Given a Feynman parameter integral, depending on a single discrete variable N and a real parameter ε, we discuss a new algorithmic framework to compute the first coefficients of its Laurent series expansion in ε. In a first step, the integrals are expressed by hypergeometric multisums by means of symbolic transformations. Given this sum format, we develop new summation tools to extract the first coefficients of its series expansion whenever they are expressible in terms of indefinite nested product-sum expressions. In particular, we enhance the known multi-sum algorithms to derive recurrences for sums with complicated boundary conditions, and we present new algorithms to find formal Laurent series solutions of a given recurrence relation.
We discuss new algorithmic strategies for multisums arising in the computation of Feynman parameter integrals, F (N, ε), for N ∈ N, ε ∈ R, and present examples of typical computations coming from the two-loop integrals described in [3] which we encountered during our work on [5]. The genesis of these integrals is described in [4] in detail. Our summation methods have been efficiently implemented in a symbolic toolbox within Mathematica containing F. Stan's package FSums [10, Chapter 3] and C. Schneider's package EvaluateMultiSums, which rely on Sigma [8], J. Ablinger's HarmonicSums [1], and K. Wegschaider's MultiSum [12].The first step of our procedure consists of rewriting Feynman parameter integrals as multisums over hypergeometric terms which fit the input class of our summation algorithms. The sum representations are of the formwhere the summand F (µ, κ.α) is proper hypergeometric in all discrete variables µ i from µ = (µ 1 , . . . , µ p ) and in all summation variables κ j , while the elements of α ∈ C l are additional parameters. For these nested sums, the summation range R ⊆ Z r does not satisfy a finite support condition. In this context, the package FSums takes the sum (1) as input and calls the package MultiSum to obtain a recurrence for its summand, using the WZ-summation strategy [13]. After summing over the algorithmically computed recurrence for the summand F, we determine an inhomogeneous recurrence relation for Sum (µ, α). The inhomogeneous part of this recurrence will contain special instances of the multisum (1) of lower nested depth. Applying the same method on these new sums recursively, we get new recurrences. The recurrences computed at the end of this procedure will have only hypergeometric terms on their right hand sides. For the next step of the method we use procedures from the Sigma package. Namely, these last inhomogeneous difference equations can be viewed in special difference fields introduced by M. Karr [7] and extended significantly by C. Schneider. In this setting, it is possible to find solutions of such recurrences [2,9]. Plugging in these answers into the recurrences from the previous level, 95
Abstract. We present proofs for typical entries from the Gradshteyn-Ryzhik Table of Integrals using the Mellin transform method and computer algebra algorithms based on WZ theory. After representing an identity from the Table in terms of multiple contour integrals of Barnes' type and nested sums, we use Wegschaider's summation algorithm to find recurrences satisfied by both sides of this identity and check finitely many initial values.
Using classical analytic methods, E. Symeonidis [5] derived explicit expressions for the Poisson kernels of geodesic balls in non-euclidean spaces. As a by-product of his work, Symeonidis obtained indirect proofs of two interesting special function identities involving Gegenbauer polynomials. In [3] the question of direct proofs was posed. We answer this question by presenting direct proofs obtained with the help of computer algebra algorithms based on WZ theory.
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