The Minkowski question mark function: explicit series for the dyadic period function and moments
aa r X i v : . [ m a t h . N T ] J a n THE MINKOWSKI QUESTION MARK FUNCTION: EXPLICIT SERIESFOR THE DYADIC PERIOD FUNCTION AND MOMENTS
GIEDRIUS ALKAUSKAS
Abstract.
Previously, several natural integral transforms of the Minkowski question markfunction F ( x ) were introduced by the author. Each of them is uniquely characterized bycertain regularity conditions and the functional equation, thus encoding intrinsic informationabout F ( x ). One of them - the dyadic period function G ( z ) - was defined as a Stieltjes transform.In this paper we introduce a family of “distributions” F p ( x ) for ℜ p ≥
1, such that F ( x ) is thequestion mark function and F ( x ) is a discrete distribution with support on x = 1. We provethat the generating function of moments of F p ( x ) satisfies the three term functional equation.This has an independent interest, though our main concern is the information it provides about F ( x ). This approach yields the following main result: we prove that the dyadic period functionis a sum of infinite series of rational functions with rational coefficients. Keywords: The Minkowski question mark function, the dyadic period function, three termfunctional equation, analytic theory of continued fractions, Julia sets, the Farey treeMathematics subject classification: Primary: 11A55, 26A30, 32A05;Secondary: 40A15, 37F50, 11F37.
Contents
1. Introduction and main result 22. p − question mark functions and p − continued fractions 103. Complex case 144. Properties of integral transforms of F p ( x ) 205. Three term functional equation 246. The proof: approach through p = 2 26Appendix A. 31A.1. Approach through p = 0 31A.2. Auxiliary lemmas 32A.3. Numerical values for the moments 34A.4. Rational functions H n ( z ) 36A.5. Rational functions Q n ( z ) 37 References 37
Acknowledgements.
The author sincerely thanks J¨orn Steuding, whose seemingly elementaryproblem, proposed at the problem session in Palanga conference in 2006, turned to be a veryrich and generous one. Also, the author thanks several colleagues for showing interest in theseresults, after preprints became available in arXiv , especially to Steven Finch, Jeffrey Lagariasand Stefano Isola. 1.
Introduction and main result
The aim of this paper to continue investigations on the moments of Minkowski ?( x ) function,begun in [1], [2] and [3]. The function ?( x ) (“the question mark function”) was introduced byMinkowski as an example of a continuous function F : [0 , ∞ ) → [0 , x itis defined by the expression F ([ a , a , a , a , ... ]) = 1 − − a + 2 − ( a + a ) − − ( a + a + a ) + ..., (1)where x = [ a , a , a , a , ... ] stands for the representation of x by a (regular) continued fraction[15]. By tradition, this function is more often investigated in the interval [0 , F (1) = 1, whereas in our case F (1) = . Accordingly, we make a con-vention that ?( x ) = 2 F ( x ) for x ∈ [0 , x , the series terminates at the last nonzeropartial quotient a n of the continued fraction. This function is continuous, monotone and sin-gular [9]. By far not complete overview of the papers written about the Minkowski questionmark function or closely related topics (Farey tree, enumeration of rationals, Stern’s diatomicsequence, various 1-dimensional generalizations and generalizations to higher dimensions, sta-tistics of denominators and Farey intervals, Hausdorff dimension and analytic properties) canbe found in [1]. These works include [5], [6], [8], [9], [10], [12], [13] (this is the only paper wherethe moments of a certain singular distribution - a close relative of F ( x ) - were considered),[11], [14], [16], [18], [20], [24], [25] [26], [27], [28], [29], [30], [31], [33]. The internet page [36]contains up-to-date and exhaustive bibliography list of papers related to Minkowski questionmark function.Recently, in Calkin and Wilf [8] defined a binary tree which is generated by the iteration ab aa + b , a + bb , starting from the root . The last two authors have greatly publicized this tree, but it was knownlong ago to physicists and mathematicians (alias, Stern-Brocot or Farey tree). Elementaryconsiderations show that this tree contains every positive rational number once and only once, XPLICIT SERIES FOR THE DYADIC PERIOD FUNCTION 3 each being represented in lowest terms. The first four iterations lead to jjjjjjjjjjjjjjj TTTTTTTTTTTTTTT uuuuuuu IIIIIII uuuuuuu IIIIIII (cid:7)(cid:7)(cid:7) (cid:7)(cid:7)(cid:7) (cid:7)(cid:7)(cid:7) (cid:7)(cid:7)(cid:7) (2)It is of utmost importance to note that the n th generation consists of exactly those 2 n − positive rational numbers, whose elements of the continued fraction sum up to n . This fact canbe easily inherited directly from the definition. First, if rational number ab is represented as acontinued fraction [ a , a , ..., a r ], then the map ab → a + bb maps ab to [ a + 1 , a ..., a r ]. Second, themap ab → aa + b maps ab to [0 , a + 1 , ..., a r ] in case ab <
1, and to [1 , a , a , ..., a r ] in case ab > F n ( x ), and F n ( x ) converges uniformly to F ( x ). The function F ( x ) as a distribution function (in the senseof probability theory, which imposes the condition of monotonicity) is uniquely determined bythe functional equation [1]2 F ( x ) = ( F ( x −
1) + 1 if x ≥ ,F ( x − x ) if 0 ≤ x < . (3)This implies F ( x ) + F (1 /x ) = 1. The mean value of F ( x ) has been investigated by severalauthors, and was proved to be 3 / F ( x ). Let M L = ∞ Z x L d F ( x ) , m L = ∞ Z (cid:16) xx + 1 (cid:17) L d F ( x ) = 2 Z x L d F ( x ) . Both sequences are of definite number-theoretical significance because M L = lim n →∞ − n X a + a + ... + a s = n [ a , a , .., a s ] L , m L = lim n →∞ − n X a + ... + a s = n [0 , a , .., a s ] L , GIEDRIUS ALKAUSKAS (the summation takes place over rational numbers represented as continued fractions; thus, a i ≥ a s ≥ M ( t ) = ∞ X L =0 M L L ! t L = ∞ Z e xt d F ( x ) , m ( t ) = ∞ X L =0 m L L ! t L = ∞ Z exp (cid:16) xtx + 1 (cid:17) d F ( x ) = 2 Z e xt d F ( x ) . One directly verifies that m ( t ) is an entire function, and that M ( t ) is meromorphic functionwith simple poles at z = log 2 + 2 πin , n ∈ Z . Further, we have M ( t ) = m ( t )2 − e t , m ( t ) = e t m ( − t ) . The second identity represents only the symmetry property, given by F ( x ) + F (1 /x ) = 1. Themain result about m ( t ) is that it is uniquely determined by the regularity condition m ( − t ) ≪ e −√ log 2 √ t , as t → ∞ , the boundary condition m (0) = 1, and the integral equation m ( − s ) = (2 e s − ∞ Z m ′ ( − t ) J (2 √ st ) d t, s ∈ R + . (4)(Here J ( ⋆ ) stands for the Bessel function J ( z ) = π R π cos( z sin x ) d x ).Our primary object of investigations is the generating function of moments. Let G ( z ) = ∞ P L =1 m L z L − . This series converges for | z | ≤
1, and the functional equation for G ( z ) (see below)implies that there exist all derivatives of G ( z ) at z = 1, if we approach this point while remainingin the domain ℜ z ≤
1. Then the integral G ( z ) = ∞ Z x + 1 − z d F ( x ) = 2 Z x − xz d F ( x ) . (5)(which is Stieltjes transform of F ( x )) extends G ( z ) to the cut plane C \ (1 , ∞ ). The generatingfunction of moments M L does not exist due to the factorial growth of M L , but this generatingfunction can still be defined in the cut plane C ′ = C \ (0 , ∞ ) by R ∞ x − xz d F ( x ). In fact, thisintegral just equals to G ( z + 1). Thus, there exist all higher derivatives of G ( z ) at z = 1, and L − L − d z L − G ( z ) (cid:12)(cid:12) z =1 = M L , L ≥
1. The following result was proved in [1].
Theorem 1.1.
The function G ( z ) , defined initially as a power series, has an analytic contin-uation to the cut plane C \ (1 , ∞ ) via (5). It satisfies the functional equation z + 1 z G (cid:16) z (cid:17) + 2 G ( z + 1) = G ( z ) , (6) XPLICIT SERIES FOR THE DYADIC PERIOD FUNCTION 5 and also the symmetry property G ( z + 1) = − z G (cid:16) z + 1 (cid:17) − z , (which is a consequence of the main functional equation). Moreover, G ( z ) → , if z → ∞ andthe distance from z to a half line [0 , ∞ ) tends to infinity. Conversely, the function having theseproperties is unique. Accordingly, this result and the specific appearance of the three term functional equationjustifies the name for G ( z ) as the dyadic period function .We wish to emphasize that the main motivation for previous research was clarification ofthe nature and structure of the moments m L . It was greatly desirable to give these constants(emerging as if from geometric chaos) some other expression than the one obtained directlyfrom the Farey (or Calkin-Wilf) tree, which could reveal their structure to greater extent. Thisis accomplished in the current work. Thus, the main result can be formulated as follows. Theorem 1.2.
There exist canonical and explicit sequence of rational functions H n ( z ) , suchthat for {| z | ≤ } ∪ {| z + | ≤ } one has an absolutely convergent series G ( z ) = ∞ Z x + 1 − z d F ( x ) = ∞ X n =0 ( − n H n ( z ) , H n ( z ) = B n ( z )( z − n +1 , where B n ( z ) is polynomial with rational coefficients of degree n − . For n ≥ it has thefollowing reciprocity property: B n ( z + 1) = ( − n z n − B n (cid:16) z + 1 (cid:17) , B n (0) = 0 . The rational function H n ( z ) are defined via implicit and rather complicated recurrence (27)(see Section 6). The following table gives initial polynomials B n ( z ). GIEDRIUS ALKAUSKAS n B n ( z ) n B n ( z )0 − − z + 53270 z − z z − z + 112675 z − z − z − z + 47029425250 z − z − z + 78760750 z z − z z − z + 27286922325625 z + 539244422325625 z − z + 4778023189375 z Remark.
