252 x + 15184 x + · · · This is the generating function of sequence A in Zagier’s tables of binomial coefficientssums (see p. 354 in [83]).The calculations of this section can straightforwardly (sometimes tediously) begeneralised to more complicated [84] nested sums of product of binomials † .Finding that a series is a diagonal of a rational function amounts to framing itinto a residue form like (34). In order to achieve this, we write the binomial (cid:0) nk (cid:1) asthe residue (cid:18) nk (cid:19) = 12 i π · Z C (1 + z ) n z k · dzz , (38)and, thus, we can rewrite S ( x ) as(2 i π ) · S ( x ) == ∞ X n =0 Z Z Z n X k =0 z z z ) k · (cid:16) (1 + z ) (1 + z ) (1 + z ) · x (cid:17) n · dz dz dz z z z = Z Z Z ∞ X n =0 − (cid:16) / ( z z z ) (cid:17) ( n +1) − (cid:16) / ( z z z ) (cid:17) · (cid:16) (1 + z ) (1 + z ) (1 + z ) · x (cid:17) n · dz dz dz z z z = − Z Z Z ∞ X n =0 z z z − z z z · (cid:16) (1 + z ) (1 + z ) (1 + z ) · x (cid:17) n · dz dz dz z z z + Z Z Z ∞ X n =0 − z z z · (cid:16) (1 + z ) (1 + z ) (1 + z ) · xz z z (cid:17) n · dz dz dz z z z = Z Z Z R ( x ; z , z , z ) · dz dz dz z z z , (39) ¶ The generating function of Harmonic numbers is H ( x ) = P H n · x n = − ln(1 − x ) / (1 − x ). § See Proposition 7.3.2 in [77]. † Not necessarily corresponding to modular forms as can be seen on (48), (49). iagonals of rational functions R ( x ; z , z , z ) reads: z z z (cid:0) − x · (1 + z )(1 + z )(1 + z ) (cid:1) (cid:0) z z z − x · (1 + z )(1 + z )(1 + z ) (cid:1) . From this last result one deduces immediately that (37) is actually the diagonal of:1 (cid:0) − z · (1 + z )(1 + z )(1 + z ) (cid:1) · (cid:0) − z z z z (1 + z )(1 + z )(1 + z ) (cid:1) . Note that, as a consequence of a combinatorial identity due to Strehl andSchmidt [85, 86, 87], S ( x ) can also be written as S ( x ) = ∞ X n =0 n X k =0 (cid:18) nk (cid:19) (cid:18) kn (cid:19) · x n = ∞ X n =0 n X k =[ n/ (cid:18) nk (cid:19) (cid:18) kn (cid:19) · x n . (40)Calculations similar to (39) on this alternative binomial representation (40), enableto express (37) as the diagonal of an alternative rational function:1 (cid:0) − z · (1 + z )(1 + z )(1 + z ) (cid:1) · (cid:0) − z z z · (1 + z )(1 + z ) (cid:1) . (41)We thus see that we can actually get explicitly , from straightforward calculations,the rational function (40) for the Calabi-Yau-like ODEs (occurring from differentialgeometry or enumerative combinatorics ) when series with nested sums of binomialstake place, and, more generally, for enumerative combinatorics problems (related ornot to Calabi-Yau manifolds) where series with nested sums of binomials take place.These effective calculations are actually algorithmic, and guarantee to obtain an explicit expression for the rational function (40). However the rational function is farfrom being unique, and worse, the number of variables, the rational function dependson, is far from being the smallest possible number. Finding the “minimal” rationalfunction (whatever the meaning of “minimal” may be) is a very difficult problem.Appendix B provides a non-trivial illustration of this fact with explicit calculationson the well-known Ap´ery series and its rewriting due to Strehl and Schmidt [85, 86, 87].We see in a crystal clear way in Appendix B that, when a given function is a diagonalof a rational function, the rational function is far from being unique, the “simplest”representation (minimal number of variables, lowest degree polynomials, ...) beinghard to find. Similar computations † show that the generating function of sequence B and E in Zagier’s list [83] are both diagonals of rationals function in four variables.All these calculations can systematically be performed on any series defined by nested sums of product of binomials . We have performed such calculations on a largenumber of the series corresponding to the list of Almkvist et al [78], that are given bysuch nested sums of product of binomials . 5. Comments and speculations In [45] (page 61 Theorem 12, see also Proposition 7 in page 50 of [62]) it is provedthat any power series with an integral representation and of maximal weight for thecorresponding Picard-Fuchs linear differential equation is the diagonal of a rationalfunction and, in particular, is globally bounded . † These results are given in section (5.1) of [19]. iagonals of rational functions n +1 F n , this result becomessomewhat trivial. More precisely, the hypergeometric function is of maximal weightif and only if b j = 1 for all j (there is only n !’s in the denominator of coefficients).In that case it is obviously the Hadamard product of algebraic functions, thereforediagonal of a rational function: n F n − ([ α , α , · · · , α n ] , [1 , , · · · , x ) (42)= (1 − x ) − α ⋆ (1 − x ) − α · · · ⋆ (1 − x ) − α n . Therefore, we now have (at least) three sets of problems yielding diagonal ofrational functions: the n -fold integrals of the form (34) with (35), the Picard-Fuchslinear ODEs with solution of maximal monodromy weight and, finally, the problemsof enumerative combinatorics where nested sums of products of binomials take place. Diagonal of rational functions, thus, occur in a quite large set of problems oftheoretical physics . At first sight, one can see the frequent appearance of diagonals ofrational functions in physics just as a mathematical curiousity † , and be surprised that,for instance, so many series in physics are, modulo a prime, algebraic functions. Beingdiagonal of rational functions is not just as a mathematical curiousity: it corresponds(see next section) to G-operators, and their rational number exponents , and can be seenas a first step to modularity properties (see sections below) in some work-in-progressintegrability. The diagonal of a rational function is globally bounded (i.e. it has non zero radius ofconvergence and integer coefficients up to one rescaling) and D-finite (i.e. solution ofa linear differential equation with polynomial coefficients) ‡ .The converse statement is the conjecture in [45] saying that any D-finite, globallybounded series is necessarily the diagonal of a rational function .A remarkable result of Chudnovski’s ([89] Chapter VIII) asserts that the minimallinear differential operator of a G -function (and in particular of a D-finite globallybounded series) is a G -operator (i.e. at least, a globally nilpotent operator) [27, 28, 29].The conjecture in [45] amounts to saying something more: if the solution of thisglobally nilpotent linear differential operator is, not only a G -series, but a globallybounded series, then it is the diagonal of a rational function .Conversely the solution, analytical at 0, of a globally nilpotent linear differentialoperator is necessarily a G -function [28, 29]. Moreover, a “classical” conjecture, withnumerous avatars, claims that any G -function comes from geometry i.e. roughlyspeaking, it has an integral representation § .To test the validity of the conjecture of [45] we look for counter-examples notcontradicting classical conjectures. For instance, we search D-finite power series with † In 1944 the occurrence of elliptic functions in Onsager’s solution of the Ising model was also seenas a mathematical curiosity ... ‡ The series expansion of the susceptibility of the isotropic 2-D Ising model can be recast into a serieswith integer coefficients (see [12, 18, 26, 88]), but it cannot be the diagonal of rational functions sincethe full susceptibility is not a D-finite function [88]. § Bombieri-Dwork conjecture see for instance [29]. iagonals of rational functions integer coefficients which are not algebraic but have an integral representation and arenot of maximal weight for the corresponding Picard-Fuchs linear ODE.As a first step let us limit ourselves to hypergeometric functions n +1 F n . Themonodromy weight W is exactly the number of 1 among the b i .When n +1 F n is globally bounded and has no integer parameters b i ( W = 0), itsminimal ODE has a p -curvature zero for almost all primes p . However, a Grothendieckconjecture, proved for F in [90], and generalised to n +1 F n in [91], asserts that,under these circumstances, the hypergeometric function is algebraic . We display inAppendix C a set of n +1 F n hypergeometric functions which yield, naturally, serieswith integer coefficients , many of them corresponding to such algebraic hypergeometricfunctions . Even if such examples are quite non-trivial, the purpose of our paper is tofocus on transcendental (non algebraic) functions.So we are looking for globally bounded hypergeometric functions satisfying1 ≤ W ≤ n − 1. In general such hypergeometric functions are G -series but are veryfar from being