Some Results of Extended Beta Function and Hypergeometric Functions by Using Wiman’s Function

In the present article, our main goal is to study the new extended beta, Gauss and confluent hypergeometric functions defined by using the two Fox-Wright functions as their kernel. Some applications of the new extended beta function to distribution theory are also discussed.

Our aim is to study and investigate the family of (p, q)-extended (incomplete and complete) elliptic-type integrals for which the usual properties and representations of various known results of the (classical) elliptic integrals are extended in a simple manner. This family of elliptic-type integrals involves a number of special cases and has a connection with (p, q)-extended Gauss’ hypergeometric function and (p, q)-extended Appell’s double hypergeometric function F_{1}. Turán-type inequalities including log-convexity properties are proved for these (p, q)-extended complete elliptic-type integrals. Further, we establish various Mellin transform formulas and obtain certain infinite series representations containing Laguerre polynomials. We also obtain some relationship between these (p, q)-extended elliptic-type integrals and Meijer G-function of two variables. Moreover, we obtain several connections with (p, q)-extended beta function as special values and deduce numerous differential and integral formulas. In conclusion, we introduce (p, q)-extension of the Epstein–Hubbell (E-H) elliptic-type integral.

The main objective of this research paper is to explore further generalization of confluent hypergeometric and Whittaker functions by introducing a new parameter $k>0$, in generalized extended confluent hypergeometric and Whittaker functions defined by Khan \emph{et al.} \cite{m7}. We also investigate some properties such as integral representations, Mallin transforms, Hankel, Laplace transformations and derivative of these new generalized extended confluent hypergeometric and Whittaker $k$-functions. We also obtain Riemann-Liouville fractional integral and Riemann-Liouville $k$-fractional integral of these new generalized extended Whittaker $k$-function.

In this article , we explore the properties and relations of the hypergeometric superhyberbolic functions in terms of generalized hypergeometric functions. Then we express the hypergeometric superhyberbolic functions with the help of hypergeometric functions to obtain an integral representation related to classical results within the theory of generalized hypergeometric functions. The work emphasises conditions of convergence, and special cases where the results can be expressed in a simple form and provide a unified framework for evaluating integrals involving hypergeometric functions.

This paper deals with extended M-series, which is extension of the generalized M-series [12]. Mittag-Leffler function, $\omega$ - hypergeometric function, generalized $\omega-$ Gauss hypergeometric function, $\omega-$ confluent hypergeometric function, Bessel-Maitland function can be deduced as special cases of our finding. Moreover, we obtain some theorem for extended M-series by using fractional calculus operators and many results associated with Riemann-Liouville, Weyl and ErdelyiKober operators. We begin our study from the following definitions.

The purpose of present paper is to introduce a new extension of Hurwitz-Lerch Zeta function by using the extended Beta function. Some recurrence relations, generating relations and integral representations are derived for that new extension.

The object of this paper is to present generalizations of some recent results ot S.S. Miller and P.T. Mocanu on the univalence and starlikeness of Kummer' confluent hypergeometric function. Some applications involving generalized hypergeometric functions associated with the Hardy space of analytic functions are also considered.

Log-convexity and log-concavity of hypergeometric-like functions

Because of the many relations connecting the special functions to each other, and to the elementary functions, it is natural to inquire whether more general functions can be developed so that the special functions and elementary functions are merely specializations of these general functions. General functions of this nature have in fact been developed and are collectively referred to as functions of the hypergeometric type. There are several varieties of these functions, but the most common are the standard hypergeometric function ( which we discuss in this chapter) and the confluent hypergeometric function (Chap. 10).

Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl–Teller–Ginocchio potential wave functions

In the paper, the authors establish the logarithmic convexity and some inequalities for the extended beta function and, by using these inequalities for the extended beta function, find the logarithmic convexity and the monotonicity for the extended confluent hypergeometric function.

Many authors defined and extended the beta function in various forms because the beta function has wide uses in different fields of science and applied science. In this article, we define a new more generalized form of the extended beta matrix function via the Wiman matrix function and describe their significant properties and special cases. Furthermore, we define an extension of the Gauss hypergeometric and confluent hypergeometric matrix functions by adopting a novel type of beta matrix function. We also derive their Laplace transform, derivative formula and transformation formulae.

The primary goal of this paper is to introduce and investigate the matrix analogues and properties of the new extended beta matrix function. Using the new extended beta matrix function, we also introduce the new extended Gauss hypergeometric matrix function and the Kummer hypergeometric matrix function. We get integral representations, recurrence relations, transformation formulas, and differential formulas for these matrix functions. We also define the new extended Appell and Lauricella matrix functions using the new extended beta matrix function, and the integral representations of these matrix functions are given.

A universal practical formula is given for calculating an integral which includes two confluent hypergeometric functions, power and exponential functions; then by means of this formula, the expressions of the radial matrix elements for a relativistic harmonic oscillator are given.

In this thesis, we find an exact formula for the weighted average of the symmetric square L-values at the center. The average is taken over a Hecke eigen basis of cusp forms of SL2(Z) with a fixed weight 2k. The weights are the n-th Fourier Coefficients of these functions. The terms in the formula involve quadratic Dirichlet L-values at the center, Confluent Hypergeometric functions, and some arithmetic functions. The main ingredient, and the starting point, is a formula due Shimura, which relates the symmetric square L-function of a Hecke eigen form f to the inner product of f with the product of the theta function, θ; and a real analytic Eisenstein series of half integral weight, E. We apply Michel-Ramakrishnan's averaging technique on Shimura's formula to write the weighted average of symmetric square L-values in terms of the Fourier coefficients of the Eisenstein series. There are two complications. First, the levels of θ x E and f are different. Second, E is not holomorphic. That is why we first take trace of θ x E, and then we take the holomorphic projection. Computing the Fourier coefficients of the resulting function gives us the exact formula desired. Finally, we deduce the asymptotic behavior of these formulas as k → ∞.