### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 052, 23 pages      arXiv:1602.07375      https://doi.org/10.3842/SIGMA.2016.052
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

### Hypergeometric Differential Equation and New Identities for the Coefficients of Nørlund and Bühring

Dmitrii Karp ab and Elena Prilepkina ab
a) Far Eastern Federal University, 8 Sukhanova Str., Vladivostok, 690950, Russia
b) Institute of Applied Mathematics, Far Eastern Branch of the Russian Academy of Sciences, 7 Radio Str., Vladivostok, 690041, Russia

Received February 25, 2016, in final form May 15, 2016; Published online May 21, 2016

Abstract
The fundamental set of solutions of the generalized hypergeometric differential equation in the neighborhood of unity has been built by Nørlund in 1955. The behavior of the generalized hypergeometric function in the neighborhood of unity has been described in the beginning of 1990s by Bühring, Srivastava and Saigo. In the first part of this paper we review their results rewriting them in terms of Meijer's $G$-function and explaining the interconnections between them. In the second part we present new formulas and identities for the coefficients that appear in the expansions of Meijer's $G$-function and generalized hypergeometric function around unity. Particular cases of these identities include known and new relations for Thomae's hypergeometric function and forgotten Hermite's identity for the sine function.

Key words: generalized hypergeometric function; hypergeometric differential equation; Meijer's $G$-function; Bernoulli polynomials; Nørlund's coefficients; Bühring's coefficients.

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