globally bounded. The hypergeometric world extends largely outsidethe world of diagonal of rational functions.Such an example in the first case n = 2, W = 1 was given in [45]: F (cid:18)(cid:20) , , (cid:21) , (cid:20) , (cid:21) ; 3 x (cid:19) = 1 + 60 x + 20475 x + 9373650 x + 4881796920 x + 2734407111744 x + 1605040007778900 x + · · · (43)The integer coefficients read with the rising factorial (or Pochhammer) symbol(1 / n · (4 / n · (5 / n (1 / n · (1) n · n ! · n = ρ ( n ) ρ (0) , (44)where: ρ ( n ) = Γ(1 / n ) Γ(4 / n ) Γ(5 / n )Γ(1 / n ) Γ(1 + n ) Γ(1 + n ) · n . (45)Note that, at first sight, it is far from clear § on (45), or on the simple recursion onthe ρ ( n ) coefficients (with the initial value ρ (0) = 1) ρ ( n + 1) ρ ( n ) = 3 · (1 + 9 n ) (4 + 9 n ) (5 + 9 n )(1 + 3 n )(1 + n ) , (46)to see that the ρ ( n )’s are actually integers. A sketch of the (quite arithmetic) proofthat the ρ ( n )’s are actually integers, is given in Appendix D.Because of the 1 / p -curvature and finding that it isnot zero [80] (see also [91, 92]). Proving that an algebraic function is the diagonal ofa rational function and proving that a solution of maximal weight for a Picard-Fuchsequation is the diagonal of a rational function use two entirely distinct ways. Thehope is to combine both techniques to conclude in the intermediate situation.This example remained for twenty years, the only “blind spot” of the conjecturein [45]. We have recently found many other F examples ‡ , such that their series § In contrast with cases where binomial (and thus integers) expressions take place. ‡ F cases are straightforward, and cannot provide counterexamples to conjecture in [45]. iagonals of rational functions integer coefficients but are not obviously diagonals of rationalfunctions. Some of these new hypergeometric examples † read for instance: F (cid:16) [ 19 , , , ] , [ 23 , , x (cid:17) = 1 + 21 x + 5544 x + 2194500 x + 1032711750 x + 535163031270 x + 294927297193620 x + 169625328357359160 x + 100668944872954458000 x + · · · or: F (cid:16) [ 17 , , , ] , [ 12 , , x (cid:17) , F (cid:16) [ 111 , , , ] , [ 12 , , x (cid:17) . (47)Unfortunately these hypergeometric examples are on the same “frustratingfooting” as Christol’s example (43): we are not able to show that one of them isactually a diagonal of a rational function, or, conversely, to show that one of themcannot be the diagonal of a rational function. 6. Integrality versus modularity: learning by examples A large number of examples of integrality of series-solutions comes from modularforms. Let us just display two such modular forms associated with HeunG functionsof the form HeunG ( a, q, , , , x ). Many more similar examples can be found in [19]. One can, for instance, rewrite the example (37) of subsection (4.2), namely HeunG ( − / , / , , , , − x ), as a hypergeometric function with two rationalpullbacks : HeunG ( − / , / , , , , − x ) = ∞ X n =0 n X k =0 (cid:18) nk (cid:19) x n (48)= (cid:16) (1 + 4 x ) · (1 + 228 x + 48 x + 64 x ) (cid:17) − / × F (cid:16) [ 112 , 512 ] , [1]; 1728 · (1 − x ) · (1 + x ) · x (1 + 228 x + 48 x + 64 x ) · (1 + 4 x ) (cid:17) = (cid:16) (1 − x ) · (1 − x + 228 x − x ) (cid:17) − / × F (cid:16) [ 112 , 512 ] , [1]; 1728 · (1 − x ) · (1 + x ) · x (1 − x + 228 x − x ) · (1 − x ) (cid:17) . The relation between the two pullbacks, that are related by the “Atkin” involution ¶ x ↔ − / /x , gives the modular curve:1953125 y z − y z · ( y + z ) + 375 y z · (16 z − y z + 16 y ) − · ( z + y ) · ( y + z + 1487 y z ) + 110592 · y z = 0 . (49) † See also [19]. ¶ In previous papers [93, 30], with some abuse of language, we called such an involution an Atkin-Lehner involution . In fact this terminology is commonly used in the mathematical community foran involution τ → − N/τ , on τ , the ratio of periods, and not for our x -involution. This is why weswitch to the wording “Atkin” involution . iagonals of rational functions self adjoint linear differential operator Ωwhere ( θ = x · D x ): x · Ω = θ − x · (7 θ + 7 θ + 2) − x · ( θ + 1) . (50) The integrality of series-solutions can be quite non-trivial like the solution of theAp´ery-like operatorΩ = x · (1 − x − x ) · D x + (1 − x − x ) · D x − ( x + 3) , (51)or: x · Ω = θ − x · (11 θ + 11 θ + 3) − x · ( θ + 1) , which can be written as a HeunG function. This (at first sight involved) HeunGfunction reads: HeunG (cid:16) − · / , − 332 + 152 · / , , , , (cid:16) − / (cid:17) · x (cid:17) = ∞ X n = 0 n X k = 0 (cid:18) nk (cid:19) (cid:18) n + kk (cid:19) · x n = 1 + 3 · x + 19 · x + 147 · x + · · · but actually corresponds to a modular form , which can be written in two differentways using two pullbacks :( x + 12 x + 14 x − x + 1) − / × F (cid:16) [ 112 , 512 ] , [1]; 1728 · x · (1 − x − x )( x + 12 x + 14 x − x + 1) (cid:17) = (1 + 228 x + 494 x − x + x ) − / (52) × F (cid:16) [ 112 , 512 ] , [1]; 1728 · x · (1 − x − x ) (1 + 228 x + 494 x − x + x ) (cid:17) . Modular form examples of series with integer coefficients displayed inAppendix A, correspond to lattice Green functions [48]. Therefore, they have n -fold integral representations † , and, after section (3.4), can be seen to be diagonals ofrational functions . 7. Integrality versus modularity Let us consider a first simple example of a hypergeometric function which issolution of a Calabi-Yau ODE, and which occurred, at least two times in thestudy of the Ising susceptibility n -fold integrals [30, 31] χ ( n ) and χ ( n ) d , namely F ([1 / , / , / , / , [1 , , x ), where we perform a (diffeomorphism ofunity) pullback: F (cid:16) [ 12 , , , 12 ] , [1 , , x c x + c x + · · · (cid:17) = 1 + 16 · x (53)+ 16 · (81 − c ) · x + 16 · (10000 + c − c − c ) · x + · · · † In contrast the modular form examples displayed in Appendix H of [19] correspond to differentialgeometry examples discovered by Golyshev and Stienstra [94], where no n -fold integral representationis available at first sight. iagonals of rational functions c n , at its denominator, are integers,one finds that the series expansion is actually a series with integer coefficients, for everysuch pullback (i.e. for every integer coefficients c n ). Furthermore, a straightforwardcalculation of the corresponding nome q ( x ) and its compositional inverse (mirror map) x ( q ), also yields series with integer coefficients : q ( x ) = x + (64 − c ) · x + ( c + 7072 − c − c ) · x + · · · , (54) x ( q ) = q + ( c − · q + ( c + 1120 + c − c ) · q + · · · , (55)when its Yukawa coupling [30], seen as a function of the nome q , K ( q ) is also a serieswith integer coefficients and is independent of the pullback : K ( q ) = 1 + 32 · q + 4896 · q + 702464 · q + · · · (56)This independence of the Yukawa coupling with regards to pullbacks, is a knownproperty, and has been proven in [75], for any pullback of the diffeomorphism of unityform p ( x ) = x + · · · Seeking for Calabi-Yau ODEs, Almkvist et al. have obtained [78] a quite large listof fourth order ODEs, which are MUM by definition and have, by construction, the integrality for the solution-series analytic at x = 0. Looking at the Yukawa couplingof these ODEs is a way to define equivalence classes up to pullbacks of ODEs sharingthe same Yukawa coupling. This “wraps in the same bag” all the linear ODEs that arethe same up to pullbacks . Let us recall how difficult it is to see if a given Calabi-YauODE has, up to operator equivalence, and up to pullback, a hypergeometric functionsolution [30, 31], because finding the pullback is extremely difficult [30, 31]. We mayhave, for the Ising model, some n +1 F n hypergeometric function prejudice [30, 31]: itis, then, important to have an invariant that is independent of this pullback that wecannot find most of the time.Finally, let us remark that the Yukawa coupling is not preserved by the operatorequivalence . Two linear differential operators, that are homomorphic, do notnecessarily have the same Yukawa coupling (see Appendix E). Another way to understand this fundamental pullback invariance , amounts to rewritingthe Yukawa coupling [75, 95], not from the definition usually given in the literature(second derivative with respect to the ratio of periods), but in terms of determinantsof solutions (Wronskians, ...) that naturally present nice covariance properties withrespect to pullback transformations (see Appendix E).We have the alternative definition for the Yukawa coupling given in Appendix E: K ( q ) = (cid:16) q · ddq (cid:17) (cid:16) y y (cid:17) = W · W W , (57)where the determinantal variables W m ’s are determinants built from the four solutionsof the MUM differential operator. This alternative definition, in terms of these