Encyclopedia of Special Functions: The Askey-Bateman Project

International Journal of Mathematical Analysis, 2016

2009

The functions of hypergeometric-type are the solutions y = yν (z) of the differential equation σ(z)y ′′ + τ (z)y ′ + λy = 0, where σ and τ are polynomials of degrees not higher than 2 and 1, respectively, and λ is a constant. Here we consider a class of functions of hypergeometric type: those that satisfy the condition λ + ντ ′ + 1 2 ν(ν − 1)σ ′′ = 0, where ν is an arbitrary complex (fixed) number. We also assume that the coefficients of the polynomials σ and τ do not depend on ν. To this class of functions belong Gauss, Kummer, and Hermite functions, and also the classical orthogonal polynomials. In this work, using the constructive approach introduced by Nikiforov and Uvarov, several structural properties of the hypergeometric-type functions y = yν (z) are obtained. Applications to hypergeometric functions and classical orthogonal polynomials are also given.

Arch Math, 1969

Experimental Mathematics, 2001

Recently, an extension of τ Gauss hypergeometric function was obtained in terms of the extended version of the pochhammer symbol[11]. We have established some properties on further generalization of the extended τ Gauss hypergeometric function containing extra parameters. We have also established some other properties and relationships involving the integral representations, derivative formulas and Mellin transforms.

Journal of Al-Qadisiyah for computer science and mathematics, 2019

IOSR Journals , 2019

Al-Gonah and Mohammed (A New Extension of Extended Gamma and Beta Functions and their Properties, Journal of Scientific and Engineering Research 5(9), 2018, 257-270) introduced a new extension of Gamma and Beta functions. In this note, we will show that a problem has been encountered regarding the Gamma function integral representations. We also studied certain results of the Gamma and Beta functions such as beta distribution, new defined Gauss and Confluent hypergeometric functions with their properties.

Comptes Rendus Mathematique, 2018

On properties and applications of (p, q)-extended τ-hypergeometric functions Sur les propriétés et applications des fonctions τ-hypergéométriques

Honam Mathematical J., 2016

The main aim of this paper is to introduce an extension of the generalized τ-Gauss hypergeometric function r F τ s (z) and investigate various properties of the new function such as integral representations , derivative formulas, Laplace transform, Mellin transform and fractional calculus operators. Some of the interesting special cases of our main results have been discussed.

Inventiones Mathematicae, 1989

Integral Equations and Operator Theory, 2000

2006

In this paper we consider some analytical relations between gamma function Γ(z) and related functions such as the Kurepa’s functionK(z) and alternating Kurepa’s functionA(z). It is well-known in the physics that the Casimir energy is defined by the principal part of the Riemann function ζ(z) (Blau, Visser, Wipf; Elizalde). Analogously, we consider the principal parts for functions Γ(z), K(z), A(z) and we also define and consider the principal part for arbitrary meromorphic functions. Next, in this paper we consider some differential-algebraic (d.a.) properties of functions Γ(z), ζ(z),K(z), A(z). As it is well-known (Holder; Ostrowski) Γ(z) is not a solution of any d.a. equation. It appears that this property of Γ(z) is universal. Namely, a large class of solutions of functional differential equations also has that property. Proof of these facts is reduced, by the use of the theory of differential algebraic fields (Ritt; Kaplansky; Kolchin), to the d.a. transcendency of Γ(z).

2006

ALVAREZ-NODARSE Abstract. The functions of hypergeometric type are the solutions y = y (z) of the dierential equation (z)y00 + (z)y0 + y = 0, where and are poly- nomials of degrees not higher than 2 and 1, respectively, and is a constant. Here we consider a class of functions of hypergeometric type: those that sat- isfy the condition +

Journal of High Energy Physics, 2007

We continue our study of the construction of analytical coefficients of the epsilon-expansion of hypergeometric functions and their connection with Feynman diagrams. In this paper, we apply the approach of obtaining iterated solutions to the differential equations associated with hypergeometric functions to prove the following result: Theorem 1. The epsilon-expansion of a generalized hypergeometric function with integer values of parameters, p F p−1 (I 1 + a 1 ε, . . . , I p + a p ε; I p+1 + b 1 ε, . . . , I 2p−1 + b p−1 ; z) , is expressible in terms of generalized polylogarithms with coefficients that are ratios of polynomials. The method used in this proof provides an efficient algorithm for calculating of the higherorder coefficients of Laurent expansion.