Computing the Inverse Mellin Transform of Holonomic Sequences using Kovacic's Algorithm
aa r X i v : . [ c s . S C ] J a n Computing the Inverse Mellin Transform ofHolonomic Sequences using Kovacic’s Algorithm
Jakob Ablinger ∗† Research Institute for Symbolic Computation (RISC)Johannes Kepler University Linz, Altenberger Straße 69, A-4040 Linz, AustriaE-mail: [email protected]
We describe how the extension of a solver for linear differential equations by Kovacic’s algorithmhelps to improve a method to compute the inverse Mellin transform of holonomic sequences. Themethod is implemented in the computer algebra package
HarmonicSums . ∗ 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). https://pos.sissa.it/ nverse Mellin Transform of Holonomic Sequences
Jakob Ablinger
1. Introduction
There have been several methods proposed to compute the inverse Mellin transform of specialsequences, for instance in [14] an algorithm (using rewrite rules) to compute the inverse Mellintransform of harmonic sums was stated. This algorithm was extended in [3] to generalized har-monic sums such as S-sums and cyclotomic sums. A different approach to compute inverse Mellintransforms of binomial sums was described in [4]. In [2] a method to compute the inverse Mellintransform of general holonomic sequences was described. That method uses holonomic closureproperties and was implemented in the computer algebra package
HarmonicSums [1, 3, 5, 6, 7].In the frame of the method a linear differential equation has to be solved. So far the differentialequations solver of
HarmonicSums was only able to find d’Alembertian solutions [8]. Recentlythe solver was generalized and therefore more general inverse Mellin transforms can be computed.In the following we repeat important definitions and properties (compare [2, 4, 11]). Let K be a field of characteristic 0. A function f = f ( x ) is called holonomic (or D-finite ) if there existpolynomials p d ( x ) , p d − ( x ) , . . . , p ( x ) ∈ K [ x ] (not all p i being 0) such that the following holonomicdifferential 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 ) = ∞ ∑ 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 constructa holonomic recurrence for the coefficients of its power series expansion. For a recent overviewof this holonomic machinery and further literature we refer to [11]. An additional property of1 nverse Mellin Transform of Holonomic Sequences Jakob Ablinger holonomic functions was given for example in [2] and [4]: if the Mellin transform of a holonomicfunction M [ f ( x )]( n ) : = Z x n f ( x ) dx = F ( n ) . (1.3)of a holonomic function is defined i.e., the integral R x n f ( x ) dx exists, then it is a holonomic se-quence. Conversely, if the Mellin transform M [ f ( x )]( n ) of a function f ( x ) is holonomic, then alsothe function f ( x ) is holonomic. In this article we will report an extension of HarmonicSums based on Kovacic’s algorithm [12] that supports the user to calculate the inverse Mellin transformin terms of iterated integrals that exceed the class of d’Alembertian solutions.The paper is organized as follows. In Section 2 we revisit the method to compute the inverseMellin transform of holonomic functions from [2], while in Section 3 we explain the generalizationof the method and in Section 4 we give some examples.
2. The Inverse Mellin Transform of Holonomic Sequences
In the following, we deal with the following problem:
Given a holonomic sequence F ( n ) . Find, whenever possible, a holonomic function f ( x ) such that for all n ∈ N (from a certain pointon) we have M [ f ( x )]( n ) = F ( n ) . In [2] a procedure was described to compute a differential equation for f ( x ) given a holonomicrecurrence for M [ f ( x )]( n ) . Given this procedure the following method to compute the inverseMellin transform of holonomic sequences was proposed in [2]:1. Compute a holonomic recurrence for M [ f ( x )]( n ) .