W m ’s, enables to understand the remarkable invariance of the Yukawa coupling bypullback transformations [31]. These determinantal variables W m quite naturally, andcanonically, yield to introduce another “Yukawa coupling” (which, in fact, correspondsto the Yukawa coupling of the adjoint operator (see E.12)). This “adjoint Yukawacoupling” is also invariant by pullbacks . It has, for the previous example, the followingseries expansion with integer coefficients: K ⋆ ( q ) = 1 + 32 · q + 4896 · q + 702464 · q + · · · (58) iagonals of rational functions F ([ , , , ] , [1 , , x )is exactly self-adjoint , and, more generally, of the fact that the order-four operator,annihilating (53), is conjugated to its adjoint by a simple function. This example, with its corresponding relations (53), (54), (56), (58) may suggesta quite wrong prejudice that the integrality of the solution of an order-four lineardifferential operator automatically yields to the integrality of the nome, mirror mapand Yukawa coupling, that we will call, for short, “ modularity ”. This is far from beingthe case , as can be seen, for instance, in the following interesting example, where thenome and Yukawa coupling K ( q ) do not correspond to globally bounded series , whenthe F solution of the order-four operator as well as the Yukawa coupling seen as afunction of x , K ( x ), are, actually, both series with integer coefficients .Let us consider the following F hypergeometric function which is clearly aHadamard product of algebraic functions and, thus, the diagonal of a rational function: F (cid:16) [ 12 , , , 34 ] , [1 , , x (cid:17) = (1 − x ) − / ⋆ (1 − x ) − / ⋆ (1 − x ) − / ⋆ (1 − x ) − / = Diag (cid:16) (1 − z ) − / (1 − z ) − / (1 − z ) − / (1 − z ) − / (cid:17) , It is therefore globally bounded: F ([ 12 , , , 34 ] , [1 , , x ) = 1 + 72 x + 45360 x + 46569600 x + 59594535000 x + 86482063571904 x + · · · (59)Its Yukawa coupling, seen as a function of x , is actually a series with integercoefficients in x : K ( x ) = 1 + 480 x + 872496 x + 1728211968 x + 3566216754432 x + 7536580798814208 x + 16177041308360579328 x + · · · (60)However, do note that the series, in term of the nome, is not globally bounded : K ( q ) = 1 + 480 q + 653616 q + 942915456 q + 1408019875200 q + · · · + 571436303929319146711343817202689132288 q 11 + · · · (61)In fact, the nome q ( x ), and the mirror map x ( q ), are also not globally bounded .Note that in this example, the non integrality appears at order twelve (for x ( q ), q ( x )and K ( q )). If the prime 11 in the denominator in (61) was the only one, one couldrecast the series into a series with integer coefficients introducing another rescaling2304 x → × x . But, in fact, we do see the appearance of an infinite numberof other primes at higher orders denominators in x ( q ), q ( x ) and K ( q ).We do not have modularity because we do not have (up to rescaling) the nomeintegrality: the nome series is not globally bounded. iagonals of rational functions ω n associated with modular forms After Maier [96] let us underline that modular forms can be written as hypergeometricfunctions with two different pullbacks , and, consequently, one can associate order-twodifferential operators to these modular forms.Let us consider the two order-two operators ω = D x + (96 x + 1)(64 x + 1) · x · D x + 4(64 x + 1) x , (62) ω = D x + (45 x + 1)(27 x + 1) · x · D x + 3(27 x + 1) x , (63)which are associated with two modular forms corresponding, on their associated nomes q , to the transformations q → q and q → q respectively (multiplication of τ , theratio of their periods by 2 and 3), as can be seen on their respective solutions: F (cid:16) [ 14 , 14 ] , [1]; − x ) = (1 + 256 x ) − / · F (cid:16) [ 112 , 512 ] , [1]; 1728 x (1 + 256 x ) (cid:17) = (1 + 16 x ) − / · F (cid:16) [ 112 , 512 ] , [1]; 1728 x (1 + 16 x ) (cid:17) (64)= 1 − x + 100 x − x + 152100 x − x + 344622096 x − x + 924193822500 x − x + · · · and: (cid:16) (1 + 27 x ) (1 + 243 x ) (cid:17) − / · F (cid:16) [ 112 , 512 ] , [1]; 1728 x (1 + 243 x ) (1 + 27 x ) (cid:17) = (cid:16) (1 + 27 x ) (1 + 3 x ) (cid:17) − / · F (cid:16) [ 112 , 512 ] , [1]; 1728 x (1 + 3 x ) (1 + 27 x ) (cid:17) (65)= F (cid:16) [ 13 , 13 ] , [1] , − x ) = 1 − x + 36 x − x + 11025 x − x + 4769856 x − x + 2391796836 x − x + · · · The relation between the two Hauptmodul pullbacks in (64) u = 1728 x (1 + 256 x ) , v = 1728 x (1 + 16 x ) = u (cid:16) x (cid:17) , (66)corresponds to the (genus-zero) fundamental modular curve:5 · u v − · · u v · ( u + v ) + 375 uv · (16 u + 16 v − uv ) − 64 ( u + v ) · ( v + 1487 uv + u ) + 2 · uv = 0 . (67)The relation between the two Hauptmodul pullbacks in (65) u = 1728 x (1 + 243 x ) (1 + 27 x ) , v = 1728 x (1 + 3 x ) (1 + 27 x ) = u (cid:16) x (cid:17) , (68)corresponds to the (genus-zero) modular curve:2 · u v · ( u + v ) + 2 u v · (27 v + 27 u − uv )+ 2 uv · ( u + v ) · ( v + 241433 uv + u ) (69)+ 729 ( u + v ) − · · ( u + v ) · u v + 15974803 · · · u v + 31 · uv · ( u + v ) − uv = 0 . iagonals of rational functions ω n associated with othermodular forms corresponding to τ → n · τ . The ω n ’s can be simply deduced fromMaier [96], for modular forms corresponding to genus-zero curves i.e. for n = 2 , , , , , , , , , , , , , 25. Since the solutions can be written as F hypergeometric up to rational pullbacks , these genus-zero ω n ’s are obviously order-two operators. After a simple rescaling, the solutions analytic at x = 0, can berewritten as series with integer coefficients .One can also consider the other ω n ’s corresponding to higher-genus modularcurves. In these cases, one does not have a rational parametrisation like (66) or (68),but one still has an identity of the same hypergeometric function with two differentpullbacks , these two pullbacks being algebraic functions and not rational functions (see (66) or (68)). These algebraic functions correspond to the so-called modularpolynomials [19].For instance for τ → · τ , one has a genus-one modular curve, the modularpolynomial reads: P ∗ ( x, H ) = (1 + 228 x + 486 x − x + 225 x ) · H (70) − · Q ( x ) · x · H + 1728 x , with: Q ( x ) = 1 − x + 1188 x − x + 69630 x − x + 133056 x + 132066 x − x + 40095 x + 24300 x − x . One has the identity F (cid:16) [ 112 , 512 ] , [1]; H (cid:17) = A ( x ) · F (cid:16) [ 112 , 512 ] , [1]; H (cid:17) , (71)where the two Hauptmoduls H and H are the two solutions of P ∗ ( x, H ) = 0,and where A ( x ) is the algebraic function such that A ( x ) + 11 A ( x ) = 211 · − x + 73206 x + 21060 x − x x + 486 x − x + 225 x , this last relation between A ( x ) and x corresponding to a genus-one curve with the same j -invariant as the genus-one curve P ∗ ( x, H ) = 0, namely j = − / .The two hypergeometric functions in (71) are actually series with integer coefficients : F ([1 / , / , [1]; H ) = 1 + 60 x − x + 614400 x − x + 16231863060 x − x + · · · (72) F ([1 / , / , [1]; H ) = 1 + 60 x + 3300 x + 110220 x + 2904660 x + 66599940 x + 1394683620 x + 27425371380 x + · · · (73)More details are given for this τ → · τ case in Appendix I of [19]. Note thatalthough F ([1 / , / , [1]; H ) and F ([1 / , / , [1]; H ) are solutions of the same order-four operator [19], one can find an appropriate algebraic function A ( x ),such that A ( x ) · F ([1 / , / , [1] , H ) is solution of an order-two operator ω (see [19] for more details).The other ω n ’s, corresponding to higher genus modular curves [97], are actually also order-two operators . The explicit expressions of ω n ’s for the elliptic values n = 17 , 19, and the hyperelliptic values [97] n = 23 , , , , , , 71 are givenin [19]. The genus of the associated modular curves [97], is respectively [19] genus-one for ˜ ω ( x ), ˜ ω ( x ), genus-two for ˜ ω ( x ), ˜ ω ( x ), ˜ ω ( x ), genus-three for ˜ ω ( x ), genus-four for ˜ ω ( x ), genus-five for ˜ ω ( x ), and genus-six for ˜ ω ( x ). iagonals of rational functions ω n ’s The two operators ω and ω have a “modularity” property: their series expansionsanalytic at x = 0, (64) and (65), as well as the corresponding nomes, mirror maps areseries with integer coefficients. The Hadamard product is a quite natural operation tointroduce because it preserves the global nilpotence of the operators , it preserves theintegrality of series-solutions , and it is a natural operation to introduce when seekingfor diagonals of rational functions ¶ . Let us perform the Hadamard product of thesetwo operators. With some abuse of language [31], the Hadamard product of the twoorder-two operators (62) and (63) H , = ω ⋆ ω = D x + 6 (2064 x − x − · x · D x + (19020 x − x − · x · D x + (4788 x − x − · x · D x + 12(1728 x − · x , (74)is defined as the (minimal order) linear differential operator having, as a solution, theHadamard product of the solution-series (64) and (65), which is, by construction, aseries with integer coefficients. This series is, of course, nothing but the expansion ofthe hypergeometric function: F ([ 14 , , , 13 ] , [1 , , x ) (75)= F ([ 14 , 14 ] , [1]; − x ) ⋆ F ([ 13 , 13 ] , [1]; − x ) . In a similar way one can consider (see [19]) H , = ω ⋆ ω (resp. H , = ω ⋆ ω ) the Hadamard product of the order-two operator (62) (resp. (63)) withitself (Hadamard square). These two operators have respectively the hypergeometricsolutions F ([ 14 , , , 14 ] , [1 , , x ) , F ([ 13 , , , 13 ] , [1 , , x ) , (76)corresponding to series expansions with integer coefficients . These operators H , , H , are MUM operators. We can, therefore, define, without any ambiguity, the nome(and mirror map) and Yukawa coupling of this order-four operator [31]. One finds outthat the nome † , and the mirror map (and the Yukawa coupling as a function of the x variable), are not globally bounded : they cannot be reduced, by one rescaling, to serieswith integer coefficients.The three linear differential operators H , , H , and H , , are MUM and oforder four, however, they are not of the Calabi-Yau type . The occurrence of Calabi-Yau type operators, that we could imagine, at first sight,to be extremely rare, is in fact quite frequent among such Hadamard products,as can be seen with other values of n and m . For instance, one can introduce ‡ H , = ω ⋆ ω , the Hadamard square of ω , which is an irreducible order-four ¶ And, consequently, has been heavily used to build Calabi-Yau-like ODEs (see Almkvist et al. [75]). † The nome of the Hadamard product of two operators has no simple relation with the nome of thesetwo linear differential operators. ‡ To get the Hadamard product of two linear differential operators use, for instance, Maple’scommand gfun[hadamardproduct] . iagonals of rational functions n -fold integrals of the decomposition of the full magnetic susceptibility ofthe Ising model [30, 31] (see also subsections (7.1) and (7.2)): F ([ 12 , , , 12 ] , [1 , , x ) (77)= F ([ 12 , 12 ] , [1]; − x ) ⋆ F ([ 12 , 12 ] , [1]; − x ) . The associated operator having (77) as a solution, obeys the “Calabi-Yaucondition” that its exterior square is of order five .Let us give in a table the orders (which go from 4 to 20) of the various H m,n = H n,m Hadamard products of the order-two operators associated with the(genus-zero) modular forms operators ω n and ω m :n \ m 2 3 4 5 6 7 8 9 10 12 13 16 18 252 4 4 4 6 4 6 4 4 10 8 10 8 12 143 4 4 6 4 6 4 4 10 8 10 8 12 144 4 ∗ ∗ ∗ ∗ 10 8 10 8 12 145 6 6 8 6 6 12 10 12 10 14 166 4 ∗ ∗ ∗ 10 8 10 8 12 147 6 6 6 12 10 12 10 14 168 4 ∗ ∗ 10 8 10 8 12 149 4 ∗ 10 8 10 8 12 1410 10 14 16 14 18 2012 8 14 12 16 1813 10 14 18 2016 8 16 1818 12 2025 14where the star ∗ denotes Calabi-Yau ODEs ♯ .The following operators are of order four: H , , H , , H , , H , , H , , H , , H , , H , , H , , H , , H , , ... Their exterior squares, which are of order six, do nothave rational solutions † .The order-four operators H , , H , , are all MUM operators ¶ , but, similarly tothe situation encountered with H , , their nome, mirror map and Yukawa couplingsare not globally bounded .The following operators are of order six: H , , H , , H , , H , , H , , H , , H , , H , , H , , H , , H , , H , , H , , H , , ... Their exterior square, which areof order fifteen, do not have rational solutions (and cannot be homomorphic to higherorder Calabi-Yau linear ODEs).Remarkably the following ten order-four operators H , , H , , H , , H , , H , , H , , H , , H , , H , , H , (with a star in the previous table) are all MUM, and are ♯ Recall that Calabi-Yau ODEs are defined by a list of constraints [75], the most important onesbeing, besides being MUM, that their exterior square are of order five . There are more exoticconditions like the cyclotomic condition on the monodromy at ∞ , see Proposition 3 in [78]. † They cannot be homomorphic to Calabi-Yau ODEs. ¶ Note that the Hadamard product of two MUM ODEs is not necessarily a MUM ODE: the order-sixoperator H , is not MUM. iagonals of rational functions such that their exterior squares are of order five § : they are Calabi-Yau ODEs . Actuallythe nome, mirror map and Yukawa coupling series are series with integer coefficientsfor all these order-four Calabi-Yau operators . Their Yukawa coupling and their adjointYukawa coupling identify. The Yukawa coupling series of these Calabi-Yau operatorsare respectively, for H , K ( q ) = K ⋆ ( q ) = 1 + 32 · q + 4896 · q + 702464 · q + 102820640 · q + 15296748032 · q + 2302235670528 · q + · · · (78)which is F ([1 / , / , / , / , [1 , , x ), and for H , K ( q ) = K ⋆ ( q ) = 1 + 20 · q + 36 · q + 15176 · q + 486564 · q + 21684020 · q + 1209684456 · q + · · · (79)which is H m,n such that their exterior squares are order five (not six as one could expect for a genericirreducible order-four operator), that actually are Calabi-Yau operators. Actuallyoperator H , is H , , H , , H , and H , are respectively [78] H , , H , and H , are respectively [78] conjugated (by an algebraic function) to their adjoints. Thus, the “adjointYukawa coupling” K ⋆ ( q ) is necessarily equal to the Yukawa coupling K ( q ) for theseoperators.On the other hand, the ten linear differential operators denoted by a star ∗ in theprevious table all share the same property : they have, as a solution, the Hadamardproduct of two HeunG functions solutions of the form HeunG ( a, q, , , , x ).Note, however, that this HeunG-viewpoint of the most interesting H m,n ’s does notreally help. Even inside this restricted set of HeunG functions solutions of the form HeunG ( a, q, , , , x ) it is hard to find exhaustively the values of the parameter a and of the accessory parameter q , such the series HeunG ( a, q, , , , x ) is globallybounded, or, just, such that the order-two operator, having HeunG ( a, q, , , , x )as a solution, is globally nilpotent [27].Many H m,n are not MUM, for instance the order-eight operator H , , or theorder-six operator H , , are not MUM . Concerning H , , and as far as its six solutionsare concerned, they are structured “like” the four solutions of an order-four MUMoperator, together with the two solutions of another order-two MUM operator, but theorder-six operator H , is not a direct-sum of an order-four and order-two operator.We have two solutions analytical at x = 0 (with no logarithmic terms), and twosolutions involving ln( x ). A linear combination of these two solutions analytical at x = 0 is, by construction a series with integer coefficients (the Hadamard productof the two series with integer coefficients which are the initial ingredients in thiscalculation), when the other linear combinations are not globally bounded . § They are conjugated to their (formal) adjoint by a function. iagonals of rational functions 8. Calabi-Yau Modularity The previous examples correspond to a “modularity” inherited from elliptic curves,more precisely Hadamard products of modular forms. Let us consider, here, twoCalabi-Yau examples that do not seem to be reducible ‡ to F hypergeometricfunctions.A first order-four Calabi-Yau operator, found by Batyrev and van Straten [77],which is self-adjoint and also corresponds to Hadamard products of simplehypergeometric functions (see (F.2)), is given in Appendix F. All the associated series(solution (F.2), nome, Yukawa coupling) are series expansion with integer coefficients .We do not have a representation of the solution (F.2), as an n -fold integral of the form(34). However, since (F.2) can be expressed as a sum of products of binomials (see(F.3)), we can conclude, again, that (F.2) is actually a diagonal of a rational function . A second example of order-four operator, corresponding to Calabi-Yau 3-folds in P × P × P × P , has been found by Batyrev and van Straten [77] (see † page 34): B = θ − x · (5 θ + 5 θ + 2) · (2 θ + 1) + 64 x · (2 θ + 3) · (2 θ + 1) · (2 θ + 2) . (80)It corresponds to the series-solution with coefficients: (cid:18) nn (cid:19) · n X k =0 (cid:18) nk (cid:19) · (cid:18) kk (cid:19) · (cid:18) n − kn − k (cid:19) . (81)Its Wronskian W is a rational function such that: x · W / = 1(1 − x ) (1 − x ) . (82)This operator is also a Calabi-Yau operator: it is MUM, and it is such that its exteriorsquare is order five . This order five property is a consequence of B being conjugatedto its adjoint: B · x = x · adjoint ( B ).The series-solution of (80) can be written as an Hadamard product S = (1 − x ) − / ⋆ HeunG (4 , / , / , / , , / 2; 16 x ) (83)= 1 + 