Expansion of hypergeometric functions in terms of polylogarithms with a nontrivial change of variables. (English. Russian original) Zbl 07908849
Theor. Math. Phys. 219, No. 3, 871-896 (2024); translation from Teor. Mat. Fiz. 219, No. 3, 391-421 (2024).
Summary: Hypergeometric functions of one and many variables play an important role in various branches of modern physics and mathematics. We often encounter hypergeometric functions with indices linearly dependent on a small parameter with respect to which we need to perform Laurent expansions. Moreover, it is desirable that such expansions be expressed in terms of well-known functions that can be evaluated with arbitrary precision. To solve this problem, we use the method of differential equations and the reduction of corresponding differential systems to a canonical basis. In this paper, we are interested in the generalized hypergeometric functions of one variable and in the Appell and Lauricella functions and their expansions in terms of the Goncharov polylogarithms. Particular attention is paid to the case of rational indices of the considered hypergeometric functions when the reduction to the canonical basis involves a nontrivial variable change. The paper comes with a Mathematica package Diogenes, which provides an algorithmic implementation of the required steps.
MSC:
| 33C50 | Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable |
| 33C65 | Appell, Horn and Lauricella functions |
| 33C70 | Other hypergeometric functions and integrals in several variables |
| 33B30 | Higher logarithm functions |
| 33F10 | Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.) |
| 33-04 | Software, source code, etc. for problems pertaining to special functions |
Software:
handyG; Mathematica; AMBRE; MBConicHulls; solvePartialLDE; HypSeries; PolyLogTools; MultiHypExp; Diogenes; Hypexp; HYPERDIRE; MBsums; epsilon; XSummer; HyperInt; Fuchsia; FeynGKZ; NumExp; LibraReferences:
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