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

Abstract

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.

arXiv comment18 pages, JHEP3.cls The code (FORM) is available via the www http://theor.jinr.ru/~kalmykov/hypergeom/hyper.html v2: Appendix and a few references added

References in corpus (43)

• Harmonic Polylogarithms
• Differential Equations for Feynman Graph Amplitudes
• Harmonic sums, Mellin transforms and Integrals
• HypExp, a Mathematica package for expanding hypergeometric functions around integer-valued parameters
• Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals
• Two-loop two-point functions with masses: asymptotic expansions and Taylor series, in any dimension
• Massive Feynman diagrams and inverse binomial sums
• Two-loop amplitudes and master integrals for the production of a Higgs boson via a massive quark and a scalar-quark loop
• New results for the epsilon-expansion of certain one-, two- and three-loop Feynman diagrams
• Analytical and numerical methods for massive two-loop self-energy diagrams
• Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity
• Two-loop QCD corrections of the massive fermion propagator
• MS Versus Pole Masses of Gauge Bosons II: Two-Loop Electroweak Fermion Corrections
• Expansion around half-integer values, binomial sums and inverse binomial sums
• O(αα_s) correction to the pole mass of the t-quark within the Standard Model
• Scalar One-Loop Integrals using the Negative-Dimension Approach
• Explicit results for all orders of the epsilon-expansion of certain massive and massless diagrams
• A New Hypergeometric Representation of One-Loop Scalar Integrals in $d$ Dimensions
• Gauss hypergeometric function: reduction, epsilon-expansion for integer/half-integer parameters and Feynman diagrams
• A magic connection between massive and massless diagrams
• Two-loop self-energy master integrals on shell
• Single-scale diagrams and multiple binomial sums
• Application of the negative-dimension approach to massless scalar box integrals
• On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory
• Single mass scale diagrams: construction of a basis for the $\epsilon$-expansion
• Application and explicit solution of recurrence relations with respect to space-time dimension
• Contiguous relations of hypergeometric series
• Master Integrals for Massless Three-Loop Form Factors: One-Loop and Two-Loop Insertions
• Hypergeometric representation of the two-loop equal mass sunrise diagram
• Some remarks on the epsilon-expansion of dimensionally regulated Feynman diagrams
• About higher order epsilon-expansion of some massive two- and three-loop master-integrals
• Effect of m_c on b quark chromomagnetic interaction and on-shell two-loop integrals with two masses
• LSJK - a C++ library for arbitrary-precision numeric evaluation of the generalized log-sine functions
• Geometrical methods in loop calculations and the three-point function
• Series and epsilon-expansion of the hypergeometric functions
• Irregular Input Data in Convergence Acceleration and Summation Processes: General Considerations and Some Special Gaussian Hypergeometric Series as Model Problems
• Hypergeometric representation of a four-loop vacuum bubble
• General massive one-loop off-shell three-point functions
• Loop Integrals, R Functions and their Analytic Continuation
• Geometrical approach to loop calculations and the epsilon-expansion of Feynman diagrams
• Representation of the Gauss hypergeometric function by multiple polylogarithms and relations of multiple zeta values
• Chen's Iterated Integral represents the Operator Product Expansion
• Algebraic Algorithms in Perturbative Calculations

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