Inverse Mellin Transform of Holonomic Sequences
aa r X i v : . [ c s . S C ] J un Inverse Mellin Transform of Holonomic Sequences
Jakob Ablinger ∗ † Research Institute for Symbolic Computation (RISC)Johannes Kepler University Linz, Altenberger Straße 69, A-4040 Linz, AustriaE-mail: [email protected]
We describe a method to compute the inverse Mellin transform of holonomic sequences, that isbased on a method to compute the Mellin transform of holonomic functions. Both methods areimplemented in the computer algebra package
HarmonicSums . Loops and Legs in Quantum Field Theory24-29 April 2016Leipzig, Germany ∗ Speaker. † This work has been supported in part by the Austrian Science Fund (FWF) grant SFB F50 (F5009-N15). c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/ nverse Mellin Transform
Jakob Ablinger
1. Introduction
In this paper we will present a method to compute the inverse Mellin transform of holonomicsequences and related to it we will revisit a method from [5] to compute the Mellin transform ofholonomic functions. We emphasize that these methods are implemented in the computer algebrapackage
HarmonicSums . Now let K be a field of characteristic 0. A function f = f ( x ) is called holonomic (or D-finite ) if there exist polynomials p d ( x ) , p d − ( x ) , . . . , p ( x ) ∈ K [ x ] (not all p i being0) such that the following holonomic differential equation holds: p d ( x ) f ( d ) ( x ) + · · · + p ( x ) f ′ ( x ) + p ( x ) f ( x ) = . (1.1)We emphasize that the class of holonomic functions is rather large due to its closure properties.Namely, if we are given two such differential equations that contain holonomic functions f ( x ) and g ( x ) as solutions, one can compute holonomic differential equations that contain f ( x ) + g ( x ) , f ( x ) g ( x ) or R x f ( y ) dy as solutions. In other words any composition of these operations over knownholonomic functions f ( x ) and g ( x ) is again a holonomic function h ( x ) . In particular, if for the innerbuilding blocks f ( x ) and g ( x ) the holonomic differential equations are given, also the holonomicdifferential equation of h ( x ) can be computed.Of special importance is the connection to recurrence relations. A sequence ( f n ) n ≥ with f n ∈ K iscalled holonomic (or P-finite ) if there exist polynomials p d ( n ) , p d − ( n ) , . . . , p ( n ) ∈ K [ n ] (not all p i being 0) such that a holonomic recurrence p d ( n ) f n + d + · · · + p ( n ) f n + + p ( n ) f n = n ∈ N (from a certain point on). In the following we utilize the fact that holonomicfunctions are precisely the generating functions of holonomic sequences: if f ( x ) is holonomic, thenthe coefficients f n of the formal power series expansion f ( x ) = ¥ (cid:229) n = f n x n form a holonomic sequence. Conversely, for a given holonomic sequence ( f n ) n ≥ , the functiondefined by the above sum (i.e., its generating function) is holonomic (this is true in the senseof formal power series, even if the sum has a zero radius of convergence). Note that given aholonomic differential equation for a holonomic function f ( x ) it is straightforward to construct aholonomic recurrence for the coefficients of its power series expansion. For a recent overview ofthis holonomic machinery and further literature we refer to [13].The paper is organized as follows. In Section 2 we revisit a method from [5] to compute theMellin transform of holonomic functions, while in Section 3 we present a method to compute theinverse Mellin transform of holonomic functions.
2. The Mellin Transform of Holonomic Functions
In the following, we deal with the problem:
Given a holonomic function f ( x ) . 1 nverse Mellin Transform Jakob Ablinger
Find, whenever possible, an expression F ( n ) given as a linear combination of indefinite nestedsums such that for all n ∈ N (from a certain point on) we have M [ f ( x )]( n ) = F ( n ) . (2.1)In [5] three different but similar methods to solve the problem above were presented. All threemethods are implemented in the Mathematica package HarmonicSums [8, 6, 7, 4]. All of thesemethods rely on the holonomic machinery sketched above. In addition the symbolic summationpackage
Sigma [15, 16] is used which is based on an algorithmic difference field theory. Hereone of the key ideas is to derive a recurrence relation that contains the Mellin transform as solutionand to execute
Sigma ’s recurrence solver that finds all solutions that can be expressed in termsof indefinite nested sums and products [14, 9, 12, 10]; these solutions are called d’Alembertiansolutions. In the following we revisit one of the methods form [5].We state the following proposition.