8 x + 168 x + 5120 x + 190120 x + 7939008 x + 357713664 x + · · · , the modular form character of HeunG (4 , / , / , / , , / 2; 16 x ) being illustratedwith identities (A.4) in Appendix A of [19]. Its nome reads: q = x + 20 x + 578 x + 20504 x + 826239 x + 36224028 x + 1684499774 x + 81788693064 x + 4104050140803 x + 211343780948764 x + · · · (84)The mirror map of (80) reads: x ( q ) = q − q + 222 q − q + 21293 q − q + 80402 q − q − q − q + · · · (85) ‡ The possibility that a solution of an order-four operator, non-trivially equivalent to these Calabi-Yau operators [77], could be written as a F hypergeometric function, up to an involved algebraicpullback , is not totally excluded. However it is extremely difficult to rule out such highly non-trivialhypergeometric scenario. † There is a small misprint in [77] page 34: (2 θ + 1) must be replaced by (2 θ + 1) in the 4 x term. iagonals of rational functions K ( q ) = K ⋆ ( q ) = 1 + 4 q + 164 q + 5800 q + 196772 q + 6564004 q + 222025448 q + 7574684408 q + 259866960036 q + · · · (86)The equality of the Yukawa coupling with the “adjoint” Yukawa coupling, K ( q ) = K ⋆ ( q ), is a straight consequence of relation B · x = x · adjoint ( B ).Recalling Batyrev and van Straten [77], and following Morrison [76], do note thatone can also write the Yukawa coupling as: K ( q ) = x ( q ) · W / y · (cid:16) qx ( q ) · dx ( q ) dq (cid:17) = W / y · (cid:16) q · dx ( q ) dq (cid:17) , (87)where W is the Wronskian (82). From this alternative expression for the Yukawacoupling, valid when the operator is conjugated to its adjoint (see (E.7)), it is obviousthat if the analytic series y ( x ), as well as the nome (84) are series with integercoefficients , then, the mirror map (85) is also a series with integer coefficients, and,therefore, y seen as a function of the nome q , as well as x W / (since it is a rationalfunction ) are also series with integer coefficients. Consequently, the Yukawa couplingis a series with integer coefficients (as a series in q or in x ).More generally, if one assumes that a linear differential operator has a globallybounded solution-series, one knows that this operator is a G -operator, necessarilyglobally nilpotent, and, consequently, its Wronskian, or the square root of theWronskian (see W / in (87)), will be a N -th root of a rational function, and, thus, will correspond to a globally bounded series . Thus, the globally bounded character of theanalytic series y ( x ) together with the nome, yields the globally bounded character ofthe mirror map, Yukawa coupling, that we associate with the modularity † . In contrastthe globally bounded character of the analytic series y ( x ), together with the globallybounded character of the Yukawa coupling (seen for instance as a series in x ) doesnot imply that the nome, or the mirror map, are globally bounded as can be seen onexample (59) (see (60) and (61)). B Let us, now, consider the order-four operator B = 256 x · θ (2 θ + 3) (2 θ + 1) − x · (2 θ + 1) (2 θ − 1) (5 θ − θ + 2) + ( θ − . (88)This operator is non-trivially ‡ homomorphic to the Calabi-Yau operator (80): B · x · (2 θ + 1) = x · (2 θ + 1) · B . (89)As a consequence of the previous intertwining relation, one immediately finds that theseries-solution analytic at x = 0 of this new MUM operator (88) is nothing but theaction of the order-one operator x · (2 θ + 1) on the series (83), and reads: x · (2 θ + 1)[ S ] = x + 24 x + 840 x + 35840 x + 1711080 x + 87329088 x + 4650277632 x + 254905896960 x + · · · (90) † Similar results can be found in Delaygue’s thesis [98], in a framework where the coefficients ofhypergeometric series are ratio of factorials (see Appendix C). ‡ The intertwiners between B and B are operators not simple functions. iagonals of rational functions integer coefficients (the action of x · (2 θ + 1) on theseries with integer coefficients is straightforwardly a series with integer coefficients).More generally, the globally bounded series remain globally bounded series by nontrivial operator equivalence, namely homomorphisms between operators (genericallythe intertwiner operators are not simple functions).The exterior square of the order-four operator (88) is an order-six operator whichis, in fact, the direct sum of an order-five operator E and an order-one operator.Operator B is non-trivially homomorphic to its adjoint: B · x · (2 θ + 3) · (2 θ + 5) = x · (2 θ + 3) · (2 θ + 5) · adjoint ( B ) . (91)The Yukawa coupling of this order-four operator (88), non-trivially homomorphic to(80), reads: K ( q ) = 1 − q − q − q − q q 3+ 88331368 q q − q 63 + · · · (92)The Yukawa coupling series (92) is not globally bounded .The “adjoint Yukawa coupling” of this order-four operator (88) reads: K ⋆ ( q ) = 1 + 12 q + 564 q + 20440 q + 865732 q + 37162444 q + 8255346664 q q 15 + 72336859374772 q 21 + · · · (93)Again, the adjoint Yukawa coupling series (93) is not globally bounded .On this example one sees that the Yukawa coupling of two non-triviallyhomomorphic operators are not necessarily equal . The Yukawa couplings of twohomomorphic operators are equal when the two operators are conjugated by a function (trivial homomorphism). The modularity property is not preserved by (non-trivial)operator equivalence: it may depend on a condition that the exterior square of theorder-four operators is of order five . The Calabi-Yau property is not preserved byoperator equivalence. To sum-up: All these examples show that the integrality (globally boundedseries) is far from identifying with modularity . 9. Conclusion Seeking for the linear differential operators for the χ ( n ) ’s, we discovered, someyears ago, that they were Fuchsian operators [11, 13], and, in fact, “special”Fuchsian operators, namely Fuchsian operators with rational exponents for all theirsingularities, and with Wronskians that are N -th roots of rational functions. Thenwe discovered that they were G -operators (or equivalently globally nilpotent [27]),and more recently, we accumulated results [31] indicating that they are “special” G -operators. There are, in fact, two quite different kinds of “special features” of these G -operators. On one side, we have the fact that one of their solutions is not only G -series, but is a globally bounded series. This special character has been addressedin this very paper, and we have seen that, in fact, this “ integrality property [99]” is aconsequence of quite general mathematical assumptions often satisfied in physics (theintegrand is not only algebraic but has an expansion at the origin of the form † (35). † Puiseux series are excluded. iagonals of rational functions G -operators, namelythe fact that they seem to be quite systematically homomorphic to their adjoints [31].We will show, in a forthcoming publication, that this last property amounts, on theassociated linear differential systems, to having special differential Galois groups , andthat their exterior or symmetric squares, have rational solutions . This last property is aproperty of a more “physical” nature than the previous one, related to an underlying Hamiltonian structure [100], or as this is the case, for instance in the Ising model,related to the underlying isomonodromic structure in the problem, which yields theoccurrence of some underlying Hamiltonian structure [100]. In general the integrality of G -operators does not imply the operator to be homomorphic to its adjoint, andconversely being homomorphic to its adjoint does not imply ‡ integrality (and even doesnot imply † the operator to be Fuchsian). Interestingly, the χ ( n ) ’s, as well as manyimportant problems of theoretical physics, correspond to G -operators that presentthese two complementary “special characters” (integrality and, up to homomorphisms,self-adjointness), and, quite often, this is seen in the framework of the emergence of“modularity”.Nomes, mirror maps, and Yukawa couplings are not D-finite functions: they aresolutions of quite involved non-linear (higher order Schwarzian) ODEs (see for instanceAppendix D in [27]). Therefore, the question of the series integrality of the nomes,mirror maps, Yukawa couplings, and other pullback-invariants (see Appendix E)requires to address the very difficult question of series-integrality for (involved) non-linear ODEs. Note, however, as seen in Section 8.1, in particular in (87), that theintegrality of the series y ( x ) and of the nome q ( x ) are sufficient to ensure , provided theoperator is conjugated to its adjoint (see (E.7)), the integrality of the other quantitiessuch as the Yukawa coupling, mirror maps. However the integrality of the nomeremains an involved problem. These questions