It is well-known that the continuous Painlevé equations PJ (J=II,. . .,VI) admit particular solutions expressible in terms of various hypergeometric functions. The coalescence cascade of hypergeometric functions, from the Gauss hypergeometric function to the Airy function, corresponds to that of Painlevé equations, from PVI to PII [1]. The similar situation is expected for the discrete Painlevé...

We construct a family of continuous biorthogonal functions related to an elliptic analogue of the Gauss hypergeometric function. The key tools used for that are the elliptic beta integral and the integral Bailey chain introduced earlier by the author. Relations to the Sklyanin algebra and elliptic analogues of the Faddeev modular double are discussed in detail.

In this paper we establish some bounds for the solution of a Cauchy-type problem for a class of fractional differential equations with a weighted sequential fractional derivative. The bounds are based on a Bihari-type inequality and a bound on Gauss hypergeometric function.

In the paper, utilizing respectively the induction, a generating function of the Lah numbers, the Chu-Vandermonde summation formula, an inversion formula, the Gauss hypergeometric series, and two generating functions of Stirling numbers of the first kind, the authors collect and provide six proofs for an identity of the Lah numbers.

In the preprint [1] one of the authors formulated some conjectures on monotonicity of ratios for exponential series sections. They lead to more general conjecture on monotonicity of ratios of Kummer hypergeometric functions and was not proved from 1993. In this paper we prove some conjectures from [1] for Kummer hypergeometric functions and its further generalizations for Gauss and generalized ...

Recently, an extended operator of fractional derivative related to a generalized Beta function was used in order to obtain some generating relations involving the extended hypergeometric functions [1]. The main object of this paper is to present a further generalization of the extended fractional derivative operator and apply the generalized extended fractional derivative operator to derive lin...

We derive all eighteen Gauss hypergeometric representations for the Ferrers function of second kind, each with a different argument. They are obtained from associated Legendre kind by using limit representation. For 18 arguments which correspond to these representations, we give geometrical descriptions corresponding convergence regions in complex plane. In addition, consider single sum Fourier...

The Gauss hypergeometric function 2F1(a, b, c; z) can be computed by using the power series in powers of z, z/(z − 1), 1 − z, 1/z, 1/(1 − z), (z − 1)/z. With these expansions 2F1(a, b, c; z) is not completely computable for all complex values of z. As pointed out in Gil, et al. [2007, §2.3], the points z = e±iπ/3 are always excluded from the domains of convergence of these expansions. Bühring [...

Abstract A new family of p -Bernoulli numbers and polynomials was introduced by Rahmani (J. Number Theory 157:350–366, 2015) with the help Gauss hypergeometric function. Motivated that paper in light recent interests finding degenerate versions, we construct generalized Bernoulli means In addition, type Eulerian as a version numbers. For numbers, express them terms Stirling second kind, -Stirli...