Proposition 1.
If the Mellin transform of a holonomic function is defined i.e., the integral Z x n f ( x ) dxexists, then it is a holonomic sequence.Proof. Let f ( x ) be a holonomic function such that the integral R x n f ( x ) dx exists. Using the prop-erties of the Mellin transform we can easily check that M [ x m f ( p ) ( x )]( n ) = ( − ) p ( n + m ) ! ( n + m − p ) ! M [ f ( x )]( n + m − p ) + p − (cid:229) i = ( − ) i ( n + m ) ! ( n + m − i ) ! f ( p − − i ) ( ) . (2.2)Finally, we apply the Mellin transform to the holonomic differential equation of f ( x ) using therelation above, and we get a holonomic recurrence for M [ f ( x )]( n ) . Now, a method to compute the Mellin transform is obvious:Let f ( x ) be a holonomic function. In order to compute the Mellin transform M [ f ( x )]( n ) , we canproceed as follows:1. Compute a holonomic differential equation for f ( x ) .
2. Use the proposition above to compute a holonomic recurrence for M [ f ( x )]( n ) .
3. Compute initial values for the recurrence.4. Solve the recurrence (with
Sigma ) to get a closed form representation for M [ f ( x )]( n ) . Note that
Sigma finds all solutions that can be expressed in terms of indefinite nested sums andproducts. Hence as long as such solutions suffice to solve the recurrence in item 4, we succeed tocompute the Mellin transform M [ f ( x )]( n ) . nverse Mellin Transform Jakob Ablinger
Example 2.
We want to compute the Mellin transform of f ( x ) : = Z x √ − t + t d t . We find that ( − + x ) f ′ ( x ) + ( − + x )( + x ) f ′′ ( x ) = Z √ − t + t d t = − ( n − ) n M [ f ( x )]( n − ) + n M [ f ( x )]( n − ) + ( n + )( n + ) M [ f ( x )]( n ) . Initial values can be computed easily and solving the recurrence leads to M [ f ( x )]( n ) = ( + ( − ) n ) R √ − t + t d t + ( − ) n (cid:18) + n (cid:229) i = ( − ) i ( ii ) (cid:19) + n − ( + n ) ( n ) ( + n )( + n ) (cid:0) nn (cid:1) . Note that this method can be extended to compute regularized Mellin transforms: given aholonomic function f ( x ) such that Z ( x n − ) f ( x ) dx exists, then we can compute M [[ f ( x )] + ]( n ) : = Z ( x n − ) f ( x ) dx using a slight extension of the method above. For example we can compute M [[ log ( x ) − x ] + ]( n ) = Z ( x n − ) log ( x ) − x dx = n (cid:229) i = i .