will certainly remain open for sometime.In contrast, and more modestly, we have shown that a very large sets of problemsin mathematical physics (see sections (3.4), (4) and (5.1)) actually corresponds todiagonals of rational functions . In particular, we have been able to show that the χ ( n ) ’s n -fold integrals of the susceptibility of the two-dimensional Ising model areactually diagonals of rational functions for any value of the integer n , thus provingthat the χ ( n ) ’s are globally bounded for any value of the integer n . As can be seen inthe “ingredients” of our simple demonstration (see (3.4)), no elliptic curves, and theirmodular forms [102], no Calabi-Yau [103], or Frobenius manifolds [100], or Shimuracurves, or arithmetic lattice assumption [104, 105] is required to prove the result. Wejust need to have an n -fold integral such that its integrand is not only algebraic , buthas an expansion at the origin of the form (35).The integrality of all the χ ( n ) ’s, consequence of the remarkable result that the all χ ( n ) ’s are diagonal of rational functions, raises the question of the modularity of the χ ( n ) ’s. Now, the full susceptibility can, formally, be seen as the diagonal of an infinitesum of rational functions . This also raises the question of defining, and addressing,modularity for non-holonomic functions § like the full susceptibility. ‡ See Appendix M and Appendix O in [19] which give an example of a (hypergeometric) family oforder-four operators satisfying the Calabi-Yau condition that their exterior square is of order five, and,even, a family of self-adjoint order-four operators, the corresponding hypergeometric solution-seriesbeing not globally bounded . † For instance the operator D nx − x D x − / § Along this line, recall Chazy’s equations [106] and their (circle) natural boundaries, and, especially, iagonals of rational functions Acknowledgments We would like to thank A. Enge and F. Morain forinteresting and detailed discussions on Fricke and Atkin-Lehner involutions. S. B.would like to thank the LPTMC and the CNRS for kind support. A. B. was supportedin part by the Microsoft Research–Inria Joint Centre. As far as physicists authors areconcerned, this work has been performed without any support of the ANR, the ERCor the MAE. Appendix A. Modular forms and series integralityFirst example : The generating function of the integers n X k =0 (cid:18) nk (cid:19) · (cid:18) kk (cid:19) · (cid:18) n − kn − k (cid:19) (A.1)= (cid:18) nn (cid:19) · F (cid:16) [ 12 , − n, − n, − n ] , [1 , , − n − 12 ]; 1 (cid:17) , is nothing else but the expansion of the square of a HeunG function HeunG (cid:16) , , , , , 12 ; 16 · x (cid:17) = 1 + 2 x + 12 x + 104 x + 1078 x + 12348 x + 150528 x + 1914432 x + · · · (A.2)solution of the order-two operator H diam = θ − · x · (10 θ + 5 θ + 1) + 16 x · (2 θ + 1) . (A.3)which corresponds to the diamond lattice [43]. This HeunG function (A.2) is actuallya modular form which can be written in two different ways: HeunG (4 , , , , , 12 ; 16 x )= (1 − x ) − / · F (cid:16) [ 16 , 13 ] , [1]; 108 x (1 − x ) (cid:17) (A.4)= (1 − x ) − / · F (cid:16) [ 16 , 13 ] , [1]; − x (1 − x ) (cid:17) . These two pullbacks are related by an “Atkin” involution x ↔ / /x . Theassociated modular curve, relating these two pullbacks (A.4) yielding the modularcurve : 4 · y z − y z · ( y + z ) + 3 y z · (4 y + 4 z − y z ) − · ( y + z ) · ( y + z + 83 y z ) + 432 y z = 0 , (A.5)which is ( y, z )-symmetric and is exactly the rational modular curve in eq. (27) alreadyfound for the order-three operator F in [31] for the five-particle contribution ˜ χ (5) ofthe magnetic susceptibility of the Ising model.This result in [41, 43] can be rephrased as follows. One introduces the order-threeoperator which has the following F solution1(4 − x ) · F (cid:16) [ 13 , , 23 ] , [1 , , x (4 − x ) (cid:17) , (A.6) Harnad and McKay paper [107] on modular solutions to equations of generalized Halphen type. iagonals of rational functions Green function of the diamond lattice . Along a modular form line lets us note that this hypergeometric function actually has two pullbacks: F (cid:16) [ 13 , , 23 ] , [1 , , x (4 − x ) (cid:17) (A.7)= x − · ( x − · F (cid:16) [ 13 , , 23 ] , [1 , , x · ( x − (cid:17) . These two pullbacks related by the “Atkin” involution x → /x : u ( x ) = 27 x (4 − x ) , v ( x ) = u ( 2 x ) = 27 x · ( x − , (A.8)corresponding, again, to the modular curve (A.5). Second example . The HeunG function HeunG ( − , , / , , , / 2; 12 · x ) (A.9)= (1 + 4 x ) − / · HeunG (cid:16) , , , , , , x x (cid:17) = 1 + 6 x + 24 x + 252 x + 2016 x + 19320 x + 183456 x + 1823094 x + 18406752 x + 189532980 x + · · · is solution of the order-two operator H fcc = θ − x · θ · (4 θ + 1) − · x · (2 θ + 1) · ( θ + 1) , (A.10)The square of (A.9) is actually the solution of an order-three operator (see equation(19) in [43]) emerging for lattice Green functions of the face-centred cubic (fcc) latticewhich is thus the symmetric square of (A.10). This hypergeometric function with apolynomial pull-back can also be written: HeunG ( − , , / , , , / 2; 12 · x )= F (cid:16) [ 16 , 13 ] , [1]; 108 · x · (1 + 4 x ) (cid:17) (A.11)= (1 − x ) − / · F (cid:16) [ 16 , 13 ] , [1]; − · x · (1 + 4 x ) (1 − x ) (cid:17) , where the involution x ↔ − / · (1 + 4 x ) / (1 − x ) takes place. The modular curverelating these two pullbacks reads exactly the rational curve (A.5) already obtainedin [31]. Third example . The HeunG function HeunG (1 / , / , / , / , , / 2; 4 x ) issolution of the order-two operator corresponding to the simple cubic lattice Greenfunction H sc = θ − x · (40 θ + 20 θ + 3) + 9 · x · (4 θ + 3) · (4 θ + 1) . The square of this HeunG function is a series with integer coefficients which identifieswith the Hadamard product of (1 − x ) − / with a modular form : HeunG (1 / , / , / , / , , / 2; 4 x ) (A.12)= (1 − x ) − / ⋆ HeunG (1 / , / , , , , x )= 1 + 6 x + 90 x + 1860 x + 44730 x + 1172556 x + 32496156 x + 936369720 x + 27770358330 x + 842090474940 x + · · · The HeunG function HeunG (1 / , / , / , / , , / 2; 4 x ) is globally bounded: theseries of HeunG (1 / , / , / , / , , / 2; 8 x ) is a series with integer coefficients. iagonals of rational functions F ([1 / , / , [1] , x )hypergeometric function up to a simple algebraic pullback (with a square root), orin terms of a F ([1 / , / , [1] , x ) hypergeometric function: HeunG (1 / , / , / , / , , / 2; 4 x ) = C / · F (cid:16) [1 / , / , [1]; P (cid:17) , with: C = 19 · (1 + 12 x ) · (cid:16) − x + 4 · (1 − x ) / (cid:17) , P = 128 · x (1 + 12 x ) · p ,p = (1 − x + 352 x − x ) + (1 − x ) · (1 − x ) · (1 − x ) / . Do note that taking the Galois conjugate (changing (1 − x ) / into − (1 − x ) / )gives the series expansion of 3 − / · HeunG (1 / , / , / , / , , / 2; 4 x ). Thisshows that there exists an identity for F ([1 / , / , [1] , x ) with two differentpullbacks , namely the previous P and its Galois conjugate, these two pullbacks beingrelated by a (symmetric genus zero) modular curve:5308416 · y z + 442368 · y z · ( y + z ) + 512 y z · (27 y + 27 z − x y )+ 192 y z · ( y + z ) · ( y + z + 10718 y z ) + y + z + 3622662 y z − · y z · ( y + z ) + 79872 · y z · ( y + z ) − · y z = 0 . (A.13) Revisiting the examples . In a recent paper [44] corresponding to spanningtree generating functions and Mahler measures, a result from Rogers (equation (36)in [44]) is given where the two following F hypergeometric functions take place: F (cid:16) [ 54 , , , , , [2 , , , , x · ( x + 3) (cid:17) , F (cid:16) [ 54 , , , , , [2 , , , , x · (1 + 3 x ) (cid:17) . (A.14)The corresponding order-five linear differential operators (annihilating these two F hypergeometric functions) are actually homomorphic (the intertwiners being order-four operators). The relation between these two pullbacks y = 256 x / / ( x + 3) and z = 256 x/ / (1 + 3 x ) , remarkably gives, again, the previous ( y, z )- symmetricmodular curve (A.13).The order-five linear differential operator, corresponding to the first F hypergeometric function, factorizes in an order-one operator, an order-three operatorand an order-one operator, the order-three operator being, in fact, exactly thesymmetric square of an order-two operator: L · Sym ( W ) · x ( x − 9) ( x + 3) · R , where the order-one operators read respectively L = D x − ddx ln (cid:16) x − x + 14 x + 9) · ( x + 3) (cid:17) , R = D x − ddx ln (cid:16) ( x + 3) x (cid:17) , and where the order-two operator W reads: W = D x + 3 (6 · x + 7 x + 3)(9 x + 14 x + 9) · x · D x + 34 · x + 2(9 x + 14 x + 9) · x . (A.15)We have a similar result for the order-five linear differential operatorcorresponding