Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order ε-expansion of generalized hypergeometric functions with one half-integer value of parameter

Kalmykov, M.Yu.; Ward, B. F. L.; Yost, S. A.

doi:10.1088/1126-6708/2007/10/048

Cited by 43 publications

(40 citation statements)

References 90 publications

(130 reference statements)

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“…For the expansion about half-integral parameters there exist also several algorithms [2,[21][22][23][24] for the all-order treatment of many types P r s , and also implementations in various computer languages [4,24]. However, not all types P r s can be expanded entirely in terms of ordinary HPLs, counterexamples involve for instance the types 2 2 0 and 3 1 2 [24].…”

Section: Discussionmentioning

confidence: 99%

“…Other, related algorithms which are valid for the integer and/or half-integer case rely on the reduction to a set of basis functions by means of recurrence relations [6,[21][22][23] or on the nested (harmonic and binomial) sums approach [1,2,[25][26][27].…”

Section: Algorithmmentioning

confidence: 99%

“…for HFs of type P i i±1 [13,16,[21][22][23]. H denotes a harmonic polylogarithm (HPL) [28], and the weights "+" and "−" are, respectively, the sum and the difference of the ordinary integer weights ±1 [29].…”

Section: Expansion Of the Basis Functionmentioning

confidence: 99%

“…Methods have been developed to expand HFs with half-integer parameters [2,[21][22][23]. Recently, we implemented a new algorithm for the expansion about half-integer parameters in the existing Mathematica package HypExp [6,24].…”

Section: Introductionmentioning

confidence: 99%

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Hypergeometric functions with rational arguments

Huber

2008

Nuclear Physics B - Proceedings Supplements

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We elaborate on the expansion of hypergeometric functions P FP −1 about rational parameters, where we focus mainly on the integer and half-integer case. The strategy and the basic steps of a recently developed algorithm for the expansion about half-integer parameters are described. The algorithm is implemented in the Mathematica package HypExp, by means of which we derive (partially new) results of selected multi-loop Feynman diagrams. Moreover, we give a new formulation of a conjecture in the context of a three-loop master integral in HQET.

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Section: Discussionmentioning

confidence: 99%

Section: Algorithmmentioning

confidence: 99%

Section: Expansion Of the Basis Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hypergeometric functions with rational arguments

Huber

2008

Nuclear Physics B - Proceedings Supplements

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“…(here S p are certain harmonic sums) were closely related to the computation of Feynman's diagrams (see [4], [7], [8]). In these papers various series with the above structure were computed in closed form in terms of certain special functions.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Power Series with Inverse Binomial Coefficients and Harmonic Numbers

Boyadzhiev

2017

Tatra Mountains Mathematical Publications

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Generating functions for series involving higher powers of inverse binomial coefficients and their applications

Şimşek

2023

Math Methods in App Sciences

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The purpose of this paper is to construct generating functions in terms of hypergeometric function and logarithm function for finite and infinite sums involving higher powers of inverse binomial coefficients. These generating functions provide a novel way of examining higher powers of inverse binomial coefficients from the perspective of these sums, assessing how several of these sums and these coefficients are related to each other. A relation between the Euler-Frobenius polynomial and B-spline associated with exponential Euler spline is reported. Moreover, with the aid of derivative operator and functional equations for generating functions, many new computational formulas involving the special finite sums of higher powers of (inverse) binomial coefficients, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Stirling numbers, the harmonic numbers, and special finite sums are derived.Moreover, a few recurrence relations containing these particular finite sums are given. Using these recurrence relations, we give a solution of the problem which was given by Charalambides. We give calculations algorithms for these finite sums. Applying these algorithms and Wolfram Mathematica 12.0, we give some plots and many values of these polynomials and finite sums.

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Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order ε-expansion of generalized hypergeometric functions with one half-integer value of parameter

Abstract: Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order ε-expansion of generalized hypergeometric functions with one half-integer value of parameter

Cited by 43 publications

References 90 publications

Hypergeometric functions with rational arguments

Hypergeometric functions with rational arguments

Power Series with Inverse Binomial Coefficients and Harmonic Numbers

Generating functions for series involving higher powers of inverse binomial coefficients and their applications

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