Extended Gauss Hypergeometric Matrix Functions

The special matrix functions have received significant attention in many fields, such as theoretical physics, number theory, probability theory, and the theory of group representations. In 2017, Dwivedi and Sahai introduced the generalized hypergeometric matrix function using matrix parameters and proved the convergence on | z | = 1 {|z|=1} . Recently, hypergeometric matrix functions and their potential applications have played a major role in mathematical physics and engineering. Motivated by aforesaid works and in order to enrich this flourishing field, we investigate the Euler-type integral representations for the generalized hypergeometric matrix function and determine various transformations in terms of hypergeometric matrix functions. Furthermore, unit and half arguments have been provided for several particular cases.

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.

ABSTRACTWe introduce q-Lauricella matrix functions of three and n-variables. The convergence of these matrix functions is examined for both and cases. The systems of matrix q-difference equations satisfied by these q-Lauricella matrix functions as well as basic Srivastava's triple hypergeometric matrix functions have been determined. Certain integral representations satisfied by these matrix functions have also been established.

In this article, we discuss certain properties for generalized gamma and Euler’s beta matrix functions and the generalized hypergeometric matrix functions. The current results for these functions include integral representations, transformation formula, recurrence relations, and integral transforms.

This paper deals with the [Formula: see text]-analogues of generalized zeta matrix function, digamma matrix function and polygamma matrix function. We also discuss their regions of convergence, integral representations and matrix relations obeyed by them. We also give a few identities involving digamma matrix function and [Formula: see text]-hypergeometric matrix series.

The present paper deals with the definition and study of new kinds of hypergeometric matrix functions within complex analysis. We also get the radius of regularity and matrix recurrence relations are then developed for l(m,n)-hypergeometric matrix function of two complex variables. We give a different approach to prove the effect of the differential operator on this function. Finally, another hypergeometric matrix function, namely, p l(m,n)-hypergeometric matrix function of two complex variables are defined, its components when the positive integer p is greater than one, provide a matrix partial differential equation satisfied by these function and some of their properties are investigated.

There emerges different extended versions of Beta function and hypergeometric functions containing extra parameters. We obtain some properties of certain functions like extended Generalized Gauss hypergeometric functions, extended Confluent hypergeometric functions including transformation formulas, Mellin transformation for the generalized extended Gauss hypergeometric function in one, two and more variables.

Through this article, we will discuss a new extension of the incomplete Wright hypergeometric matrix function by using the extended incomplete Pochhammer matrix symbol. First, we give a generalization of the extended incomplete Wright hypergeometric matrix function and state some integral equations and differential formulas about it. Next, we obtain some results about fractional calculus of these extended incomplete Wright hypergeometric matrix functions. Finally, we discuss an application of the extended incomplete Wright hypergeometric matrix function in the kinetic equations.

An efficient and accurate contour integration method is presented for large-scale electronic structure calculations based on the Green function. By introducing a continued fraction representation of the Fermi-Dirac function derived from a hypergeometric function, the Matsubara summation is generalized with respect to distribution of poles so that the integration of the Green function can converge rapidly. Numerical illustrations, evaluation of the density matrix for a simple model Green function and a total energy calculation for aluminum bulk within density functional theory, clearly show that the method provides remarkable convergence with a small number of poles, indicating that the method can be applied to not only the electronic structure calculations, but also a wide variety of problems.

On the meaning and interpretation of tomography in abstract Hilbert spaces

In this paper we consider a matrix Hypergeometric differential equation, which are special matrix functions and solution of a specific second order linear differential equation. The aim of this work is to extend a well known theorem on Hypergeometric function in the complex plane to a matrix version, and we show that the asymptotic expansions of Hypergeometric function in the complex plane ” that are given in the literature are special members of our main result. Background and motivation are discussed.

We construct an algorithm that allows us to determine an effective generalized factorization of a special class of matrix functions. We use the same algorithm to analyze the spectrum of a self-adjoint operator which is related to the obtained generalized factorization.

For the study presented in this paper, the concepts specific to strong differential subordination theory, introduced for the second-order differential subordination case, are extended for the third-order differential subordination case. The general results presented in the first theorem proved in this investigation are applied considering the prolific operators recently introduced as the fractional integral of confluent (Kummer) hypergeometric function and as the fractional integral of Gaussian hypergeometric function, respectively, adapted considering the extra parameter imposed by the strong differential subordination theory and new third-order differential subordinations are investigated. Simple examples are also constructed but it is hoped that the results given here would inspire future similar studies for applying the new third-order strong differential subordination results using other known operators adapted in order to fit this new context.

In the present paper, a new relation including hypergeometric matrix function between Laguerre and Hermite matrix polynomials presented in [2,3] is derived.

In recent years, various results of the special matrix functions have been established in many papers. In the present paper, we have developed certain properties involving generalised hypergeometric matrix functions, such as, integral representations and reduction formulae. Also, open problems concerning the generalised hypergeometric matrix functions are stated.