3. The Inverse Mellin Transform of Holonomic Sequences
In the following, we deal with the problem:
Given a holonomic sequence F ( n ) . Find, whenever possible, an expression f ( x ) given as a linear combination of indefinite iteratedintegrals such that for all n ∈ N (from a certain point on) we have M [ f ( x )]( n ) = F ( n ) . As a first step we want to compute a differential equation for f ( x ) given a holonomic recurrencefor M [ f ( x )]( n ) . Analyzing (2.2) we see that M [( − ) p x m + p f ( p ) ( x )]( n ) = ( n + m + p ) ! ( n + m ) ! M [ f ( x )]( n + m )+ p − (cid:229) i = ( − ) i + p ( n + m + p ) ! ( n + m + p − i ) ! f ( p − − i ) ( ) . (3.1)3 nverse Mellin Transform Jakob Ablinger
Hence we get n p M [ f ( x )]( n + m ) = M [( − ) p x m + p f ( p ) ( x )]( n ) − a ( n ) M [ f ( x )]( n + m ) − p − (cid:229) i = ( − ) i + p ( n + m + p ) ! ( n + m + p − i ) ! f ( p − − i ) ( ) , (3.2)where a ( n ) ∈ K [ n ] with deg ( a ( n )) < p . We can use this observation to compute the differentialequation recursively: Let p d ( n ) f n + d + · · · + p ( n ) f n + + p ( n ) f n = M [ f ( x )]( n ) . Let k : = max ≤ i ≤ d ( deg ( p i ( x ))) and let c be the coefficientof n k in the recurrence i.e., c = d (cid:229) i = c i f n + i for some c i ∈ K . For 0 ≤ i ≤ d we replace c i n k f n + i by c i n k f n + i + c i ( − ) k x k + i f ( k ) ( x ) − c i M [( − ) k x k + i f ( k ) ( x )]( n ) and apply (2.2). Considering (3.2) we conclude that we reduced the degree of n . We apply thisstrategy until we have removed all appearences of f n + i . At this point we have an equation of theform q l ( x ) f ( l ) ( x ) + · · · + q ( x ) f ′ ( x ) + q ( x ) f ( x ) + k − (cid:229) j = r j ( n ) f ( j ) ( ) = . where r i ( n ) ∈ K [ n ] . If all r i ( n ) = , we are done. If not, we differentiate the differential equation.In both cases we end up with a holonomic differential equation for f ( x ) . Let us illustrate this strategy using an example.
Example 3.
Consider the recurrence ( + n ) f n + − f n + − ( n + ) f n = . (3.4)The maximal degree of the coefficients f n + i with 0 ≤ i ≤ n of the lefthand side of (3.4) is f n + − f n . We substitute n f n + → n f n + − x f ′ ( x ) + M [ x f ′ ( x )]( n ) − n f n → − n f n + x f ′ ( x ) − M [ x f ′ ( x )]( n ) in (3.4) and apply (2.2). This yields ( − x + x ) f ′ − f n + − f n + = . (3.5)since M [ x f ′ ( x )]( n ) = − ( n + ) f n + + f ( ) and M [ x f ′ ( x )]( n ) = − ( n + ) f n + f ( ) . Next we sub-stitute − f n + → − f n + − x f ( x ) + M [ x f ( x )]( n ) nverse Mellin Transform Jakob Ablinger − f n + → − f n + − x f ( x ) + M [ x f ( x )]( n ) in (3.5) and apply (2.2). Since M [ x f ( x )]( n ) = f n + and M [ x f ( x )]( n ) = f n + , this yields thedifferential equation ( − x + x ) f ′ ( x ) − ( x + x ) f ( x ) = . (3.6)Our strategy to compute the inverse Mellin transform of holonomic sequences can be summa-rized as follows:1. Compute a holonomic recurrence for M [ f ( x )]( n ) .
2. Use the method above to compute a holonomic differential equation for f ( x ) .
3. Compute a linear independent set of solutions of the holonomic differentialequation (using
HarmonicSums ).4. Compute initial values for M [ f ( x )]( n ) .
5. Combine the initial values and the solutions to get a closed form representation for f ( x ) . Note that
HarmonicSums finds all solutions that can be expressed in terms of iterated integralsover hyperexponential alphabets [14, 9, 12, 11, 5]; these solutions are called d’Alembertian solu-tions. Hence as long as such solutions suffice to solve the differential equation in item 3 we succeedto compute f ( x ) . Example 4.
