The main aim of this research paper is to introduce a new extension the Gauss hypergeometric function and confluent by using an extended beta function. Some functional relations, summation integral representations, linear transformation formulas, derivative formulas for these functions are derived. We also logarithmic convexity some important inequalities

In this paper, we introduce and investigate a new extension of the beta function by means an integral operator involving product Bessel-Struve kernel functions. We also define well-known distribution, Gauss hypergeometric confluent in terms our extended function. addition, some useful properties these functions are indicated systematic way.

This paper introduces the hypergeo package of R routines for numerical calculation of hypergeometric functions. The package is focussed on efficient and accurate evaluation of the Gauss hypergeometric function over the whole of the complex plane within the constraints of fixed-precision arithmetic. The hypergeometric series is convergent only within the unit circle, so analytic continuation mus...

To solve globally bounded order 3 linear differential equations with rational function coefficients, this thesis introduces a partial 3F2-solver (Section 3.2) and F1-solver (Chapter 4) where 3F2 is the hypergeometric function 3F2(a1, a2, a3; b1, b2 |x) and F1 is the Appell’s F1(a, b1, b2, c |x, y). To investigate the relations among order 3 multivariate hypergeometric functions, this thesis pre...

In this paper our aim is to deduce some Turán type inequalities for q-hypergeometric and q-confluent hypergeometric functions. In order to obtain the main results we apply the methods developed in the case of classical Kummer and Gauss hypergeometric functions. c ⃝ 2013 Elsevier Inc. All rights reserved.

A family of continuous biorthogonal functions related to an elliptic analog of the Gauss hypergeometric function is constructed. The key tools used for that are the elliptic beta integral and the integral Bailey chain introduced earlier by the author. The relationship with the Sklyanin algebra and elliptic analogs of the Faddeev modular double are discussed in detail. §

This paper deals with some multidimensional integral operators involving the Gauss hypergeometric function in the kernel and generating the multidimensional modified fractional calculus operators introduced in [8]. Some mapping properties, weighted inequalities, a formula of integration by parts and index laws are obtained.

In this course we will study multivariate hypergeometric functions in the sense of Gel’fand, Kapranov, and Zelevinsky (GKZ systems). These functions generalize the classical hypergeometric functions of Gauss, Horn, Appell, and Lauricella. We will emphasize the algebraic methods of Saito, Sturmfels, and Takayama to construct hypergeometric series and the connection with deformation techniques in...

Integral form expressions are obtained for the Mathieu-type series and their associated alternating versions, terms of which contain a (p,ν)-extended Gauss hypergeometric function. Contiguous recurrence relations found with respect to two parameters, finally, particular cases related bounding inequalities established.