The constant can be replaced by any constant less than 1 . − (the latter comesexactly from Lemma A.3). Unfortunately, our method does not allow to prove an absoluteconvergence in the disk | z | ≤
1. In fact, apparently the true region of convergence of the seriesin question is the half plane ℜ z ≤
1. Take, for example, z = + 4 i . Then by (6) and symmetryproperty one has G ( z ) = 12 G ( z − − z − G (cid:16) z − (cid:17) − z −
1) = − z − G (cid:16) z − z − (cid:17) − z − G (cid:16) z − (cid:17) − z − − z − . Both arguments under G on the right belong to the unit circle, and thus we can use Taylorseries for G ( z ). Using numerical values of m L , obtained via the method described in AppendixA.2., we obtain: G ( z ) = 0 . + + 0 . + i , with all digits exact. On the other hand, theseries in Theorem 1.2 for n = 60 gives X n =0 ( − n H n ( z ) = 0 . + + 0 . + i. Finally, based on the last integral in (5), we can calculate G ( z ) as a Stieltjes integral. If wedivide the unit interval into N = 3560 equal subintervals, and use Riemann-Stieltjes sum, weget an approximate value G ( z ) ≈ . + + 0 . + i . All evaluations match very well. Experimental observation 1.3.
We conjecture that the series in Theorem 1.2 convergesabsolutely for ℜ z ≤ . With a slight abuse of notation, we will henceforth write f ( L − ( z ) instead of d L − d z L − f ( z ) (cid:12)(cid:12) z = z . Corollary 1.4.
The moments m L can be expressed by the convergent series of rational numbers: m L = lim n →∞ − n X a + a + ... + a s = n [0 , a , a , ..., a s ] L = 1( L − ∞ X n =0 ( − n H ( L − n (0) . XPLICIT SERIES FOR THE DYADIC PERIOD FUNCTION 7
The speed of convergence is given by the following estimate: (cid:12)(cid:12)(cid:12) H ( L − n (0) (cid:12)(cid:12)(cid:12) ≪ n M , for every M ∈ N . The implied constant depends only on L and M . Thus, m = P ∞ n =0 ( − n H ′ n (0) = 0 . + . Regarding the speed, numerical calcula-tions show that in fact the convergence is geometric. Theorem 1.2 in case z = 1 gives M = G (1) = 1 + 0 + ∞ X n =0 (cid:16) (cid:17) n = 32 , which we already know (see Corollary 4.5; the above is a Taylor series for M ( p ) in powers of p −
2, specialized at p = 1). Geometric convergence would be the consequence of the factthat analytic functions m L ( p ) extend beyond p = 1 (see below). This is supported by thephenomena represented as Experimental observation 1.5. Meanwhile, we are able to prove onlythe given rate. If we were allowed to use the point z = 1, Theorem 1.2 would give a convergentseries for the moments M L as well. This is exactly the same as the series in the Corollary 1.4,only one needs to use a point z = 1 instead of z = 0. Experimental observation 1.5.
For L ≥ , the series M L ( p ) = 1( L − · ∞ X n =0 ( p − n H ( L − n (1) , M L (1) = M L , has exactly − L √ as a radius of convergence. To this account, Proposition 4.3 endorse this phenomena, which is highly supported by nu-merical calculations, and which does hold for L = 1.The following two tables give starting values for the sequence H ′ n (0). n H ′ n (0) n H ′ n (0) n H ′ n (0)0 14 5 − · · − · · · · · − · ·
11 7972095369765570794232 · · · · · ·
312 148 7 2389012 · · · · · · · · · · −
172 8 − · · · − · · · · · · · · · · ·
17 14 72261916360136752928335145486035163954998992 · · · · · · · GIEDRIUS ALKAUSKAS n H ′ n (0)15 − · · · · · · · · · · · · · · · · · · − · · · · · · · · · · − . . − . P Nn =0 ( − n H ′ n (0) = 0 . + (where N = 17), whereas N = 40 gives 0 . + ,and N = 50 gives 0 . + . Note that the manifestation of Fermat and Mersenne primesin the denominators at an early stage is not accidental, minding the exact value of the deter-minant in Lemma 6.1, Chapter 6 (see below). Moreover, the prime powers of every odd prime,which divides the denominator, increase each time by 1 while passing from H ′ n (0) to H ′ n +1 (0).The pattern for the powers of 2 is more complicated. More thorough research of the linearmap in Lemma 6.1 can thus clarify prime decomposition of denominators; numerators remainsmuch more complicated.As will be apparent later, the result in Theorem 1.2 is derived from the knowledge of p − derivatives of G ( p , z ) at p = 2 (see below). On the other hand, since there are two points( p = 2 and p = 0) such that all higher p − derivatives of G ( p , z ) are rational functions in z ,it is not completely surprising that the approach through p = 0 also gives convergent seriesfor the moments, though in this case we are forced to use Borel summation. At this point,the author does not have a strict mathematical proof of this result (since the function G ( p , z )is meanwhile defined only for ℜ p ≥ Experimental observation 1.6.
Define the rational functions (with rational coefficients) Q n ( z ) , n ≥ , by Q ( z ) = − z , and recurrently by Q n ( z ) = 12 n − X j =0 j ! · ∂ j ∂z j Q n − j − ( − · (cid:16) z j − z j +2 (cid:17) . Then m L = lim n →∞ − n X a + a + ... + a s = n [0 , a , a , ..., a s ] L = 1( L − ∞ X r =0 (cid:16) ∞ X n =0 Q ( L − n ( − n ! · r +1 Z r t n e − t d t (cid:17) . (7) XPLICIT SERIES FOR THE DYADIC PERIOD FUNCTION 9
Moreover, Q n ( z ) = ( z + 1)( z − D n ( z ) z n +1 , n ≥ , where D n ( z ) are polynomials with rational coefficients ( Q p integers for p = 2 ) of degree n − with the reciprocity property D n ( z ) = z n − D n (cid:16) z (cid:17) . Note the order of summation in the series for m L , since the reason for introducing exponentialfunction is because we use Borel summation. For example,“1 − − −
32 + ... ” Borel = ∞ X r =0 (cid:16) ∞ X n =0 ( − n n ! · r +1 Z r t n e − t d t (cid:17) = 13 . The following table gives initial results. n D n ( z ) n D n ( z )1 (2 z − z + 6 z − z + 6 z − z + 2)2 ( z + 1) 5 ( z − z + 4 z − z + 4 z − z + 4 z − z + 1)3 ( z − z + z − z + 1) 6 (2 z − z + 12 z − z + 37 z −− z + 37 z − z + 12 z − z + 2)The next table gives Q ′ n ( −
1) = 2( − n D n ( −
1) explicitly: these constants appear in the seriesdefining the first non-trivial moment m . Also, since these numbers are p − adic integers for p = 2, there is a hope for the successful implementation of the idea from the last chapter in [2];that is, possibly one can define moments m L as p − adic rationals as well. n Q ′ n ( − n Q ′ n ( − n Q ′ n ( − n Q ′ n ( − − − − − − − − −
12 30365 20 −
16 13 − − − − − − − The final table in this section lists float values of the constants ϑ r = ∞ X n =0 Q ′ n ( − n ! · r +1 Z r t n e − t d t, r ∈ N , ∞ X r =0 ϑ r = m , appearing in Borel summation. r ϑ r r ϑ r . . . . . . . . − . . . . P r =0 ϑ r = 0 . + = m + 0 . + .This paper is organized as follows. In Section 2, for each p , 1 ≤ p < ∞ , we introducea generalization of the Farey (Calkin-Wilf) tree, denoted by Q p . This leads to the notion of p − continued fractions and p − Minkowski question mark functions F p ( x ). Though p − continuedfractions are of independent interest (one could define a transfer operator, to prove an analogueof Gauss-Kuzmin-L´evy theorem, various metric results and introduce structural constants), weconfine to the facts which are necessary for our purposes and leave the deeper research for thefuture. In Section 3 we extend these results to the case of complex p , | p − | ≤
1. The crucialconsequence of these results is the fact that a function X ( p , x ) (which gives a bijection betweentrees Q and Q p ) is a continuous function in x and an analytic function in p for | p − | ≤
1. InSection 4 we introduce exactly the same integral transforms of F p ( x ) as was done in a special(though most important) case of F ( x ) = F ( x ). Also, in this section we prove certain relationsamong the moments. In Section 5 we give the proof of the three term functional equation for G p ( z ) and the integral equation for m p ( t ). Finally, Theorem 1.2 is proved in Section 6. Thehierarchy of sections is linear, and all results from previous ones is used in Section 6. AppendixA. contains: derivation for the series (7); MAPLE codes to compute rational functions H n ( z )and Q n ( z ); description of high-precision method to calculate numerical values for the constants m L ; auxiliary lemmas for the Section 3. The paper also contains graphs of some p − Minkowskiquestion mark functions F p ( x ) for real p , and also pictures of locus points of elements of trees Q p for certain characteristic values of p .2. p − question mark functions and p − continued fractions In this section we introduce a family of natural generalizations of the Minkowski questionmark function F ( x ). Let 1 ≤ p <
2. Consider the following binary tree, which we denote by
XPLICIT SERIES FOR THE DYADIC PERIOD FUNCTION 11 Q p . We start from the root x = 1. Further, each element (“root”) x of this tree generates two“offsprings” by the following rule: x p xx + 1 , x + 1 p . We will use the notation T p ( x ) = x +1 p , U p ( x ) = p xx +1 . When p is fixed, we will sometimes discardthe subscript. Thus, the first four generations lead to p hhhhhhhhhhhhhhhhhhh p VVVVVVVVVVVVVVVVVVV p p +2 tttttt p +22 p EEEEE pp +2 yyyyy p +2 p JJJJJJ p p + p +2 tttt p + p +2 p +2 p p +2 p p +2 zzz p +22 p BBB p p +2 ||| p +2 p +2 p DDD p +2 pp + p +2 p + p +2 p KKKK (8)We refer the reader to the paper [11], where authors consider a rather similar construction,though having a different purpose in mind (see also [6]). Denote by T n ( p ) the sequence ofpolynomials, appearing as numerators of fractions of this tree. Thus, T ( p ) = 1, T ( p ) = p , T ( p ) = 2. Directly from the definition of this tree we inherit that T n ( p ) = p T n ( p ) for n ≥ ,T n − ( p ) = T n − ( p ) + p − ǫ T n ( p ) for n ≥ , where ǫ = ǫ ( n ) = 1 if n is a power of two, and ǫ = 0 otherwise. Thus, the definition ofthese polynomials is almost the same as it appeared in [17] (these polynomials were namedStern polynomials by the authors), with the distinction that in [17] everywhere one has ǫ = 0.Naturally, this difference produces different sequence of polynomials.There are 2 n − positive real numbers in each generation of the tree Q p , say a ( n ) k , 1 ≤ k ≤ n − .Moreover, they are all contained in the interval [ p − , p − ]. Indeed, this holds for the initialroot x = 1, and p − ≤ x ≤ p − ⇔ p − ≤ p xx + 1 ≤ , p − ≤ x ≤ p − ⇔ ≤ x + 1 p ≤ p − . This also shows that the left offspring is contained in the interval [ p − , , p − ]. The real numbers appearing in this tree have intrinsic relation with p − continued fractions algorithm. The definition of the latter is as follows. Let x ∈ ( p − , p − ). Consider the following procedure: R p ( x ) = T − ( x ) = p x − , if 1 ≤ x < p − , I ( x ) = x , if p − < x < , STOP , if x = p − . Then each such x can be uniquely represented as p − continued fraction x = [ a , a , a , a , .... ] p , where a i ∈ N for i ≥