We want to compute the inverse Mellin transform of f n : = ( − ) n n (cid:229) i = ( − ) i (cid:229) ij = j i − ¥ (cid:229) i = ( − ) i (cid:229) ij = j i ! We find that 0 = ( n + )( n + ) f n − ( n + ) (cid:0) n + n + (cid:1) f n + + (cid:0) − n − n − n + (cid:1) f n + + ( n + ) f n + which leads to the differential equation0 = − ( x − ) ( x + ) x f ( ) ( x ) − ( x − )( x − )( x + ) x f ′′ ( x ) − ( x − )( x − ) x f ′ ( x ) − ( x − ) x f ( x ) that has the general solution s ( x ) = c x + + c x + Z x y − dy + c x + Z x log ( y ) y − dy , for some constants c , c , c . In order to determine these constants we compute Z x s ( x ) dx = c log ( ) + c log ( ) − z + c z − log ( ) z , nverse Mellin Transform Jakob Ablinger Z x s ( x ) dx = c ( − log ( )) + c − log ( ) + z − + c log ( ) z − z + , Z x s ( x ) dx = c ( ) − + c ( ) − z + + c − ( ) z + z − . Since f = − ¥ (cid:229) i = ( − ) i (cid:229) ij = j i ; f = + ¥ (cid:229) i = ( − ) i (cid:229) ij = j i ; f = − − ¥ (cid:229) i = ( − ) i (cid:229) ij = j i we can deduce that c = , c = c = f n = M (cid:20) x + Z x log ( y ) y − dy (cid:21) ( n ) . Note that the method above only works if the result is of the form c n M [ f ( x )]( n ) + d for some c , d ∈ R . However, in general we will find results of the form k (cid:229) i = c ni M [ f i ( x )]( n ) + d for some c i , d ∈ R . Hence in order to deal with more general functions we refine our approach andcompute f i ( x ) and c i for i = i = k one after another. We illustrate this using the followingexample. Example 5.
We want to compute the inverse Mellin transform of f n : = n (cid:229) i = ( − ) i i i (cid:229) j = j j . We find that0 = ( + n )( + n ) f n − ( + n )( + n ) f n + + (cid:0) − n − n (cid:1) f n + + ( + n ) f n + which leads to the differential equation0 = (cid:0) − x + x (cid:1) f ( x ) + (cid:0) x − x + x (cid:1) f ′ ( x ) + (cid:0) x − x − x + x (cid:1) f ′′ ( x )+ (cid:0) x − x − x + x (cid:1) f ( ) ( x ) with the following three linear independent solutions12 + x , − log ( − x ) + x , ( + x ) (cid:18) p + ( ) + ( − x ) log ( − x ) − ( − x )+ p log (cid:16) x − (cid:17) + log (cid:18) ( − + x ) x (cid:19) − (cid:18) − x (cid:19) + ( x − ) (cid:19) . nverse Mellin Transform Jakob Ablinger
We know that f n has to appear in at least one of the Mellin transforms of these solutions. Indeedwe get M [ − log ( − x ) + x ]( n ) = ( − ) n (cid:18) n (cid:229) i = ( − ) i i i (cid:229) j = j j + log ( ) (cid:18) log ( ) − n (cid:229) i = ( − ) i i + n (cid:229) i = ( − ) i i (cid:19) + p − ( ) − Li (cid:18) (cid:19)(cid:19) . Now we can write f n = f n + ( − ) n M [ − log ( − x ) + x ]( n ) − (cid:18) n (cid:229) i = ( − ) i i i (cid:229) j = j j + log ( ) (cid:18) log ( ) − n (cid:229) i = ( − ) i i + n (cid:229) i = ( − ) i i (cid:19) + p − ( ) − Li (cid:18) (cid:19)(cid:19) = ( − ) n M [ − log ( − x ) + x ]( n ) − log ( ) (cid:18) log ( ) − n (cid:229) i = ( − ) i i + n (cid:229) i = ( − ) i i (cid:19) − p + ( ) + Li (cid:18) (cid:19) Next we compute the inverse Mellin transform of g n = n (cid:229) i = ( − ) i i . We find that0 = − ( + n ) g n + + ( + n ) g n + + ( + n ) g n + which leads to the differential equation0 = x ( − x ) g ( x ) + x ( − x − x ) g ′ ( x ) with the solution + x . Note that M [ + x ]( n ) = ( − ) n log ( ) − log ( ) + n (cid:229) i = ( − ) i i ! , hence we can write f n = − ( − ) n M [ log ( − x ) + x ]( n ) +
18 log ( ) − p + Li (cid:18) (cid:19) + log ( ) n (cid:229) i = ( − ) i i . It remains to compute the inverse Mellin transform of h n = (cid:229) ni = ( − ) i i . We derive the recurrence0 = − ( + n ) h n + h n + + ( + n ) h n + which gives rise to the differential equation0 = x ( − x ) h ( x ) + x ( − x ) h ′ ( x ) with the solution + x . Since M [ + x ]( n ) = ( − ) n log ( ) + n (cid:229) i = ( − ) i i ! we finally get f n = − ( − ) n M [ log ( − x ) + x ]( n ) + ( − ) n log ( ) M [ + x ]( n ) + ( ) − p + Li (cid:18) (cid:19) . nverse Mellin Transform Jakob Ablinger
Acknowledgements
I want to thank C. Schneider, C. Raab and J. Blümlein for useful discussions.