2. Use the method mentioned 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 ) . In our applications we usually apply this method on expressions in terms of nested sums, howeveras long as there is a method to compute the holonomic recurrence for a given expression (i.e., item 1can be performed) this proposed method can be used. Another possible input would be a holonomicrecurrence together with sufficient initial values. Note that until recently
HarmonicSums couldfind all solutions of holonomic differential equations that can be expressed in terms of iteratedintegrals over hyperexponential alphabets [4, 9, 10, 13]; these solutions are called d’Alembertiansolutions [8]. Hence as long as such solutions were sufficient to solve the differential equation initem 3 we succeeded to compute f ( x ) . In case d’Alembertian solutions do not suffice to solve thedifferential equation in item 3, we have to extend the solver for differential equations.2 nverse Mellin Transform of Holonomic Sequences
Jakob Ablinger
3. Beyond d’Alembertian solutions of linear differential equations
Until recently only d’Alembertian solutions of linear differential equations could be found in us-ing
HarmonicSums (compare [2]), but in order to treat more general problems the differentialequation solver had to be extended. In [12] an algorithm to solve second order linear homogeneousdifferential equations is described. We will refer to this algorithm as Kovacic’s algorithm.Consider the holonomic differential equation ( p i ( x ) ∈ C [ x ] ) p ( x ) f ′′ ( x ) + p ( x ) f ′ ( x ) + p ( x ) f ( x ) = . (3.1)Kovacic’s algorithm decides whether (3.1) • has a solution of the form e R ω where ω ∈ C ( x ) ; • has a solution of the form e R ω where ω is algebraic over C ( x ) of degree 2 and the previouscase does not hold; • all solutions are algebraic over C ( x ) and the previous cases do not hold; • has no such solutions;and finds the solutions if they exist. Note that the solutions Kovacic’s algorithm can find are calledLiouvillian solutions [10]. In case Kovacic’s algorithm finds a solution, it is straightforward tocompute a second solution which will again be Liouvillian. This algorithm was implementedin HarmonicSums . Example 1.
Consider the following differential equation: (cid:18) x (cid:0) − x + x (cid:1) + x (cid:0) − x + x (cid:1) D x + x (cid:0) − x + x (cid:1) D x (cid:19) f ( x ) = , with the implementation of Kovacic’s algorithm in HarmonicSums we find the following twosolutions: f ( x ) = − p − √ − x − x p + √ − x − x p ( − x ) √ x ( − + x ) , f ( x ) = − p + √ − x − x p − √ − x − x p ( − x ) √ x ( − + x ) . Suppose we are given the linear differential equation ( q i , p i ∈ C [ x ] ; d > (cid:16) q d ( x ) D dx + · · · + q ( x ) (cid:17) f ( x ) = , (3.2)which factorizes linearly into d first-order factors. Then this yields d linearly independent solutionsof the form f ( x ) , f ( x ) Z f ( x ) f ( x ) dx , f ( x ) Z f ( x ) f ( x ) Z f ( x ) f ( x ) dxdx , . . . , f ( x ) Z f ( x ) f ( x ) · · · Z f d ( x ) f d − ( x ) dx · · · dx , nverse Mellin Transform of Holonomic Sequences Jakob Ablinger where the f i are hyperexponential functions (i.e., D x f i ( x ) f i ( x ) ∈ K ( x ) ∗ ). These solutions are also calledd’Alembertian solutions of (3.2), compare [13, 8].Now suppose that a given differential equation does not factorize linearly, but contains inbetween second-order factors, which can be solved, e.g., by Kovacic’s algorithm. Let the followingdifferential equation correspond to a second order factor: (cid:0) p ( x ) D x + p ( x ) D x + p ( x ) (cid:1) f ( x ) = , (3.3)then we can compose the solutions of the first order and second order factors as follows. Let s ( x ) be solution of (3.2) and let g ( x ) and g ( x ) be solutions of (3.3). Then s ( x ) , s ( x ) Z g ( x ) s ( x ) dx and s ( x ) Z g ( x ) s ( x ) dx are solutions of (cid:0) p ( x ) D x + p ( x ) D x + p ( x ) (cid:1) (cid:16) q d ( x ) D dx + · · · + q ( x ) (cid:17) f ( x ) = . In addition, if we define w ( x ) : = p ( x )( g ′ ( x ) g ( x ) − g ( x ) g ′ ( x )) then g ( x ) , g ( x ) and g ( x ) Z s ( x ) w ( x ) g ( x ) dx − g ( x ) Z s ( x ) w ( x ) g ( x ) dx are solutions of (cid:16) q d ( x ) D dx + · · · + q ( x ) (cid:17) (cid:0) p ( x ) D x + p ( x ) D x + p ( x ) (cid:1) f ( x ) = .