1, and a ∈ N ∪ { } . This notation means that in the course of iterations R ∞ p ( x ) we apply T − ( x ) exactly a times, then once I , then we apply T − exactly a times,then I , and so on. The procedure terminates exactly for those x ∈ ( p − , p − ), which are themembers of the tree Q p (“ p -rationals”). Also, direct inspection shows that if procedure doesterminate, the last entry a s ≥
2. Thus, we have the same ambiguity for the last entry exactlyas is the case with ordinary continued fractions. At this point it is straightforward to showthat the n th generation of Q p consists of x = [ a , a , ..., a s ] p such that P sj =0 a j = n , exactly asin the case p = 1 and tree (2).Now, consider a function X p ( x ) with the following property: X p ( x ) = x , where x is a rationalnumber in the Calkin-Wilf tree (2), and x is a corresponding number in the tree (8). In otherwords, X p ( x ) is simply a bijection between these two trees. First, if x < y , then x < y . Also, allpositive rationals appear in the tree (2) and they are everywhere dense in R + . Moreover, T and U both preserve order, and [ p − , p − ) is a disjoint union of T [ p − , p − ) and U [ p − , p − ).Now it is obvious that the function X p ( x ) can be extended to a continuous monotone increasingfunction X p ( ⋆ ) : R + → [ p − , p − (cid:17) , X p ( ∞ ) = 1 p − . Thus, X p (cid:0) [ a , a , a , a ... ] (cid:1) = [ a , a , a , a ... ] p . As can be seen from the definitions of both trees (2) and (8), this function satisfies functionalequations X p ( x + 1) = X p ( x ) + 1 p , X p (cid:16) xx + 1 (cid:17) = p X p ( x ) X p ( x ) + 1 , (9) X p (cid:16) x (cid:17) = 1 X p ( x ) . XPLICIT SERIES FOR THE DYADIC PERIOD FUNCTION 13
The last one (symmetry property) is a consequence of the first two. We are not aware whetherthis notion of p − continued fractions is new or not. For example,1 + √ p p = [1 , , , , , , ... ] p = X p (cid:16) √ (cid:17) , √ , , , , , , , , , , , , , , , , , , , , , ... ] , , , , , , √ . Now fix p , 1 ≤ p <
2. The following proposition follows immediately from the properties of F ( x ). Proposition 2.1.
There exists a limit distribution of the n th generation of the tree Q p as n → ∞ , defined as F p ( x ) = lim n →∞ − n +1 { k : a ( n ) k < x } . This function is continuous, F p ( x ) = 0 for x ≤ p − , F p ( x ) = 1 for x ≥ p − , and it satisfiestwo functional equations: F p ( x ) = ( F p ( p x −
1) + 1 , if ≤ x ≤ p − ,F p ( x p − x ) , if p − ≤ x ≤ . (10) Additionally, F p ( x ) + F p (cid:16) x (cid:17) = 1 for x > . The explicit expression for F p ( x ) is given by F p ([ a , a , a , a , ... ] p ) = 1 − − a + 2 − ( a + a ) − − ( a + a + a ) + .... We will refer to the last functional equation as the symmetry property . As was said, it is aconsequence of the other two, though it is convenient to separate it.
Proof.
Indeed, as it is obvious from the observations above, we simply have F p (cid:0) X p ( x ) (cid:1) = F ( x ) , x ∈ [0 , ∞ ) . Therefore, two functional equations follow from (3) and (9). All the other statements are im-mediate and follow from the properties of F ( x ). (cid:3) Equally important, consider the binary tree (8) for p >
2. In this case analogous propositionholds.
Proposition 2.2.
Let p > . Then there exists a limit distribution of the n th generation as n → ∞ . Denote it by f p ( x ) This function is continuous, f p ( x ) = 0 for x ≤ p − , f p ( x ) = 1 for x ≥ p − , and it satisfies two functional equations: f p ( x ) = ( f p ( p x − if ≤ x ≤ p − ,f p ( x p − x ) + 1 if p − ≤ x ≤ , and f p ( x ) + f p (cid:16) x (cid:17) = 1 for x > . Proof.
The proof is analogous to the one of Proposition 2.1, only this time we use equivalences p − ≤ x ≤ p − ⇔ ≤ p xx + 1 ≤ p − , p − ≤ x ≤ p − ⇔ p − ≤ x + 1 p ≤ p − . (cid:3) For the sake of uniformity, we introduce F p ( x ) = 1 − f p ( x ) for p >
2. Then F p ( x ) satisfiesexactly the same functional equations (3), with a slight difference that F p ( x ) = 1 for x ≤ p − and F p ( x ) = 0 for x ≥ p −
1. Consequently, we will not separate these two cases and all oursubsequent results hold uniformly. To this account it should be noted that, for example, incase p > R p − ⋆ d ⋆ should be understood as − R p − ⋆ d ⋆ . Figure 1 gives graphicimages of typical cases for F p ( x ). 3. Complex case
After dealing the case of real p , 1 ≤ p < ∞ , let us consider a tree (8) when p ∈ C . Forour purpose we will concentrate on the case | p − | ≤
1. It should be noted that the methodwhich we use allows to extend these result to the case ℜ p ≥
1. The question in determiningthe set in the complex plain where similar results are valid remains open. More importantly,the problem to determine all p ∈ C for which there exists an analytic function G p ( z ), whichsatisfied (22), seems to be much harder and interesting. Here and below [0 , ∞ ] stands for acompactification of [0 , ∞ ). In the sequel, the notion of a function f ( z ) to be analytic in theclosed disc | z − | ≤ z = 1, | z − | ≤
1, this function is analytic in a certainsmall neighborhood of z . If z = 1, this means that there exist all higher derivatives, if oneapproaches the point z = 1 while remaining in the disc | z − | ≤ Theorem 3.1.
There exists a unique function X p ( x ) = X ( p , x ) : {| p − | ≤ } × { [0 , ∞ ] } → C ∪ {∞} , having these properties: ( i ) X ( p , x ) satisfies functional equations (9); XPLICIT SERIES FOR THE DYADIC PERIOD FUNCTION 15 p = 1 . x ∈ [0 . , p = 3, x ∈ [0 . , p = 10, x ∈ [0 . , p = 25, x ∈ [0 , Figure 1.
Functions F p ( x ).( ii ) For fixed p = 1 , X ( p , x ) : [0 , ∞ ] → C is a continuous function, and the image (denote itby I p ) is thus a bounded curve; it is contained in the domain { C \ {| z + 1 | ≤ } ; ( iii ) For every p , | p − | ≤ , p = 1 , in some neighborhood of p there exists the derivative ∂∂ p X ( p , x ) , which is a continuous function for x ∈ [0 , ∞ ] ; ( iv ) There exist all derivatives S N ( x ) = ∂ N ∂ p N X ( p , x ) | p =1 : [0 , ∞ ) → R (the derivatives are takeninside | p − | ≤ ). These functions are uniformly continuous for irrational x in any finiteinterval. Moreover, S N ( x ) ≪ N x N +1 for x ≥ , and S N ( x ) ≪ N for x ∈ (0 , . The curve I p has a natural fractal structure: it decomposes into two parts, namely I p +1 p and p I p I p +1 , with a common point z = 1. Additionally, I p = I p . As a consequence, 0 / ∈ I p for p = 1. Figures 2-4 show the images of I p for certain characteristic values of p .The investigations of the tree Q p deserve a separate paper. I am very grateful to my col-leagues Jeffrey Lagarias and Stefano Isola, who sent me various references, also informing aboutthe intrinsic relations of this problem with: Julia sets of rational maps of the Riemann sphere;iterated function systems; forward limit sets of semigroups; various topics from complex dynam-ics and geometry of discrete groups. Thus, the problem is much more subtle and involved thanit appears to be. This poses a difficult question on the limit set of the semigroup generated bytransformations U p and T p , or any other two “conjugate” analytic maps of the Riemann sphere(say, two analytic maps A and B are “conjugate”, if A ( α ) = α , B ( β ) = β , A ( β ) = B ( α ) forsome two points α and β on the Riemann sphere). Possibly, certain techniques from complexdynamics do apply here. As pointed out by Curtis McMullen, the property of boundednessof I p can be reformulated in a coordinate-free manner. It appears that this curve consists ofthe closure of the attracting fixed points of the elements of the semigroup hT p , U p i . Then theproperty for the curve being bounded and being bounded away from z = 0 means that it doesnot contain a repelling fixed point of T p ( z = ∞ ) and a repelling fixed point of U p ( z = 0).It also does not contain neither of the repelling fixed points of the elements of this semigroup.Note that T (1) = U (1) = 1, T ′ (1) = U ′ (1) = 1 /
2. Thus, there exists a small ball D around z = 1, such that T ( D ) ⊂ D , U ( D ) ⊂ D , and the last two maps are contractions in D . Thisstrict containment is an open condition on p , and thus there exists a neighborhood of p = 2such that Theorem 3.1 does hold. I am grateful to Curtis McMullen for this remark: we getthe result almost for free. Yet, the full result for | p − | ≤ z sz ±
1, forfixed s , | s | <
1, and investigates a closure of a set of all attracting fixed points. For example,for | s | > − / this set is connected. Further development of this problem can be seen in [32].On the other hand, the case of one rational map is rather well understood, and it is treated in[4]. Thus, though the machinery of complex dynamics can greatly clarify our understanding ofthe structure of the curve I p , we will rather employ the techniques from the analytic theoryof continued fractions. The main source is the monograph by H.S. Wall [34]. (Lemmas A.1,A.2 and A.3 can be found in the Appendix A.2.) Proof of Theorem 3.1 . We need the following two results.