References [1] J. Ablinger,
The package HarmonicSums: Computer Algebra and Analytic aspects of Nested Sums ,PoS LL (2014) 019 [arXiv:1407.6180 [cs.SC]].[2] J. Ablinger and J. Blümlein,
Harmonic Sums, Polylogarithms, Special Numbers, and theirGeneralizations , in
Computer Algebra in Quantum Field Theory: Integration, Summation and SpecialFunctions, Texts & Monographs in Symbolic Computation , Springer, Wien, 2013. [arXiv:1304.7071[math-ph]].[3] J. Ablinger, J. Blümlein and C. Schneider,
Generalized Harmonic, Cyclotomic, and Binomial Sums,their Polylogarithms and Special Numbers , J. Phys. Conf. Ser. (2014) 012060doi:10.1088/1742-6596/523/1/012060 [arXiv:1310.5645 [math-ph]].[4] J. Ablinger,
Computer Algebra Algorithms for Special Functions in Particle Physics , PhD Thesis,J. Kepler University Linz , 2012. arXiv:1305.0687 [math-ph].[5] J. Ablinger, J. Blümlein, C. G. Raab and C. Schneider,
Iterated Binomial Sums and their AssociatedIterated Integrals , J. Math. Phys. (2014) 112301 doi:10.1063/1.4900836 [arXiv:1407.1822[hep-th]].[6] J. Ablinger, J. Blümlein and C. Schneider, Harmonic Sums and Polylogarithms Generated byCyclotomic Polynomials , J. Math. Phys. (2011) 102301 doi:10.1063/1.3629472 [arXiv:1105.6063[math-ph]].[7] J. Ablinger, J. Blümlein and C. Schneider, Analytic and Algorithmic Aspects of GeneralizedHarmonic Sums and Polylogarithms , J. Math. Phys. (2013) 082301 doi:10.1063/1.4811117[arXiv:1302.0378 [math-ph]].[8] J. Ablinger, A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics ,arXiv:1011.1176 [math-ph].[9] S.A. Abramov and M. Petkovšek,
D’Alembertian solutions of linear differential and differenceequations , in proceedings of
ISSAC’94 , ACM Press, 1994.[10] J. Blümlein, M. Kauers, S. Klein and C. Schneider,
Determining the closed forms of the O ( a s ) anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra ,Comput. Phys. Commun. (2009) 2143 doi:10.1016/j.cpc.2009.06.020 [arXiv:0902.4091[hep-ph]].[11] M. Bronstein, Linear Ordinary Differential Equations: breaking through the order 2 barrier , inproceedings of
ISSAC’92 , ACM Press, 1992.[12] P.A. Hendriks and M.F. Singer,
Solving difference equations in finite terms , J. Symbolic Comput. ,1999.[13] M. Kauers and P. Paule, The Concrete Tetrahedron , Text and Monographs in Symbolic Computation,Springer, Wien, 2011.[14] M. Petkovšek,
Hypergeometric solutions of linear recurrences with polynomial coefficients , J. Symbolic Comput. , 1992.[15] C. Schneider, Symbolic summation assists combinatorics , Sém. Lothar. Combin. , 2007. nverse Mellin Transform Jakob Ablinger[16] C. Schneider,
Simplifying Multiple Sums in Difference Fields , in
Computer Algebra in Quantum FieldTheory: Integration, Summation and Special Functions, Texts & Monographs in SymbolicComputation , Springer, Wien, 2013. [arXiv:11304.4134 [cs.SC]]., Springer, Wien, 2013. [arXiv:11304.4134 [cs.SC]].