On the all-order epsilon-expansion of generalized hypergeometric functions with integer values of parameters

Abstract

We continue our study of the construction of analytical coefficients of the epsilon-expansion of hypergeometric functions and their connection with Feynman diagrams. In this paper, we apply the approach of obtaining iteratated solutions to the differential equations associated with hypergeometric functions to prove the following result (Theorem 1): The epsilon-expansion of a generalized hypergeometric function with integer values of parameters is expressible in terms of generalized polylogarithms with coefficients that are ratios of polynomials. The method used in this proof provides an efficient algorithm for calculatiing of the higher-order coefficients of Laurent expansion.

12 pages, Latex + amsmath, JHEP3 class packages. This revision adds references 1 and 19. The FORM code is available via the WWW at http://theor.jinr.ru/~kalmykov/hypergeom/hyper.html

References in corpus (26)

• Harmonic Polylogarithms
• Connection between Feynman integrals having different values of the space-time dimension
• Analytical Result for Dimensionally Regularized Massless On-shell Double Box
• Non-planar massless two-loop Feynman diagrams with four on-shell legs
• HypExp, a Mathematica package for expanding hypergeometric functions around integer-valued parameters
• Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals
• Massive Feynman diagrams and inverse binomial sums
• Feynman Diagrams and Differential Equations
• New results for the epsilon-expansion of certain one-, two- and three-loop Feynman diagrams
• Analytic two-loop results for selfenergy- and vertex-type diagrams with one non-zero mass
• - XSummer - Transcendental Functions and Symbolic Summation in Form
• 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
• Symbolic Expansion of Transcendental Functions
• O(αα_s) correction to the pole mass of the t-quark within the Standard Model
• Explicit results for all orders of the epsilon-expansion of certain massive and massless diagrams
• Gauss hypergeometric function: reduction, epsilon-expansion for integer/half-integer parameters and Feynman diagrams
• Single-scale diagrams and multiple binomial sums
• Single mass scale diagrams: construction of a basis for the $\epsilon$-expansion
• All order epsilon-expansion of Gauss hypergeometric functions with integer and half/integer values of parameters
• Contiguous relations of hypergeometric series
• Some remarks on the epsilon-expansion of dimensionally regulated Feynman diagrams
• Effect of m_c on b quark chromomagnetic interaction and on-shell two-loop integrals with two masses
• 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
• Series and epsilon-expansion of the hypergeometric functions

Cited by in corpus (19)

• HypExp 2, Expanding Hypergeometric Functions about Half-Integer Parameters
• Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms
• Structural Relations of Harmonic Sums and Mellin Transforms up to Weight w = 5
• On the reduction of generalized polylogarithms to ${\text{Li}}_n$ and ${\text{Li}}_{2,2}$ and on the evaluation thereof
• Towards all-order Laurent expansion of generalized hypergeometric functions around rational values of parameters
• Differential reduction of generalized hypergeometric functions from Feynman diagrams: One-variable case
• On the field theory expansion of superstring five point amplitudes
• NumExp: Numerical epsilon expansion of hypergeometric functions
• A new approach to the epsilon expansion of generalized hypergeometric functions
• $\epsilon$-Expansion of Multivariable Hypergeometric Functions Appearing in Feynman Integral Calculus
• "Sixth root of unity" and Feynman diagrams: hypergeometric function approach point of view
• Two-loop kite master integral for a correlator of two composite vertices
• Expansion of hypergeometric functions in terms of polylogarithms with nontrivial variable change
• Hypergeometric functions with rational arguments
• Two-loop Feynman integrals for ${\varphi }^4$ theory with long-range correlated disorder
• One loop integration with hypergeometric series by using recursion relations
• New Evaluations of Inverse Binomial Series via Cyclotomic Multiple Zeta Values
• Chinese Magic in Loop Integrals
• Hyper-Mahler measures via Goncharov-Deligne cyclotomy