Theorem 3.2. ( [34] p. 57.) Let v ν , ν ∈ N be positive numbers such that v < , v ν + v ν +1 ≤ , for ν ≥ . (11) Suppose given complex numbers e ν , ν ∈ N , such that | e ν +1 | − ℜ ( e ν +1 ) ≤ v ν , ν ≥ . (12) XPLICIT SERIES FOR THE DYADIC PERIOD FUNCTION 17
Define the sequence b ν by the recurrence b = 1 , e ν +1 = b ν b ν +1 , ν ≥ . Then the continuedfraction F = 11 + e e e . . . (13) converges if, and only if, ( a ) some e ν vanishes, or ( b ) e ν = 0 for ν ≥ and the series P ∞ ν =1 | b ν | diverges. Moreover, if e ν ( z ) : K → K are analytic functions of a complex variable, K and K are compact sets, (11) and (12) are satisfied, and the above series diverges uniformly, thenthe continued fraction converges uniformly for all z ∈ K . Theorem 3.3. ( [34] , p. 60.) If all v ν = , and the conditions ( a ) and ( b ) of Theorem 3.2 hold,then |F − | ≤ , F 6 = 0 . For a, b ∈ N , p ∈ C , | p − | ≤
1, define rational functions W a ( p ) = p a − p a +1 − p a ,T a,b ( p ) = W − a ( p ) W − b ( p ) p − a = ( p − p b ( p a − p b − , T a, ∞ ( p ) = ( p − ( p a − . Since, for fixed p = 1, W a ( p ) → p −
1, as a → ∞ , then there exist two constants k = k ( p )and k = k ( p ), such that0 < k ≤ | W a ( p ) | ≤ k < + ∞ , a ∈ N . (14)Let x ≥ x = [ a , a , a , ... ], be an irrational number, a i ∈ N . Let us consider the continuedfraction F ( p , x ) = F ( p , a , a , ... ) = 11 + T a ,a ( p )1 + T a ,a ( p )1 + T a ,a ( p ). . . . (15) If x = [ a , a , ..., a κ ] ≥ F ( p , x ) = F ( p , a , a , ..., a κ ) = 11 + T a ,a ( p )1 + T a ,a ( p )1 + . . .1 + T a κ , ∞ . From the definition, this continued fraction obeys the following rule F ( p , a , a , ... ) = 11 + T a ,a ( p ) · F ( p , a , a ... ) . We will now apply Theorem 3.2 to F ( p , a , a , a , ... ). Suppose x is irrational. Thus, let e ν = T a ν − ,a ν ( p ), ν ≥
2. Let us define constants µ ( a, b ) = sup p ∈ C , | p − |≤ | T a,b ( p ) | − ℜ ( T a,b ( p )) . By Lemma A.1, µ ( a, b ) + µ ( b, c ) < . a, b, c ∈ N . Further, from the definition in Theorem3.2 it follows that b ν = W a ( p ) W a ν ( p ) p a ν − − ... + a − a + a ,b ν +1 = W − a ( p ) W a ν +1 ( p ) p a ν − ... − a + a − a . (16)It is obvious that the series P ∞ ν =1 | b ν | diverges. Hence, Theorem 3.2 tells that the continuedfraction converges, and that for fixed irrational x = [ a , a , ... ] > F ( p , a , a , ... ) is ananalytic function in p in some small neighborhood of p . For rational x this is in fact arational function.As it is shown in [34], the ν th convergent of the continued fraction (13) (denote it by A ν B ν ) isequal to the ν th convergent (denote it by P ν Q ν ) of the continued fraction1 b + 1 b + 1 b + 1. . . . Moreover, since (11) and (12) are satisfied, we have that, for certain positive constant k = k ( b , b , b ) ([34], p.55-56), | Q ν | ≥ k (1 + | b | + | b | + ..., + | b ν | ) , | Q ν +1 | ≥ k (1 + | b | + | b | + ..., + | b ν +1 | ) , (17) (cid:12)(cid:12)(cid:12) A ν +1 B ν +1 − A ν B ν (cid:12)(cid:12)(cid:12) = 1 | Q ν Q ν +1 | . XPLICIT SERIES FOR THE DYADIC PERIOD FUNCTION 19
Now we have
Proposition 3.4.
Fix p ∈ C , | p − | ≤ , p = 1 . Let x = [ a , a , ... ] ≥ be a real number.The function F ( p , x ) : [1 , ∞ ] → C is continuous.Proof. Fix irrational x >
1. Let δ >
0, and y ≥ | x − y | < δ . Then there exists N such that the first N partial quotients of x and y coincide, N = N ( δ ) → ∞ as δ → F ( p , x ) and F ( p , y ) be respectively A B , A B , ..., A N B N , A N +1 B N +1 , A N +2 B N +2 , ... ; and A B , A B , ..., A N B N , A ′ N +1 B ′ N +1 , A ′ N +2 B ′ N +2 ... Now, combining (14), (16) and (17) we see that | Q ν Q ν +1 | > k k k − × (cid:16) | p | a + | p | a − a + a + ... + | p | a ν − − ... + a − a + a (cid:17) × (cid:16) | p | a − a + | p | a − a + a − a + ... + | p | a ν − ... − a + a − a (cid:17) . Denote c = k k k − . Let | p | a ℓ − − ... + a − a + a = λ ℓ , 1 ≤ ℓ ≤ ν . The above inequality and thearithmetic-harmonic mean inequality give | Q ν Q ν +1 | > c ( λ + λ + ... + λ ν ) · ( | p | a λ − + | p | a λ − + ... + | p | a ν λ − ν ) ≥ | p | c ( λ + λ + ... + λ ν ) · ( λ − + λ − + ... + λ − ν ) ≥ | p | c ν ν ≥ . Analogously we prove that | Q ν − Q ν | > | p | c ν , ν ≥
2. Thus, | Q ν Q ν +1 | > cν for certain real c > ν ≥
2. We see that (17) yield (cid:12)(cid:12)(cid:12) F ( p , x ) − A N B N (cid:12)(cid:12)(cid:12) < ∞ X ν = N | Q ν Q ν +1 | ≤ ∞ X ν = N c − ν < c − N − (cid:12)(cid:12)(cid:12) F ( p , y ) − A N B N (cid:12)(cid:12)(cid:12) < c − N − . This implies |F ( p , x ) − F ( p , y ) | < c − N − . In case x is rational we argue in a similar way. In thiscase note that real numbers close to x = [ a , a , ..., a κ ] are of the form or [ a , a , ..., a κ , T, ... ],either [ a , a , ..., a κ − , , T, ... ] for T sufficiently large. The case x = ∞ is analogous. Thisestablishes the validity of the Proposition. (cid:3) Eventually, for real number x ≥ x = [ a , a , a , ... ], let us define X ( p , [ a , a , ... ]) = W a ( p ) + p − a W a ( p ) + p − a W a ( p ) + p − a W a ( p ) + . . . . After an equivalence transformation ([34], p.19), this can be given an expression X ( p , [ a , a , ... ]) = W a ( p ) + p − a W − a ( p ) · F ( p , a , a , a , ... ) . (18)From the very construction, this function satisfies the functional equations (9), is continuousat x = 1 and thus is continuous for x ∈ [0 , ∞ ]. Obviously, (9) determine the values of X ( p , x )at rational x uniquely, hence a continuous solution to (9) is unique. We are left to show thatthe image of the curve I p is contained outside the circle | z + 1 | ≤ . This is equivalent to thestatement that p I p I p +1 is contained inside the circle | z − p | ≤ p . But the points on p I p I p +1 areexactly the point on the curve I p with a = 0. Thus, we need to show that | p − X ( p , [0 , a , a , ... ]) − | = | p − W − a F ( p , a , a , ... ) − | ≤ . (19)Unfortunately, we cannot apply Theorem 3.3 directly to all p , | p − | ≤
1, since the table aboveLemma A.1 shows that µ (1 , b ) > for infinitely many b . The maximum values µ (1 , b ) (see thedefinition of this constant) are produced by points p close to χ = 2 + e πi/ , or to χ . For thisreason let us introduce µ ⋆ ( a, b ) = sup p ∈ C , | p − |≤ , | p − χ |≥ . , | p − χ |≥ . | T a,b ( p ) | − ℜ ( T a,b ( p )) . Then indeed µ ⋆ ( a, b ) < for all a, b ∈ N . Thus, Theorem 3.3 gives |F ( p , a , a , ... ) − | ≤ | p − | ≤ | p − χ | < .
19 (or | p − χ | < .
19) we use another theorem by Wall ([34], p. 152), which describes the valueregion of a continued fraction (13), provided elements e ν belong to the compact domain in theparabolic region | z | − ℜ ( ze iφ ) ≤ h cos φ , for certain fixed − π < φ < + π , 0 < h ≤ . Weomit the details. This proves part ( ii ). In a similar fashion we prove part ( iii ). Finally, adirect inspection shows that slightly modified proofs remain valid in case p = 1, if we define afunction to be analytic at p = 1, if it possesses all higher p − derivatives, while remaining insidethe disc | p − | ≤ (cid:3) Definition 3.5.