to the second F hypergeometric function. iagonals of rational functions x + 3) x · Z x − x + 3) · x · F (cid:16) [ 14 , , 34 ] , [1 , , x · ( x + 3) (cid:17) · dx. (A.16)The expansion of the F hypergeometric function in (A.16) is globally bounded(change x → x to get a series with integer coefficients).Recalling the two previous pullbacks we have, in fact, the following identity:3 · (1 + 3 x ) · F (cid:16) [ 14 , , 34 ] , [1 , , x · ( x + 3) (cid:17) = ( x + 3) · F (cid:16) [ 14 , , 34 ] , [1 , , x · (1 + 3 x ) (cid:17) . (A.17)However this F hypergeometric function is nothing but the square of a F hypergeometric function F (cid:16) [ 14 , , 34 ] , [1 , , x (cid:17) = F (cid:16) [ 18 , 38 ] , [1] , x (cid:17) . (A.18)Thus, the previous identity (A.17) is nothing but the identity on a F hypergeometricfunction with two different pullbacks :(1 + 3 x ) / · F (cid:16) [ 18 , 38 ] , [1] , x · ( x + 3) (cid:17) = (cid:16) x (cid:17) / · F (cid:16) [ 18 , 38 ] , [1] , x · (1 + 3 x ) (cid:17) . (A.19)The expansion of (A.19) is globally bounded. One gets a series with positive integer coefficients using the simple rescaling x → · x . Note that the two pullbacks can beexchanged by the simple “Atkin” involution x ↔ /x , being related by the modularcurve occurring for the simple cubic lattice, namely (A.13).We have a similar result for the other F hypergeometric functions popping outin [44].For instance, for the diamond lattice one gets an expression (see eq. (50) in [44])where the two following F hypergeometric functions take place ‡ : F (cid:16) [ 53 , , , , , [2 , , , , − x · (1 − x ) (cid:17) , F (cid:16) [ 53 , , , , , [2 , , , , x (4 − x ) (cid:17) . (A.20)These two pullbacks can be exchanged by the simple “Atkin” involution x ↔ /x .These two pullbacks have been seen to be related by the (genus-zero) ( y, z )-symmetricmodular curve (A.5):4 y z − y z · ( y + z ) + 3 y z (cid:0) y + 4 z − y z (cid:1) − · ( y + z ) · ( y + z + 83 y z ) + 432 y z = 0 . (A.21)Similarly to (A.17) we have an identity between two F hypergeometric functions(namely F ([2 / , / , / , [1 , , z )) with the two pullbacks (A.20), and these F ‡ Note a small misprint in eq. (50) of [44]: one should read − z / / (1 − z ) instead of − z / / (1 − z ) . iagonals of rational functions F hypergeometric functions, one findsthat the “deus ex machina” is the identity similar to (A.19):(1 − x ) / · F (cid:16) [ 13 , 16 ] , [1] , x (4 − x ) (cid:17) = (1 − x / · F (cid:16) [ 13 , 16 ] , [1] , − x · (1 − x ) (cid:17) . (A.22)The series expansion of (A.22) is globally bounded. Rescaling the x variable as x → x , the series expansion becomes a series with positive integer coefficients (upto the first constant term).For the face-centred cubic lattice one gets an expression (see eq. (52) in [44])where the two following F hypergeometric functions take place † : F (cid:16) [ 53 , , , , , [2 , , , , x · ( x + 3) ( x − (cid:17) , F (cid:16) [ 53 , , , , , [2 , , , , x · ( x + 3)4 (cid:17) . (A.23)This example is nothing but the previous diamond lattice example (A.20) with thechange of variable x → − x / ( x − 4) in (A.23). Therefore, the two pullbacks in(A.23) are, again, related by the modular curve (A.5). The two pullbacks in (A.23)can actually be seen directly in the following identity (equivalent to (A.22)): F (cid:16) [ 13 , 16 ] , [1] , x · ( x + 3) ( x − (cid:17) = (1 − x ) / · F (cid:16) [ 13 , 16 ] , [1] , x · ( x + 3)4 (cid:17) . Finally, the equation (17) of [44] on Mahler measures, the two following F hypergeometric functions take place: F (cid:16) [ 53 , , , , [2 , , , x ( x − (cid:17) , F (cid:16) [ 53 , , , , [2 , , , x ( x + 4) (cid:17) . (A.24)These two previous pullbacks can be exchanged by an “Atkin” involution x ↔ − /x and are related by the (genus-zero) ( y, z )-symmetric modular curve:8 y z − y z · ( y + z ) + 3 y z · (2 y + 2 z + 13 y z ) − ( y + z ) · ( y + z + 29 y z ) + 27 y z = 0 . (A.25)The underlying identity on F hypergeometric functions with the two pullbacks(A.24) read: − · ( x − · F (cid:16) [ 13 , 23 ] , [1] , x ( x + 4) (cid:17) = ( x + 4) · F (cid:16) [ 13 , 23 ] , [1] , x ( x − (cid:17) . (A.26)The series expansion of (A.26) is globally bounded. Rescaling the x variable as x → − x , the series expansion becomes a series with positive integer coefficients. † There is one more misprint in [44]: the pullback − x ( x + 3) / ( x − must be changed into x ( x + 3) / ( x − . iagonals of rational functions Appendix B. Seeking for the “minimal” rational function The effective calculations of section (4.2) guarantee to obtain an explicit expression forthe rational function associated with (40), however the rational function is far frombeing unique. Recalling the well-known Ap´ery series A ( x ), and its rewriting due toStrehl and Schmidt [85, 86, 87], A ( x ) = ∞ X n =0 n X k =0 (cid:18) nk (cid:19) (cid:18) n + kk (cid:19) · x n = ∞ X n =0 n X k =0 k X j =0 (cid:18) nk (cid:19) (cid:18) n + kk (cid:19)(cid:18) kj (cid:19) · x n = 1 + 5 x + 73 x + 1445 x + 33001 x + · · · , (B.1) A ( x ) is known to be the diagonal of the rational function in five variables 1 /R /R where R , R read [60]: R = 1 − z , R = (1 − z )(1 − z )(1 − z )(1 − z ) − z z z , as well as the diagonal of the rational function in five variables 1 /Q /Q where Q , Q read [61, 45]: Q = 1 − z z z z , Q = (1 − z )(1 − z ) − z · (1 + z )(1 + z ) , and also the diagonal of the rational function in six variables 1 /P /P /P where P , P , P read [60]: P = 1 − z z , P = 1 − z − z − z z z , P = 1 − z − z − z z z . A yet different diagonal representation for the Ap´ery series, due to Delaygue † , isprovided by the diagonal of the rational function in eight variables:1(1 − z z z z ) · (1 − z · (1 + z )) · (1 − z · (1 + z )) · (1 − z − z ) · (1 − z − z ) . Calculations similar to (39) on these new binomial expressions provide two newrational functions such that (B.1) can be written as the diagonal of one of these tworational functions. One is a rational function of five variables, of the form 1 /Q (5)1 /Q (5)2 Q (5)1 = 1 − z z z z z · (1 + z ) (1 + z ) (1 + z ) (1 + z ) ,Q (5)2 = 1 − z · (1 + z ) (1 + z ) (1 + z ) (1 + z ) , (B.2)and the other one, is a rational function of six variables, of the form 1 /Q (6)1 /Q (6)2 /Q (6)3 Q (6)1 = 1 − z z z z · (1 + z ) (1 + z ) (1 + z ) (1 + z ) (1 + z ) ,Q (6)2 = 1 − z z z z z z · (1 + z ) (1 + z ) ,Q (6)3 = 1 − z · (1 + z ) (1 + z ) (1 + z ) (1 + z ) (1 + z ) . (B.3) Appendix C. Hypergeometric series with coefficients ratio of factorials As a consequence of the classification by Beukers and Heckman [91] of all algebraic n F n − ’s, the F hypergeometric series F (cid:18)(cid:20) , , , , , , , (cid:21) , (cid:20) , , , , , , (cid:21) , x (cid:19) , † Personal communication. iagonals of rational functions integer coefficients , and is an algebraic function . The Galois group belonging tothis function is the Weyl group W ( E ) which has 696729600 elements [108]. It isan algebraic series of degree 483840. More precisely, it was noticed by Rodriguez-Villegas [109] that the previous power series reads: ∞ X n =0 (30 n )! n !(15 n )! (10 n )! (6 n )! · x n , (C.1)which is closely related to the series introduced by Chebyshev in his work [110] onthe distribution of prime numbers to establish the estimate [19] on the prime countingfunction π ( x ).Considering hypergeometric series such that their coefficients are ratio offactorials, reference [109] gives the conditions of these factorials for the hypergeometricseries to be algebraic (all the coefficients are thus integers). A simple example is, forinstance the algebraic function: F (cid:16) [ 14 , , 34 ] , [ 13 , 23 ]; 2 · x (cid:17) = ∞ X n =0 (cid:18) nn (cid:19) · x n . (C.2)Along this line it is worth recalling Delaygue’s Thesis [98] (see alsoBober [111]) which gives some results ♯ for series expansions ¶ such thattheir coefficients are ratio of factorials , namely F ([1 / , / , [1]; 27 x ) , and F ([1 / , / , / , / , [1 , , x ), giving respectively the series ∞ X n =0 (3 n )!( n !) · x n , ∞ X n =0 ((2 n )!) ( n !) · x n , and: F (cid:16) [ 12 , , , 56 ] , [1 , , · x (cid:17) = ∞ X n =0 (6 n )! (2 n )!