4. Examples
Example 2.
We want to compute the inverse Mellin transform of f n : = (cid:18) (cid:19) n (cid:18) nn (cid:19) . We find that − ( n + )( n + ) f n + ( n + )( n + ) f n + = , which leads to the differential equation ( x − ) f ( x ) + x ( x − ) f ′ ( x ) + x ( x − ) f ′′ ( x ) = , for which we find with the help of Kovacic’s algorithm the general solution s ( x ) = c p + √ − x − x √ − x √ x + c p − √ − x − x √ − x √ x , for some constants c and c . In order to determine these constants we compute Z x s ( x ) dx = c (cid:18) − + π √ + ( ) (cid:19) + c (cid:18) + π √ − ( ) (cid:19) , nverse Mellin Transform of Holonomic Sequences Jakob Ablinger Z x s ( x ) dx = c (cid:18) − + π √ +
320 log ( ) (cid:19) + c (cid:18) + π √ −
320 log ( ) (cid:19) . Since f = / f = /
243 we can deduce that c = c = √ π and hence f n = √ π M " p − √ − x − x + p + √ − x − x √ − xx / ( n ) . Example 3.
During the Computation of the inverse Mellin transform of n ∑ i = (cid:18) ii (cid:19) i we have to solve the following differential equation:0 = ( x − ) f ( x ) + (cid:0) x − x + (cid:1) f ′ ( x ) + x (cid:0) x − x + (cid:1) f ′′ ( x )+ ( x − ) x ( x − ) f ( ) ( x ) . We are able to find the general solution of that differential equation using
HarmonicSums : c x − + c R x ( −√ − τ ) / √ + √ − τ √ − τ d τ x − + c R x √ −√ − τ ( + √ − τ ) / √ − τ d τ x − . Given this general solution we find: n ∑ i = (cid:18) ii (cid:19) i = (cid:18) (cid:19) n + Z (cid:18) x n − n n (cid:19) x − dx − √ π Z (cid:18) x n − n n (cid:19) Z x p − √ − τ (cid:0) + √ − τ (cid:1) / √ − τ d τ x − dx − √ π Z (cid:18) x n − n n (cid:19) Z x p + √ − τ (cid:0) − √ − τ (cid:1) / √ − τ d τ x − dx ! . Finally, we list several examples that could be computed using
HarmonicSums : Example 4. (cid:18) n n (cid:19) = n √ π Z x n ( + √ x + √ x − ) q √ x + √ x − √ − x x dx , n (cid:0) n n (cid:1) = n √ Z x n (cid:0) + √ x − + √ x (cid:1)q √ x − + √ x √ − xx dx , n (cid:0) nn (cid:1) = (cid:0) (cid:1) n √ Z x n (cid:16) + (cid:0) √ x − + √ x (cid:1) / (cid:17) q √ x − + √ x √ − xx dx , nverse Mellin Transform of Holonomic Sequences Jakob Ablinger n ∑ i = (cid:18) ii (cid:19) = √ (cid:0) (cid:1) n + π Z (cid:16) x n − (cid:0) (cid:1) n (cid:17) (cid:16)(cid:16) p − √ − x − x + p + √ − x − x (cid:17) √ x (cid:17) √ − x ( x − ) dx , n ∑ i = (cid:18) i i (cid:19) i = n + √ π Z x n − − n − x Z x + √ y − + √ y q √ y − + √ y √ − yy / dydx − Z x n − − n − x dx ! n ∑ i = (cid:18) ii (cid:19) i = (cid:18) (cid:19) n + √ π Z x n − (cid:0) (cid:1) n x − Z x y Z y p − √ − z + p + √ − z √ − zz / dz dy dx , − Z x n − (cid:0) (cid:1) n x − ( x ) dx − (cid:18) (cid:19) Z x n − (cid:0) (cid:1) n x − dx ! . Acknowledgements
I want to thank C. Schneider, C. Raab and J. Blümlein for useful discussions.
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