We define Minkowski p − question mark function F p ( x ) : I p → [0 , , by F p ( X ( p , x )) = F ( x ) , x ∈ [0 , ∞ ] . Properties of integral transforms of F p ( x )For given p , | p − | ≤
1, we define χ n = p + p n − − p n − ( p − , I n = [ χ n , χ n +1 ] = X ( p , [ n, n + 1]) for n ∈ N . Complex numbers χ n stand for the analogue of non-negative integers on the curve I p . Inother words, χ n = U n ( p − I n as part of the curve I p contained betweenthe points χ n and χ n +1 . Thus, χ = p − χ = 1, and the sequence χ n is “increasing”, in XPLICIT SERIES FOR THE DYADIC PERIOD FUNCTION 21 the sense that χ j as a point on a curve I p is between χ i and χ k if i < j < k . Moreover, ∞ S n =0 I n S { p − } = I p . Proposition 4.1.
Let ω ( x ) : I p → C be a continuous function. Then Z I p ω ( x ) d F p ( x ) = ∞ X n =0 n +1 Z I p ω (cid:16) x p n − ( x + 1) + p n − p n +1 − p n (cid:17) d F p ( x ) . Proof.
Indeed, using (10) we obtain Z I p ω ( x ) d F p ( x ) = ∞ X n =0 Z I n ω ( x ) d F p ( x ) = ∞ X n =0 Z T n ( I ) ω ( x ) d F p ( x ) x →T n x = ∞ X n =0 n Z I ω ( T n x ) d F p ( x ) x →U x = ∞ X n =0 n +1 Z I p ω ( T n U x ) d F p ( x ) , and this is exactly the statement of the Proposition. (cid:3) For
L, T ∈ N let us introduce B L,T ( p ) = ∞ X n =0 n +1 p T n (cid:16) p n − p n +1 − p n (cid:17) L . For example, B ,T = p T p T − , B ,T ( p ) = p T (2 p T − p T +1 − ,B ,T ( p ) = p T (2 p T +1 + 1)(2 p T +2 − p T +1 − p T − ,B ,T ( p ) = p T (4 p T +3 + 4 p T +2 + 4 p T +1 + 1)(2 p T +3 − p T +2 − p T +1 − p T − ,B ,T ( p ) = p T (2 p T +2 + 1)(4 p T +4 + 6 p T +3 + 8 p T +2 + 6 p T +1 + 1)(2 p T +4 − p T +3 − p T +2 − p T +1 − p T − . As it is easy to see, B L,T ( p ) are rational functions in p for L, T ∈ N . Indeed, B L,T ( p ) = 1( p − L · ∞ X n =0 p T n n +1 (cid:16) − p n (cid:17) L = 12( p − L · L X s =0 ( − s (cid:18) Ls (cid:19) ∞ X n =0 n p n ( s + T ) = p T ( p − L · L X s =0 ( − s (cid:18) Ls (cid:19) p s p s + T − p T R L,T ( p )(2 p T + L − p T + L − − · ... · (2 p T +1 − p T − , where R L,T ( p ) are polynomials. This follows from the observation that p = 1 is a root ofnumerator of multiplicity not less than L . As in case p = 1, our main concern are the moments of distributions F p ( x ), which are definedby m L ( p ) = 2 Z I x L d F p ( x ) = Z I p (cid:16) p xx + 1 (cid:17) L d F p ( x )= 2 Z X L ( p , x ) d F ( x ) = lim n →∞ − n X a + a + ... + a s = n [0 , a , a , .., a s ] L p .,M L ( p ) = Z I p x L d F p ( x ) . Thus, if sup z ∈ I p | z | = ρ p >
1, which is finite for ℜ p ≥ p = 1 (see Section 3), then M L ( p ) ≤ ρ L p . Proposition 4.2.
The function m L ( p ) is analytic in the disc | p − | ≤ , including its boundary.In particular, if in this disc b m L ( p ) := m L ( p ) p L = ∞ X v =0 η v,L ( p − v , then for any M ∈ N , one has the estimate η v,L ≪ v − M as v → ∞ .Proof. The function X ( p , x ) possesses a derivative in p for ℜ p ≥ | p − | ≤
1, and these arebounded and continuous functions for x ∈ R + . Therefore m L ( p ) has a derivative. For p = 1,there exists d M d p M X ( p , x ) ≪ x M +1 , and it is a continuous function for irrational x . Additionally, F ′ ( x ) = 0 for x ∈ Q + . This proves the analyticity of m L ( p ) in the disc | p − | ≤
1. Then anestimate for the Taylor coefficients is the standard fact from Fourier analysis. In fact, η v,L = Z b m L (2 + e πiϑ ) e − πivϑ d ϑ. The function b m L (2 + e πiϑ ) ∈ C ∞ ( R ), hence the iteration of integration by parts implies theneeded estimate. (cid:3) Proposition 4.3.
Functions M L ( p ) and m L ( p ) are related via rational functions B L,T ( p ) inthe following way: M L ( p ) = L X s =0 m s ( p ) B L − s,s ( p ) (cid:18) Ls (cid:19) . Proof.
Indeed, this follows from the definitions and Proposition 4.1 in case ω ( x ) = x L . (cid:3) XPLICIT SERIES FOR THE DYADIC PERIOD FUNCTION 23
Let us introduce, following [1] in case p = 1, the following generating functions: m p ( t ) = ∞ X L =0 m L ( p ) t L L ! = 2 Z I e xt d F p ( x ) = Z I p exp (cid:16) p xtx + 1 (cid:17) d F p ( x ); G p ( z ) = ∞ X L =1 m L ( p ) p L z L − = Z I p x + 1 − z d F p ( x ) = ∞ Z X ( p , x ) + 1 − z d F ( x ) . (20)The situation p = 2 is particularly important, since all these functions can be explicitlycalculated, and it provides the case where all the subsequent results can be checked directlyand the starting point in proving Theorem 1.2. Thus, m ( t ) = e t , G ( z ) = 12 − z . By the definition, expressions m L ( p ) / p L are Taylor coefficients of G p ( z ) at z = 0. Differen-tiation of L − G p ( z ), and substitution z = 1 gives G ( L − p (1) = ( L − Z I p x L d F p ( x ) = ( L − M L ( p ) ⇒ G p ( z + 1) = ∞ X L =0 M L ( p ) z L − , (21)with a radius of convergence equal to ρ − p . As was proved in [1] and mentioned before, in case p = 1 ( ρ = ∞ ) this must be interpreted that there exist all derivatives at z = 1. The nextProposition shows how symmetry property reflects in m p ( t ). Proposition 4.4.
One has m p ( t ) = e p t m p ( − t ) . Proof.
Indeed, m p ( t ) = Z I p exp (cid:16) p xtx + 1 (cid:17) d F p ( x ) = Z I p exp (cid:16) p t − p tx + 1 (cid:17) d F p ( x ) = e p t Z I p exp (cid:16) − p tx + 1 (cid:17) d F p ( x ) x → x = e p t m p ( − t ) . (cid:3) This result allows to obtain linear relations among moments m L ( p ) and the exact value of thefirst (trivial) moment m ( p ). Corollary 4.5.
One has m ( p ) = p , M ( p ) = p + 24 p − . Proof.
Indeed, the last propositions implies m L ( p ) = L X s =0 (cid:18) Ls (cid:19) ( − s m s ( p ) p L − s , L ≥ . For L = 1 this gives the first statement of the Corollary. Additionally, Proposition 4.3 for L = 1reads as M ( p ) = p p − · m ( p ) + 12 p − , and we are done. (cid:3) Three term functional equation
Theorem 5.1.
The function G p ( z ) can be extended to analytic function in the domain C \ ( I p + 1) . It satisfies the functional equation z + p z G p (cid:16) p z (cid:17) + 2 G p ( z + 1) = p G p ( p z ) , for z / ∈ I p + 1 p . (22) Its consequence is the symmetry property G p ( z + 1) = − z G p (cid:16) z + 1 (cid:17) − z . Moreover, G p ( z ) → if dist ( z, I p ) → ∞ .Conversely - the function satisfying this functional equation and regularity property is unique.Proof. Let w ( x, z ) = x +1 − z . Then it is straightforward to check that w ( x + 1 p , z + 1) = p · w ( x, p z ) ,w ( p x + 1 , z + 1) = − p z w ( x, p z ) − z . Thus, for | p − | ≤ p = 2,2 G p ( z + 1) = 2 Z I w ( x, z + 1) d F p ( x ) + 2 Z I p \ I w ( x, z + 1) d F p ( x )= 2 Z I p w ( p xx + 1 , z + 1) d F p (cid:16) p xx + 1 (cid:17) + 2 Z I p w ( x + 1 p , z + 1) d F p (cid:16) x + 1 p (cid:17) = Z I p w ( p x + 1 , z + 1) d F p ( x ) + Z I p w ( x + 1 p , z + 1) d F p ( x )= − z − p z G p (cid:16) p z (cid:17) + p G p ( p z ) . (In the first integral we used a substitution x → x ). The functional equation holds in case p = 2 as well, which can be checked directly. The holomorphicity of G p ( z ) follows exactly as XPLICIT SERIES FOR THE DYADIC PERIOD FUNCTION 25 in case p = 1 [1]. All we need is the first integral in (20) and the fact that I p is a closed set.As was mentioned, the uniqueness of a function satisfying (22) for p = 1 was proved in [1].Thus, the converse implication follows from analytic continuation principle for the function intwo complex variables ( p , z ) (see Lemma 6.2 below, where the proof in case p = 2 is presented.Similar argument works for general p ). (cid:3) Corollary 5.2.
Let p = 1 , and C be any closed smooth contour which rounds the curve I p + 1 once in the positive direction. Then πi I C G p ( z ) d z = − . Proof.
Indeed, this follows from the functional equation (22), as well as from the symmetryproperty. It is enough to take a sufficiently large circle C = {| z | = R } such that C − + 1 iscontained in a small neighborhood of z = 1, for which ( C − + 1) ∩ ( I p + 1) = ∅ . This ispossible since 0 / ∈ I p (see Theorem 3.1). (cid:3) We finish with providing an integral equation for m p ( t ). We indulge in being concise sincethe argument directly generalizes the one used in [1] to prove the integral functional equationfor m ( t ) (in our notation, this is m ( t )). Proposition 5.3.
