Formula for the Product of Gauss Hypergeometric Functions and Applications
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Journal of Mathematical Sciences, Vol. 249, No. 6, September, 2020
FORMULA FOR THE PRODUCT OF GAUSS HYPERGEOMETRIC FUNCTIONS AND APPLICATIONS D. V. Artamonov Lomonosov Moscow State University Moscow 119991, Russia [email protected]
UDC 517.588
We deduce a formula for the product of two Γ-series in four variables, connected with the lattice B = Z(1, −1, −1, 1). As a consequence, we obtain a formula for the product of Gauss hypergeometric functions F2,1 , which can be interpreted as a part of the explicit decomposition of the tensor product of two representations of gl3 into the direct sum of irreducible representations. Bibliography: 12 titles.
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Introduction
Hypergeometric type functions in the form of series, called Γ-series, in variables z1 , . . . , zN were introduced in [1, 2] (cf. also [3]). The Γ-series is defined by a lattice B ⊂ ZN and a fixed vector γ ∈ CN . A system of differential equations satisfied by this series was obtained by Gel’fand, Kapranov, and Zelevinskij [2]. Their approach to defining hypergeometric functions of many variables is not the most general one (for example, there is a more general approach due to Horn (cf. Introduction to [3]), but it yields a class of functions possessing many good properties. For example, the corresponding Gel’fand–Kapranov–Zelevinskij system has a finite-dimensional space of solutions, and the solutions to this system can be explicitly described [4]. This class turns out to be sufficiently large, i.e., such functions arise in problems of complex geometry [5], when explicitly solving systems of algebraic equations [6], and in equations of mathematical physics (cf. Appendix to [7]). In the particular case N = 4, B = Z(1, −1, −1, 1) the Gel’fand–Kapranov–Zelevinskij series is expressed through the Gauss hypergeometric series F2,1 , and the corresponding system is closely connected with the Gauss hypergeometric equation [6]. Only such Γ-series will be considered in this paper, except for formulas (3.1) and (3.4). Nevertheless, in the case under consideration, the Γ-series have an important advantage over F2,1 : the parameters of the Γ-series can be easily restored from the series itself. Namely, the parameters provide asymptotics of the Γ-series near one of the irreducible components of the singular set of the Gel’fand–Kapranov–Zelevinskij system. This fact is used in this paper to
Translated from Problemy Matematicheskogo Analiza 103, 2020, pp. 3-10. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2496-0817
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obtain a formula for the product of Γ-series, where one of the factors has integer parameters (Theorem 3.1). We note that the derivation of such formulas is a typical problem in the theory of special functions (cf., for example, the classical handbook [8]). As a consequence of the formula for the product of Γ-series, we obtain a new formula for the product of F2,1 (Corollary 3.1). We emphasize that this formula is absent in the handbooks [8, 9], as well as in the reference systems of packages MAPLE and Matematica, where only the fo
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