A new approach to the epsilon expansion of generalized hypergeometric functions

Abstract

Assumed that the parameters of a generalized hypergeometric function depend linearly on a small variable $\epsilon$, the successive derivatives of the function with respect to that small variable are evaluated at $\epsilon =0$ to obtain the coefficients of the $\epsilon$-expansion of the function. The procedure, quite naive, benefits from simple explicit expressions of the derivatives, to any order, of the Pochhammer and reciprocal Pochhammer symbols with respect to their argument. The algorithm may be used algebraically, irrespective of the values of the parameters. It reproduces the exact results obtained by other authors in cases of especially simple parameters. Implemented numerically, the procedure improves considerably the numerical expansions given by other methods.

Some formulae added

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Cited by in corpus (9)

• HYPERDIRE: HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions: $F_D$ and $F_S$ Horn-type hypergeometric functions of three variables
• $\epsilon$-Expansion of Multivariable Hypergeometric Functions Appearing in Feynman Integral Calculus
• New Series Representations for the Two-Loop Massive Sunset Diagram
• Analytic Periods via Twisted Symmetric Squares
• q-derivatives of multivariable q-hypergeometric function with respect to their parameters
• Coulomb expectation values in $D=3$ and $D=3-2\epsilon$ dimensions
• Expansion of hypergeometric functions in terms of polylogarithms with nontrivial variable change
• Scalar 1-loop Feynman integrals as meromorphic functions in space-time dimension d, II: Special kinematics
• Two-loop Feynman integrals for ${\varphi }^4$ theory with long-range correlated disorder