Let ≤ p < ∞ be real. Then the function m p ( t ) satisfies the boundarycondition m p (0) = 1 , regularity property m p ( − t ) ≪ e −√ t log 2 , and the integral equation m p ( − s ) = ∞ Z m ′ p ( − t ) (cid:16) e s J (2 √ p st ) − J (2 √ st ) (cid:17) d t, s ∈ R + . For instance, in the case p = 1 this reduces to (4), and in the case p = 2 this reads as2 e s ∞ Z e − t J (2 √ st ) d t = 2 e s e − s = e − s + e − s = e − s + ∞ Z e − t J (2 √ st ) d t, which is an identity [35]. Proof.
Indeed, the functional equation for G p ( z ) in the region ℜ z < − m ′ p ( t ) readsas 1 z = ∞ Z m ′ p ( − t ) (cid:16) z + 1 e p tz +1 + 1 z e tz − z e tz (cid:17) d t. Now, multiply this by e − sz and integrate over ℜ z = − σ < −
1, where s > (cid:3)
Remark. If p = 1, the regularity bound is easier than in case p = 1. Take, for example,1 < p <
2. Then | m p ( t ) | ≤ p − Z p − (cid:12)(cid:12)(cid:12) exp (cid:16) p xtx + 1 (cid:17)(cid:12)(cid:12)(cid:12) d F p ( x ) < p − Z p − e t d F p ( x ) = e t . Thus, Proposition 4.4 gives | m p ( − t ) | < e (1 − p ) t . The same argument shows that for p > | m p ( − t ) | < e − t . 6. The proof: approach through p = 2Let us rewrite the functional equation for G p ( z ) = G ( p , z ) as1 z + p z G (cid:16) p , p z (cid:17) + 2 G ( p , z + 1) = p G ( p , p z ) . (23)Direct induction shows that the following “chain-rule” holds: ∂ n ∂ p n (cid:16) p G ( p , p z ) (cid:17) = X i + j = n (cid:18) nj (cid:19) p ∂ i ∂ j ∂ p i ∂z j G ( p , p z ) z j + X i + j = n − n (cid:18) n − j (cid:19) ∂ i ∂ j ∂ p i ∂z j G ( p , p z ) z j , (24)where in the summation it is assumed that i, j ≥ G ( p , z ) in terms ofpowers of ( p −
2) and certain rational functions. The function G ( p , z ) is analytic in {| p − | ≤ } × {| z | ≤ } . This follows from Theorem 3.1 and integral representation (20). Thus, for {| p − | < } × {| z | ≤ } it has a Taylor expansion G ( p , z ) = ∞ X L =1 ∞ X v =0 η v,L · z L − ( p − v . (25)Moreover, the function G (2 + e πiϑ , e πiϕ ) ∈ C ∞ ( R × R ), and it is double-periodic. Thus, η v,L = (cid:16) (cid:17) L − Z Z G (2 + e πiϑ , e πiϕ ) e − πivϑ − πi ( L − ϕ d ϑ d ϕ, v ≥ , L ≥ . A standard trick from Fourier analysis (using iteration of integration by parts) shows that η v,L ≪ M (4 / L · ( Lv ) − M for any M ∈ N . Thus, (25) holds for ( p , z ) ∈ {| p − | ≤ } × {| z | ≤ / } .Our idea is a simple one. Indeed, let us look at (20). This implies the Taylor series for m L ( p ) / p L = P ∞ v =0 η v,L ( p − v , convergent in the disc | p − | ≤
1. Due to the absolute
XPLICIT SERIES FOR THE DYADIC PERIOD FUNCTION 27 convergence, the order of summation in (25) is not essential. This yields G ( p , z ) = ∞ X v =0 ( p − v (cid:16) ∞ X L =1 η v,L · z L − (cid:17) . Therefore, let 1 n ! ∂ n ∂ p n G ( p , z ) (cid:12)(cid:12)(cid:12) p =2 = H n ( z ) = ∞ X L =1 η n,L · z L − . We already know that H ( z ) = − z . Though m L ( p ) are obviously highly transcendental (andmysterious) functions, the series for H n ( z ) is in fact a rational function in z , and this is themain point of our approach. Moreover, we will show that H n ( z ) = B n ( z )( z − n +1 , where B n ( z ) is a polynomial with rational coefficients of degree n − B n ( z + 1) = ( − n z n − B n ( z + 1), B n (0) = 0. We argue by induction on n . First weneed an auxiliary lemma.Let Q [ z ] n − denote the linear space of dimension n of polynomials of degree ≤ n − L n − : Q [ z ] n − → Q [ z ] n − , defined by L n − ( P )( z ) = P ( z + 1) − n +1 P (2 z ) + ( − n +1 n +1 P (cid:16) z (cid:17) z n − . Lemma 6.1. det ( L n − ) = 0 . Accordingly, L n − is the isomorphism.Remark. Let m = (cid:2) n (cid:3) . Then it can be proved that indeed det( L n − ) = Q mi =1 (4 i − m m . Proof.
Suppose P ∈ ker( L n − ). Then a rational function H ( z ) = P ( z )( z − n +1 satisfies the threeterm functional equation H ( z + 1) − H (2 z ) + H (cid:16) z (cid:17) z = 0 , z = 1 . (26)Also, H ( z ) = o (1), as z → ∞ . Now the result follows from the next Lemma 6.2.
Let Υ( z ) be any analytic function in the domain C \ { } . Then if H ( z ) is asolution of the equation H ( z + 1) − H (2 z ) + H (cid:16) z (cid:17) z = Υ( z ) , such that H ( z ) → as z → ∞ , H ( z ) is analytic in C \ { } , then such H ( z ) is unique. Proof.
All we need is to show that with the imposed diminishing condition, homogeneousequation (26) admits only the solution H ( z ) ≡
0. Indeed, let H ( z ) be such a solution. Put z → n z + 1. Thus, H (2 n z + 2) − H (2 n +1 z + 2) + 1(2 n z + 1) H (cid:16) n z + 1 (cid:17) = 0 . This is valid for z = 0 (since H ( z ) is allowed to have a singularity at z = 2). Now sum this over n ≥
0. Due to the diminishing assumption, one gets (after additional substitution z → z − H ( z ) = − ∞ X n =0 n z − n +1 + 1) H (cid:16) n z − n +1 + 1 (cid:17) . For clarity, put z → − z and consider a function b H ( z ) = H ( − z ). Thus, b H ( z ) = − ∞ X n =0 n z + 2 n +1 − b H (cid:16) n z + 2 n +1 − (cid:17) . Consider this for z ∈ [0 , b H ( z ) = ∞ X n,m =0 n + m +1 z + 2 n + m +2 − n z − n +1 + 1) b H (cid:16) ω m ◦ ω n ( z ) (cid:17) , where ω n ( z ) = n z +2 n +1 − . As said, ω m ◦ ω n ( z ) ∈ [0 ,
2] for z ∈ [0 , b H ( z )is continuous in the interval [0 , z ∈ [0 ,
2] be such that M = | b H ( z ) | = sup z ∈ [0 , | b H ( z ) | .Consider the above expression for z = z . Thus, M = | b H ( z ) | ≤ ∞ X n,m =0 (cid:12)(cid:12)(cid:12) n + m +1 z + 2 n + m +2 − n z − n +1 + 1) b H (cid:16) ω m ◦ ω n ( z ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ M ∞ X n,m =0 n + m +2 − n +1 + 1) = 0 . + M. This is contradictory unless M = 0. By the principle of analytic continuation, H ( z ) ≡
0, andthis proves the Lemma. (cid:3)
Remark.
Direct inspection of the proof reveals that the statement of Lemma still holds witha weaker assumption that H ( z ) is real-analytic function on ( −∞ , XPLICIT SERIES FOR THE DYADIC PERIOD FUNCTION 29
Now, let us differentiate (23) n times with respect to p , use (24) and afterwards substitute p = 2. This gives n X j =1 j ! ∂ j ∂z j H n − j (2 z ) z j + n − X j =0 j ! ∂ j ∂z j H n − j − (2 z ) z j − n X j =1 j ! ∂ j ∂z j H n − j (cid:16) z (cid:17) z j +2 − n − X j =0 j ! ∂ j ∂z j H n − j − (cid:16) z (cid:17) z j +2 =2 H n ( z + 1) − H n (2 z ) + 2 H n (cid:16) z (cid:17) z . (27)We note that this implies the reciprocity property H n ( z + 1) = − z H n (cid:16) z + 1 (cid:17) , n ≥ . A posteriori , this clarifies how the identity F ( x ) + F (1 /x ) = 1 reflects in the series for G ( z ),as stated in Theorem 1.2: reciprocity property (non-homogeneous for n = 0 and homogeneousfor n ≥
1) is reflected in each of the summands separately, whereas the three term functionalequation heavily depends on inter-relations among H n ( z ).Now, suppose we know all H j ( z ) for j < n . Lemma 6.3.
