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HYPERDIRE: HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions pFq, F1,F2,F3,F4

Vladimir V. Bytev, Mikhail Yu. Kalmykov, Bernd A. Kniehl

TL;DR

HYPERDIRE addresses the problem of symbolically reducing Horn-type hypergeometric functions to a finite basis to facilitate analytic studies and Feynman-diagram evaluations. The authors implement this via two Mathematica-based packages, pfq and AppellF1F4, which realize Takayama-style differential-reduction operators and PDE-based reductions for ${}_{p+1}F_p$ and Appell functions ${F_1,...,F_4}$. Key contributions include explicit operator formulas, handling of both non-exceptional and exceptional parameter values, and a structured output format with examples reducing ${}_{p+1}F_p$, ${}_pF_{p-1}$, and Appell functions. The approach permits reduction prior to the $\\varepsilon$-expansion, with potential for extension to other Horn-type hypergeometric functions, thereby aiding analytic work on Feynman diagrams and related systems.

Abstract

HYPERDIRE is a project devoted to the creation of a set of Mathematica based programs for the differential reduction of hypergeometric functions. The current version includes two parts: one, pfq, is relevant for manipulations of hypergeometric functions_{p+1}F_p, and the second one, AppellF1F4, for manipulations with Appell hypergeometric functions F_1,F_2,F_3,F_4 of two variables.

HYPERDIRE: HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions pFq, F1,F2,F3,F4

TL;DR

HYPERDIRE addresses the problem of symbolically reducing Horn-type hypergeometric functions to a finite basis to facilitate analytic studies and Feynman-diagram evaluations. The authors implement this via two Mathematica-based packages, pfq and AppellF1F4, which realize Takayama-style differential-reduction operators and PDE-based reductions for and Appell functions . Key contributions include explicit operator formulas, handling of both non-exceptional and exceptional parameter values, and a structured output format with examples reducing , , and Appell functions. The approach permits reduction prior to the -expansion, with potential for extension to other Horn-type hypergeometric functions, thereby aiding analytic work on Feynman diagrams and related systems.

Abstract

HYPERDIRE is a project devoted to the creation of a set of Mathematica based programs for the differential reduction of hypergeometric functions. The current version includes two parts: one, pfq, is relevant for manipulations of hypergeometric functions_{p+1}F_p, and the second one, AppellF1F4, for manipulations with Appell hypergeometric functions F_1,F_2,F_3,F_4 of two variables.

Paper Structure

This paper contains 26 sections, 79 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Differential-reduction algorithm for generalized hypergeometric function ${}_{p+1}F_{p}$
  3. General consideration
  4. Differential reduction
  5. Differential reduction of Appell hypergeometric functions
  6. Appell hypergeometric functions: system of differential equations
  7. Appell hypergeometric function $F_1$
  8. General consideration
  9. Differential reduction of $F_1$
  10. Appell hypergeometric function $F_2$
  11. General consideration
  12. Differential reduction of $F_2$
  13. Appell hypergeometric function $F_3$
  14. General consideration
  15. Differential reduction of $F_3$
  16. ...and 11 more sections