All order epsilon-expansion of Gauss hypergeometric functions with integer and half/integer values of parameters

B.F.L.Ward; JINR); M.Yu.Kalmykov (Baylor U. & Dubna; S.Yost (Baylor U.)

All order epsilon-expansion of Gauss hypergeometric functions with integer and half/integer values of parameters

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arxiv hep-th/0612240 v2 pith:V4CBJEJB submitted 2006-12-21 hep-th cs.SChep-phmath-phmath.CAmath.MPphysics.comp-ph

M.Yu.Kalmykov (Baylor U. & Dubna , JINR) , B.F.L.Ward , S.Yost (Baylor U.) This is my paper · ORCID

classification hep-thcs.SChep-phmath-phmath.CAmath.MPphysics.comp-ph

keywords epsilonarbitraryfunctionsgausshypergeometricintegercoefficientsexpansion

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It is proved that the Laurent expansion of the following Gauss hypergeometric functions, 2F1(I1+a*epsilon, I2+b*ep; I3+c*epsilon;z), 2F1(I1+a*epsilon, I2+b*epsilon;I3+1/2+c*epsilon;z), 2F1(I1+1/2+a*epsilon, I2+b*epsilon; I3+c*epsilon;z), 2F1(I1+1/2+a*epsilon, I2+b*epsilon; I3+1/2+c*epsilon;z), 2F1(I1+1/2+a*epsilon,I2+1/2+b*epsilon; I3+1/2+c*epsilon;z), where I1,I2,I3 are an arbitrary integer nonnegative numbers, a,b,c are an arbitrary numbers and epsilon is an arbitrary small parameters, are expressible in terms of the harmonic polylogarithms of Remiddi and Vermaseren with polynomial coefficients. An efficient algorithm for the calculation of the higher-order coefficients of Laurent expansion is constructed. Some particular cases of Gauss hypergeometric functions are also discussed.

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$\texttt{PrecisionLauricella}$: package for numerical computation of Lauricella functions depending on a parameter
cs.MS 2025-02 conditional novelty 4.0 of 10

PrecisionLauricella is a Mathematica package that computes epsilon-expansions of Lauricella F_A, F_B, and F_D functions for n up to 3 using Frobenius-series analytic continuation.

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