The left hand side of the equation (27) is of the forml.h.s. = J n ( z )( z − n +1 , where J n ( z ) ∈ Q [ z ] n − .Proof. First, as it is clear from the appearance of l.h.s., we need to verify that z does notdivide a denominator, if l.h.s. is represented as a quotient of two co-prime polynomials. Indeed,using the symmetry property in (23) for the term G ( p , p z ), we obtain the three term functionalequation of the form − p − z − p ( p − z ) G (cid:16) p , pp − z (cid:17) + 2 G ( p , z + 1) = p G ( p , p z ) . Let us perform the same procedure which we followed to arrive at the equation (27). Thus,differentiation n times with respect to p and substitution p = 2 gives the expression of theform l.h.s. = 2 H n ( z + 1) − H n (2 z ) − H n (cid:16) − z (cid:17) − z ) , where lh.s. is expressed in terms of H j ( z ) for j < n . Nevertheless, this time the commondenominator of l.h.s. is of the form ( z − n +1 ( z − n +2 . As a corollary, z does not divide it. Finally, due to the reciprocity property, for n ≥ H n (cid:16) − z (cid:17) − z ) = − H n (cid:16) z (cid:17) z . This shows that actually l.h.s. = l.h.s. , and therefore if this is expressed as a quotient of twopolynomials in lowest terms, the denominator is a power of ( z − n + 1, and one easily verifies that deg J n ( z ) ≤ n −
1. (Possibly, J n ( z )can be divisible by ( z − (cid:3) Proof of Theorem 1.2 . Now, using Lemma 6.1, we inherit that there exists a uniquepolynomial B n ( z ) of degree ≤ n − B n ( z ) = L − n − ( J n )( z ). Summarizing, H n ( z ) = B n ( z )( z − n +1 solves the equation (27). On the other hand, Lemma 6.2 implies that the solution of(27) we obtained is indeed the unique one. This reasoning proves that for | p − | ≤ | z | ≤ G ( p , z ) = ∞ X n =0 ( p − n · H n ( z ) . This finally establishes the validity of Theorem 1.2. Note also that each summand satisfies thesymmetry property. The series converges absolutely for any z , | z | ≤ /
4, and if this holds for z , the same does hold for zz − , which gives the circle | z + 9 / | ≤ / (cid:3) Curiously, one could formally verify that the function defined by this series does indeedsatisfy (22). Indeed, using (27), we get:2 G ( p , z + 1) = 2 H ( z + 1) + 2 ∞ X n =1 ( p − n H n ( z + 1) =2 H ( z + 1) + ∞ X n =1 ( p − n n X j =0 j ! ∂ j ∂z j H n − j (2 z ) z j + n − X j =0 j ! ∂ j ∂z j H n − j − (2 z ) z j − n X j =0 j ! ∂ j ∂z j H n − j (cid:16) z (cid:17) z j +2 − n − X j =0 j ! ∂ j ∂z j H n − j − (cid:16) z (cid:17) z j +2 ! Denote n − j = s . Then interchanging the order of summation for the first term of the sum inthe brackets, we obtain:2 ∞ X n =1 ( p − n n X j =0 j ! ∂ j ∂z j H n − j (2 z ) z j = 2 ∞ X s =0 ∞ X j =0 ( p − j + s j ! ∂ j ∂z j H s (2 z ) z j − H (2 z ) =2 ∞ X s =0 ( p − s H s (2 z + ( p − z ) − H (2 z ) = 2 G ( p , p z ) − H (2 z ) . XPLICIT SERIES FOR THE DYADIC PERIOD FUNCTION 31
The same works for the second sum: ∞ X n =1 ( p − n n − X j =0 j ! ∂ j ∂z j H n − j − (2 z ) z j = ( p − G ( p , p z ) . We perform the same interchange of summation for the second and the third summand respec-tively. Thus, this yields2 G ( p , z + 1) = p G ( p , p z ) − p z G (cid:16) p , p z (cid:17) + 2 H ( z + 1) − H (2 z ) + 2 z H (cid:16) z (cid:17) = p G ( p , p z ) − p z G (cid:16) p , p z (cid:17) − z . On the other hand, it is unclear how one can make this argument to work. This would requirerather detailed investigation of the linear map L n − and recurrence (27), and this seems to bevery technical. Appendix
A.A.1.
Approach through p = 0 . With a slight abuse of notation, we will use the expression ∂ s ∂ p s G (0 , z ) to denote ∂ s ∂ p s G ( p , z ) (cid:12)(cid:12) p =0 for s ∈ N . Though the function G ( p , z ) is defined onlyfor ℜ p ≥ z / ∈ ( I p + 1), assume that we are able to prove that it is analytic in p in a certainwider domain containing a disc | p | < ̟ , ̟ >
0. These are only formal calculations, but theyunexpectedly yield series (7) (see Section 1), and numerical calculations do strongly confirmthe validity of it.Thus, substitution p = 0 into (23) gives G (0 , z ) = − z ) . Partial differentiation of (23) withrespect to p , and consequent substitution p = 0 gives1 z G (0 ,
0) + 2 ∂∂ p G (0 , z + 1) = G (0 , ⇒ ∂∂ p G (0 , z ) = ( z − − z − . In the same fashion, differentiating the second time, we obtain ∂ ∂ p G (0 , z ) = ( z − − z − . Ingeneral, differentiating (23) n ≥ p , using (24), and substituting p = 0,we obtain: 2 ∂ n ∂ p n G (0 , z + 1) = X i + j = n − n (cid:18) n − j (cid:19) ∂ i ∂ j ∂ p i ∂z j G (0 , (cid:16) z j − z j +2 (cid:17) . Let 1 n ! · ∂ n ∂ p n G (0 , z ) = Q n ( z ) . Then 2 Q n ( z + 1) = n − X j =0 j ! ∂ j ∂z j Q n − j − (0) (cid:16) z j − z j +2 (cid:17) . Consequently, we have a recurrent formula to compute rational functions Q ( z ). Let Q n ( z ) = Q n ( z + 1). Thus, Q n ( z ) = ( z + 1)( z − D n ( z ) z n +1 , n ≥ , where D n are polynomials of degree 2 n − D n ( z ) = z n − D n (cid:16) z (cid:17) (this is obvious from the recurrence relation which defines Q n ( z )). Moreover, the coefficientsof D n are Q p integers for any prime p = 2. These calculations yield a following formal result: G ( p , z )“ = ” ∞ X n =0 p n · Q n ( z −
1) = ∞ X n =0 p n z ( z − D n ( z − z − n +1 . This produces the “series” for the second and higher moments of the form m ( p ) = p · ∞ X n =0 p n Q ′ n ( − . In particular, inspection of the table in Section 1 (where the initial values for Q ′ n ( −
1) arelisted) shows that this series for p = 1 does not converge. However, the Borel sum is properlydefined and it converges exactly to the value m . This gives empirical evidence for the validityof (7). The principles of Borel summation also suggest the mysterious fact that indeed G ( p , z )analytically extends to the interval p ∈ [0 , n p | Q ′ n ( − | ismonotonically increasing (apparently, tends to ∞ ), while n log | Q ′ n ( − | − log n monotonicallydecreases (apparently, tends to −∞ ). Thus, A n < | Q ′ n ( − | < ( cn ) n , for c = 0 . A = 3 . n ≥ c = c ( n ) →
0, as n → ∞ , then the functionΛ( t ) = ∞ X n =0 Q ′ n ( − n ! t n is entire, and in case L = 2, result (7) is equivalent to the fact that ∞ Z Λ( t ) e − t d t = m . A.2.
Auxiliary lemmas.
These lemmas are needed in Section 3. For a, b ∈ N , p ∈ C , | p − | ≤
1, define rational functions W a ( p ) = p a − p a +1 − p a , T a,b ( p ) = W − a ( p ) W − b ( p ) p − a = ( p − p b ( p a − p b − . XPLICIT SERIES FOR THE DYADIC PERIOD FUNCTION 33
Let us define constants µ ( a, b ) = sup p ∈ C , | p − |≤ | T a,b ( p ) | − ℜ ( T a,b ( p )) . The following table provides some initial values for constants µ ( a, b ), computed numerically. b \ a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . · · · · · · · · · · · · · · · · · · · · ·∞ . . . . . . b →∞ µ ( a, b ), and µ ( a, b ) → b , as a → ∞ . Thus, thetable above and some standard evaluations give the following Lemma A.1.
Let a, b, c ∈ N . Then µ ( a, b ) + µ ( b, c ) ≤ µ (1 ,
1) + µ (1 , < . . (cid:3) Lemma A.2.
There exists an absolute constant c > such that for all p ∈ C , ℜ p ≥ , andall a ∈ N , on has (cid:12)(cid:12) p a − p − (cid:12)(cid:12) > c .Proof. Consider a contour, consisting of the segment [1 − iT, iT ], and a semicircle 1 + T e iφ , − π ≤ φ ≤ π . For sufficiently large T , p a − p − will be large on the semicircle. Moreover, thisfunction never vanishes inside or on the contour. Thus, from the maximum-minimum principle,its minimal absolute value is obtained on the segment [1 − iT, iT ]. Thus, let p = ψ e iψ , − π < ψ < π . Without loss of generality, let ψ ≥
0. Consider the case π a ≤ ψ < π . Then (cid:12)(cid:12)(cid:12) p a − p − (cid:12)(cid:12)(cid:12) = a ψ − aψ cos a ψ + 1 ψ − ≥ a ψ − a ψ + 1 ψ − ρ a − ρ − Y ( ρ ) , ρ = 1cos ψ . The function Y ( ρ ) is an increasing function in ρ for ρ ≥
1. It is obvious that we may considera case of a sufficiently large. Thus, (cid:12)(cid:12)(cid:12) p a − p − (cid:12)(cid:12)(cid:12) ≥ Y (cid:16) π a (cid:17) = (cid:0) a π a − (cid:1) tan π a = (cid:16) (1 + π a + O (1) a ) a − (cid:17) π a + O (1) a = π a + O (1) a π a + O (1) a = π
16 + O (1) a . Let now 0 ≤ ψ < π a . First, consider a function y log cos( yψ ) := V ( y ). It is a decreasingfunction for 0 < y < π ψ . Indeed, this is equivalent to the inequality − tan x · x − log cos x < , for 0 < x < π . The function on the left is itself a decreasing function, with maximum value attained at x = 0.Thus, V (1) ≥ V ( a ), which means cos aψ ≤ cos a ψ , and this gives (cid:12)(cid:12)(cid:12) p a − p − (cid:12)(cid:12)(cid:12) ≥ a ψ − ψ − ≥ . (cid:3) Therefore, Lemma A.2 implies that the function p − W − a ( p ) is uniformly bounded:sup a ∈ N , | p − |≤ | p − W − a ( p ) | = c < + ∞ . This shows the validity of the following Lemma (apart from a numerical bound, which is theoutcome of computer calculations).
Lemma A.3.
One has sup | p − |≤ ,a ∈ N , | z − |≤ | p − W − a ( p ) z − | < . . (cid:3) A.3.
Numerical values for the moments.