(3 n )! ( n !) · x n . (C.3)These ratio of factorials are all integer numbers . Appendix D. Proof of integrality of series (43) Let us sketch the proof of the integrality of series (43), namely, the integrality ofcoefficients (45). For each power of the integer number q = p n a term like 4 + 9 n is periodically divisible (period p ) by q . In order to have the ratio (46) be an integer,one needs the numerator to be divisible by this factor q before the denominator. Thecase p = 3 is an easy one. The other prime p do not divide 9. One needs to findthe first case of divisibility, namely the first integer n such that 4 + 9 n = k q (thiscorresponds to the smallest k ). If d q = 1, mod. k = 4 d , mod. 9. In otherwords, the smallest k is the rest of 4 d , mod. 9. Consequently, we have replaced thecalculations, for every integer q , by a finite set of calculations for d = 1 , , , , , Remark: The terms n + 1 are always the last to be divisible by q . Hence, onecan forget its factors. However, one needs as many factors at the numerator than at ♯ Necessary and sufficient conditions for the integrality of the mirror maps series. ¶ These series are not algebraic functions. iagonals of rational functions d · a gives thecomplete proof: . Appendix E. Yukawa couplings Appendix E.1. Yukawa couplings as ratio of determinants Consider an order-four MUM linear differential operator. Let us introduce thedeterminantal variables W m = det( M m ) which are the determinants ‡ of the following m × m matrices M m , m = 1 , . . . , 4, with entries expressed in terms of derivatives ofthe four solutions y ( x ), y ( x ), y ( x ) and y ( x ) of the MUM linear differential operator(see Section (4.1) for the definitions). One takes W ( x ) = y ( x ) and: M = " y y y ′ y ′ , M = y y y y ′ y ′ y ′ y ′′ y ′′ y ′′ , M = y y y y y ′ y ′ y ′ y ′ y ′′ y ′′ y ′′ y ′′ y ′′′ y ′′′ y ′′′ y ′′′ , where: y ′ i = ddx y i , y ′′ i = d dx y i , y ′′′ i = d dx y i . (E.1)Since q is equal to q = exp( y /y ), and its derivative verifies q · ddq = W W · ddx = y W · ddx , (E.2)we have that (cid:16) q · ddq (cid:17) = y W · d dx + 2 y W dy dx · ddx − y W dW dx · ddx . (E.3)We deduce, after some simple algebra, an alternative definition for the Yukawacoupling : K ( q ) = (cid:16) q · ddq (cid:17) (cid:16) y y (cid:17) = W · W W = y · W W . (E.4)to be compared with the other previous alternative expression previously given (87)for the Yukawa coupling K ( q ) = x ( q ) · W / y · (cid:16) qx ( q ) · dx ( q ) dq (cid:17) = W / y · (cid:16) q · dx ( q ) dq (cid:17) . (E.5) ‡ For an order-four operator the Wronskian is W . iagonals of rational functions (cid:16) q · dx ( q ) dq (cid:17) = W W = y W , (E.6)and, so, (E.5) is compatible with (E.4) if the following identity is verified : W = W · y = W · W . (E.7)This identity is in fact specific of order-four operators conjugated to their adjoints (seebelow (E.17)). Therefore we prefer to use definition (E.4) for the Yukawa coupling,instead of the more restricted definition (E.5).Let us assume that the pullback p ( x ) has a series expansion of the form p ( x ) = λ · x r · A ( x ) , (E.8)where the exponent r is an integer, where λ is a constant, and where A ( x ) is afunction analytic at x = 0 with the series expansion: A ( x ) = 1 + α · x + α · x + · · · The determinantal variables W m transform very nicely under pullbacks p ( x ) of theform (E.8):( W ( x ) , W ( x ) , W ( x ) , W ( x )) −→ (E.9) (cid:16) W ( p ( x )) , p ′ r · W ( p ( x )) , p ′ r · W ( p ( x )) , p ′ r · W ( p ( x )) (cid:17) , p ′ = dp ( x ) dx . One can show that the nome (36) of an order- N operator transforms under apullback p ( x ): q ( x ) −→ Q ( x ) with: λ · Q ( x ) r = q ( p ( x )) . (E.10)From the covariance property (E.9), and from the previous transformation q → λ · q r for the nome, one easily gets the transformation of the Yukawa couplingseen as a function of the nome K ( q ) → K ( λ · q r ): K ( q ( x )) = W ( x ) · W ( x ) W ( x ) (E.11) −→ W ( p ( x )) · W ( p ( x )) W ( p ( x )) = K ( q ( p ( x ))) = K ( λ · Q ( x ) r ) . For λ = 1 and r = 1 (i.e. when the pullback is a deformation of theidentity transformation), one recovers the known invariance of the Yukawa couplingby pullbacks (see Proposition 3 in [112]).One finds another pullback invariant ratio, namely: K ⋆ = W · W W · W , (E.12)which is, in fact, nothing but the Yukawa coupling for the adjoint of the originaloperator.Another invariance property is worth noting. Let us consider two lineardifferential operators Ω and Ω of order N that are equivalent, in the sense ofthe equivalence of linear differential operators. This means that there exists lineardifferential operators intertwiners I , I , J , J , of order at most N − · I = I · Ω , and: J · Ω = Ω · J . (E.13) iagonals of rational functions = ρ ( x ) · Ω · ρ ( x ) − , (E.14)which correspond to changing the four solutions as follows: y i → ρ ( x ) · y i . In sucha case the previous determinant variables transform, again, very nicely under the“gauge” function ρ ( x ):( W , W , W , W ) → ( ρ ( x ) · W , ρ ( x ) · W , ρ ( x ) · W , ρ ( x ) · W ) . (E.15)It is straightforward to see that the Yukawa coupling and the “dual Yukawa”, K ⋆ , are invariant by such a transformation ¶ . Two conjugated operators (E.14)automatically have the same Yukawa coupling.Do note that the Yukawa couplings for two operators, which are non triviallyhomomorphic to each other (intertwiners of order one, two, ...), are actually different. The (pullback invariant) Yukawa coupling is not preserved by operator equivalence (seesubsection (8.2)). Remark: The definition of these determinantal variables W i heavily relies onthe MUM structure of the operator between the four solutions the definition of W i ’s,in particular the log-ordering of the solutions. It is worth noting that if one permutesthe four solution y i , one would get 24 other sets of ( W , W , W , W ) which areactually also nicely covariant by pullbacks , thus yielding a finite set of other “Yukawacouplings” or adjoint Yukawa coupling K ⋆ also invariant by pullbacks .In fact these “Yukawa couplings” (E.4), and other adjoint Yukawa K ⋆ (E.12),can even be defined when the linear differential operator is not MUM, and they arestill invariant by pullbacks. Appendix E.2. Pullback-invariants for higher order ODEs These simple calculations can straightforwardly be generalised to higher order lineardifferential equations. We give here the invariants for higher order linear differentialoperators.Let us give, for the n -th order linear differential operator the list of the K n invariants by pullback transformations: K = W · W W , K = W · W W , K = W · W W , K = W · W W , · · · K n = W a n · W n W b n , with: a n = n · ( n − , b n = n · ( n − . (E.16)A n -th order linear differential operator has K n as an invariant by pullbacktransformation, as well as all the K m with m ≤ n . K is the Yukawa coupling,and one remarks, for the order-four operators, that the other pullback invariant K ⋆ (see (E.12)), which is actually also the Yukawa coupling of the adjoint operator, isnothing but K /K .For order-four operators conjugated to their adjoint (see (E.14)) (i.e. operatorshomomorphic to their adjoint, the intertwiner being an order zero differential operator, ¶ K and K ⋆ (and their combinations) are the only monomials W n W n W n W n to be invariantby (E.9) and (E.15). iagonals of rational functions 43a function), one has the equality K = K , i.e. K = K ⋆ , or W = W · W , (E.17)to be compared with the equality in Almkvist et al. (see Proposition 2 in [75]) y y ′ − y y ′ = y y ′ − y y ′ , (E.18)which is satisfied when the Calabi-Yau condition that the exterior square is of orderfive is satisfied.If a linear differential operator Ω verifies condition (E.17), its conjugate by afunction, ρ ( x ) · Ω · ρ ( x ) − , also verifies condition (E.17) (their Yukawa couplings areequal).The condition (E.17) is not satisfied for linear differential operators homomorphicto their adjoint with non-trivial intertwiner (of order greater than zero). For instancethe order-four operator (88) does not satisfy condition (E.17). Appendix F. More Hadamard products: a Batyrev and van StratenCalabi-Yau ODE An order-four operator has been found by Batyrev and van Straten [77] associatedwith a Calabi-Yau three-fold on P × P B = θ − x · (7 θ + 7 θ + 2) · (3 θ + 1) · (3 θ + 2) (F.1) − x · (3 θ + 5) · (3 θ + 4) · (3 θ + 2) · (3 θ + 1) . This operator is conjugated to its adjoint: B · x = x · adjoint ( B ).Operator (F.1) is a Calabi-Yau operator [77]: it is MUM, and it is such that itsexterior square is of order five . It has a solution analytical at x = 0 which is actuallythe Hadamard product of the previous selected hypergeometric F : F (cid:16) [ 13 , 23 ] , [1]; 27 x (cid:17) ⋆ (cid:16) 11 + 4 x · F (cid:16) [ 13 , 23 ] , [1]; 27 · x (1 + 4 x ) (cid:17)(cid:17) . (F.2)The coefficients of the series expansion of (F.2) are integers: they can actually bewritten as a sum of product of binomials :(3 n )!( n !) · n X k =0 (cid:18) nk (cid:19) = (cid:18) nn (cid:19) · (cid:18) nn (cid:19) · n X k =0 (cid:18) nk (cid:19) . 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