Dmitrii Karp, Elena Prilepkina, “Hypergeometric Differential Equation and New Identities for the Coefficients of Nørlund and Bühring”, SIGMA, 2016, Volume 12,052, 23 pp.

This article is cited in 20 papers

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

Dmitrii Karp^ab, Elena Prilepkina^ba

^a Institute of Applied Mathematics, Far Eastern Branch of the Russian Academy of Sciences, 7 Radio Str., Vladivostok, 690041, Russia
^b Far Eastern Federal University, 8 Sukhanova Str., Vladivostok, 690950, Russia

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.

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

MSC: 33C20; 33C60; 34M35

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

Language: English

DOI: 10.3842/SIGMA.2016.052