An algorithmic approach to the Mellin transform method

Kohl, Karen T.; Stan, Flavia

doi:10.1090/conm/517/10142

Cited by 2 publications

(4 citation statements)

References 14 publications

(26 reference statements)

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“…In joint work with K. Kohl [38], we contributed with an approach based on the Mellin transform method for rewriting definite integration problems in terms of nested Mellin-Barnes integrals. Viewing the identities in [32] from the perspective of the Mellin transform method seems natural, especially since most entries from the table of Mellin transforms [46] are also found there.…”

Section: Back To Proving Special Functions Identitiesmentioning

confidence: 99%

“…After distinguishing between odd and even values of the parameter k, for an arbitrary Ising-class integral C n,k , n, k ≥ 1, one obtains two representations of the form 38) where µ = k 2 , respectively, µ = k−1 2 such that µ ∈ N. In both cases the integrand Ψ (µ, t) is proper hypergeometric in µ ≥ 0 and in all integration variables t j from t = (t 1 , . .…”

Section: Recurrences For the Integrals C Nkmentioning

confidence: 99%

“…Our procedure to determine homogeneous and inhomogeneous recurrences for contour integrals of this type eliminates the need to find sum representations which was a crucial non-algorithmic step in the computational approach presented in Chapter 3. We also apply this new algorithmic technique to prove typical entries from the Gradshteyn-Ryzhik table of integrals [32] using the Mellin transform [38] and to find recurrences for a class of Ising integrals [65].…”

Section: Introductionmentioning

confidence: 99%

“…The work presented in this thesis has lead to four papers [15,38,65,66], three of which are accepted for publication.…”

Section: Introductionmentioning

confidence: 99%

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Algorithms for special functions

Stan¹

2011

ACM Commun. Comput. Algebra

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In this thesis we present our contributions to symbolic summation, extending WZFasenmyer methods to handle definite hypergeometric sums with nonstandard boundary conditions and to compute recurrences for multiple Mellin-Barnes integrals over hypergeometric terms. We also include concrete applications of these methods to Feynman integral calculus, as well as for proving identities involving definite integrals and special functions.First we give a short introduction to WZ-summation methods, including K. Wegschaider's approach to this method and its implementation in the package MultiSum. Inspired by work of Sister Celine Fasenmyer, these techniques were introduced by H. Wilf and D. Zeilberger to algorithmically compute recurrences for multiple sums over hypergeometric terms. Their procedure is based on finding a certificate recurrence satisfied by the hypergeometric summand and summing over this difference equation to obtain a recurrence for the nested sum. Our proofs of two nontrivial special function identities involving Gegenbauer polynomials, provide classic applications of the method.As part of the collaboration between RISC and DESY coordinated by C. Schneider, we developed an algorithmic approach to compute Feynman parameter integrals after rewriting them as multisums over hypergeometric terms to fit the input class of classic summation algorithms.Since these definite sums have nonstandard boundary conditions, the WZ-method delivers inhomogeneous recurrence relations. We designed a recursive procedure to determine the inhomogeneous parts of these recurrences and implemented it in the Mathematica package FSums which builds on the already existing packages, MultiSum and C. Schneider's Sigma.Another approach to evaluate Feynman integrals is by representating them in terms of nested Mellin-Barnes integrals. These complex contour integrals can also be viewed as sums of residues at certain poles of the integrands and they are connected to the inversion formula for the Mellin transform.In the last part, we show how WZ-methods can be used to compute recurrences for multiple Mellin-Barnes integrals over hypergeometric terms, eliminating the need to search for sum representations. We applied this new algorithmic technique to prove typical entries from the Gradshteyn-Ryzhik

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Section: Back To Proving Special Functions Identitiesmentioning

confidence: 99%

Section: Recurrences For the Integrals C Nkmentioning

confidence: 99%