Unfortunately, Corollary 1.4 is not very usefulin finding exact decimal digits of m . In fact, the vector ( m , m , m ... ) is the solution ofan (infinite) system of linear equations, which encodes the functional equation (6) (see [1],Proposition 6). Namely, if we denote c L = P ∞ n =1 12 n n L = Li L ( ), we have a linear system for m s which describes the coefficients m s uniquely: m s = ∞ X L =0 ( − L c L + s (cid:18) L + s − s − (cid:19) m L , s ≥ . XPLICIT SERIES FOR THE DYADIC PERIOD FUNCTION 35
Note that this system is not homogeneous ( m = 1). We truncate this matrix at sufficientlyhigh order to obtain float values. The accuracy of this calculation can be checked on the testvalue m = 0 .
5. This approach yields (for the matrix of order 325): m = 0 . + ,m = 0 . + m = 0 . + . with all 58 digits exact (note that 3 m − m = 0 . m L for 1 ≤ L ≤ m L = p π log 2 · c · L / C √ L + O ( C √ L L − / ) , (28)where c = R x (1 − F ( x )) d x = 1 . + . So obtained numerical values for highermoments tend to deviate from this expression rather quickly.Kinney [16] proved that the Hausdorff dimension of growth points of ?( x ) is equal to α = 12 (cid:16) Z log (1 + x ) d?( x ) (cid:17) − . Lagarias [19] gives the following estimates: 0 . < α < . α ≈ . α ≈ . − x )+?( x ) =1): A := Z log(1 + x ) d?( x ) = Z log (cid:16) − − x (cid:17) d?( x ) + Z log 2 d?( x ) = − ∞ X L =1 L · L Z (1 − x ) L d?( x ) + log 2 = − ∞ X L =1 m L L · L + log 2 . Thus, we are able to present much more precise result: α = log 22 A = 0 . ... with all 36 digits exact. The author of this paper have contacted the authors of [26] inquiringabout the error bound for the numerical value of α they obtained. It appears that for thispurpose 10 generations of (2) were used. The authors of [26] were very kind in agreeing toperform the same calculations with more generations. Thus, if one uses 18 generations, this gives 0 . < α < . c in (28) is given by c = Z x (1 − F ( x )) d x = m (log 2)2 log 2 = 12 ∞ X L =0 m L L ! (log 2) L − . This series is fast convergent, and we obtain c = 1 . ... A.4.
Rational functions H n ( z ) . The following is MAPLE code to compute rational functions H n ( z )= h[n] and coefficients H ′ n (0)= alpha[n] for 0 ≤ n ≤ > restart;> with(LinearAlgebra):> U:=50:> h[0]:=1/(2-z):> for n from 1 to U do> j[n]:=1/2*simplify(> add( unapply(diff(h[n-j],z$j),z)(2*z)*2/j!*(z^(j)),j=1..n)+> add( unapply(diff(h[n-j-1],z$j),z)(2*z)*1/j!*(z^(j)),j=1..n-1)+> unapply(h[n-1],z)(2*z) ):> k[n]:=simplify((z-1)^(n+1)*(unapply(j[n],z)(z)-> unapply(j[n],z)(1/z)/z^2)):> M[n,1]:=Matrix(n,n):M[n,2]:=Matrix(n,n): M[n,3]:=Matrix(n,n):> for tx from 1 to n do for ty from tx to n do> M[n,1][ty,tx]:=binomial(n-tx,n-ty)> end do: end do:> for tx from 1 to n do M[n,2][tx,tx]:=2^(n-tx) end do:> for tx from 1 to n do M[n,3][tx,n+1-tx]:=2^(tx-1) end do:> Y[n]:=M[n,1]-1/2^(n+1)*M[n,2]+(-1)^(n+1)/2^(n+1)*M[n,3]:> A[n]:=Matrix(n,1):> for tt from 1 to n do A[n][tt,1]:=coeff(k[n],z,n-tt) end do:> B[n]:=MatrixMatrixMultiply(MatrixInverse(Y[n]),A[n]):> h[n]:=add(z^(n-s)*B[n][s,1](s,1),s=1..n)/(z-2)^(n+1):> end do:>> for n from 0 to U do alpha[n]:=unapply(diff(h[n],z$1),z)(0) end do; XPLICIT SERIES FOR THE DYADIC PERIOD FUNCTION 37
It causes no complications to compute h[n] on a standard home computer for 0 ≤ n ≤ n > Rational functions Q n ( z ) . This program computes Q n ( z ) = q[n] and the values Q ′ n ( −
1) = beta[n] for 0 ≤ n ≤ > restart;>q[0]:=-1/(2*z);>N:=50:>q[1]:=simplify(1/2*unapply(q[0],z)(-1)*(1-1/z^2)):> for n from 1 to N do> q[n]:=1/2*simplify(> add(unapply(diff(q[n-j-1],z$j),z)(-1)/j!*(z^(j)-1/z^(j+2)),j=1..n-1)+> unapply(q[n-1],z)(-1)*(1-1/z^2)> ):end do:> for w from 0 to N do beta[w]:=unapply(diff(q[w],z$1),z)(-1) end do; References [1]
G. Alkauskas , The moments of Minkowski question mark function: the dyadic period function (submitted); arXiv:0801.0051 .[2]
G. Alkauskas , Generating and zeta functions, structure, spectral and analytic properties of the momentsof the Minkowski question mark function,
Involve (to appear); arXiv:0801.0056 .[3]
G. Alkauskas , An asymptotic formula for the moments of Minkowski question mark function in the interval[0 , Lith. Math. J. (4) 2008, 357-367.[4] A. F. Beardon , Iteration of rational functions , Springer-Verlag, 1991.[5]
C. Bonanno, S. Graffi, S. Isola , Spectral analysis of transfer operators associated to Farey fractions,
Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. (1) (2008), 1-23.[6] C. Bonanno, S. Isola , Orderings of the rationals and dynamical systems,
Colloq. Math. (to appear); arXiv:0805.2178 .[7]
T. Bousch , Connexit´e locale et par chemins h¨olderiens pour les syst`emes it´er´es de fonctions; Available at http://topo.math.u-psud.fr/~bousch (1993) (unpublished).[8]
N. Calkin, H. Wilf , Recounting the rationals,
Amer. Math. Monthly (4) (2000), 360-363.[9]
A. Denjoy , Sur une fonction r´eelle de Minkowski,
J. Math. Pures Appl. (1938), 105-151.[10] A. Dushistova, N.G. Moshchevitin , On the derivative of the Minkowski question mark function ?( x ); arXiv:0706.2219 [11] M.D. Esposti, S. Isola, A. Knauf , Generalized Farey trees, transfer operators and phase transitions,
Comm. Math. Phys. (2) (2007), 297-329. [12]
R. Girgensohn , Constructing singular functions via Farey fractions,
J. Math. Anal. Appl. (1) (1996),127-141.[13]
P. J. Grabner, P. Kirschenhofer, R. Tichy , Combinatorial and arithmetical properties of linearnumeration systems,
Combinatorica (2) (2002), 245-267.[14] M. Kesseb¨ohmer, B.O. Stratmann , Fractal analysis for sets of non-differentiability of Minkowski’squestion mark function;
J. Number Theory (9) (2008), 2663-2686.[15]
A. Ya. Khinchin , Continued fractions , The University of Chicago Press, 1964.[16]
J.R. Kinney , Note on a singular function of Minkowski,
Proc. Amer. Math. Soc. (5) (1960), 788-794.[17] S. Klavˇzar, U. Milutinovi´c, C. Petr , Stern Polynomials.
Adv. in Appl. Math. (1) (2007), 86-95.[18] J.C. Lagarias , The Farey shift and the Minkowski ?-function, (preprint).[19]
J. C. Lagarias , Number theory and dynamical systems,
The unreasonable effectiveness of number theory(Orono, ME, 1991) , Proc. Sympos. Appl. Math., 46,
Amer. Math. Soc. (1992), 35-72.[20]
J.C. Lagarias, C.P. Tresser , A walk along the branches of the extended Farey tree,
IBM J. Res.Develop. (3) May (1995), 788-794.[21] M. Lamberger , On a family of singular measures related to Minkowski’s ?( x ) function, Indag. Math.(N.S.) , (1) (2006), 45-63.[22] J.B. Lewis , Spaces of holomorphic functions equivalent to the even Maass cusp forms,
Invent. Math. (2) (1997), 271-306.[23]
J.B. Lewis, D. Zagier , Period functions for Maass wave forms. I,
Ann. of Math. (2) (1) (2001),191-258.[24]
H. Okamoto, M. Wunsch , A geometric construction of continuous, strictly increasing singular functions,
Proc. Japan Acad. Ser. A Math. Sci. (7) (2007), 114-118.[25] G. Panti , Multidimensional continued fractions and a Minkowski function,
Monatsh. Math. (3) (2008),247-264.[26]
J. Parad´ıs, P. Viader, L. Bibiloni , The derivative of Minkowski’s ?( x ) function. J. Math. Anal. Appl. (1) (2001), 107-125.[27]
J. Parad´ıs, P. Viader, L. Bibiloni , A new light on Minkowski’s ?( x ) function, J. Number Theory (2) (1998), 212-227.[28] G. Ramharter , On Minkowski’s singular function,
Proc. Amer. Math. Soc. , (3) (1987), 596-597.[29] S. Reese , Some Fourier-Stieltjes coefficients revisited.
Proc. Amer. Math. Soc. (2) (1989), 384-386.[30]
F. Ryde , On the relation between two Minkowski functions,
J. Number Theory , (1) (1983), 47-51.[31] R. Salem , On some singular monotonic functions which are strictly increasing,
Trans. Amer. Math. Soc. (3) (1943), 427-439.[32] B. Solomyak , On the ‘Mandelbrot set’ for pairs of linear maps: asymptotic self-similarity,
Nonlinearity (5) (2005), 1927-1943.[33] R. F. Tichy, J. Uitz , An extension of Minkowski’s singular function,
Appl. Math. Lett . (5) (1995),39-46.[34] H. S. Wall , Analytic theory of continued fractions , D. Van Nostrand Company, Inc., New York, N. Y.,1948.[35]
G. N. Watson , A treatise on the theory of Bessel functions, 2nd ed.
Cambridge University Press, 1996.[36] An exhaustive bibliography on the Minkowski question mark function, Available at
XPLICIT SERIES FOR THE DYADIC PERIOD FUNCTION 39