scholarly journals Some Results on the Extended Hypergeometric Matrix Functions and Related Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Mohamed Abdalla ◽  
Sahar Ahmed Idris ◽  
Ibrahim Mekawy
Keyword(s):  
Recurrence Relations ◽  
Integral Transforms ◽  
Integral Representations ◽  
Transformation Formula ◽  
Matrix Functions ◽  
Generalized Gamma ◽  
Beta Matrix

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.

scholarly journals A new generalization of confluent hypergeometric function and whittaker function

2018 ◽  
Vol 38 (2) ◽  
pp. 9-26
Author(s):  
Nabiullah Khan ◽  
Talha Usman ◽  
Mohd Ghayasuddin
Keyword(s):  
Hypergeometric Function ◽  
Recurrence Relations ◽  
Integral Transforms ◽  
Whittaker Function ◽  
Confluent Hypergeometric Function ◽  
Integral Representations ◽  
Extra Parameter

In this article, we introduce a further generalizations of the confluent hypergeometric function and Whittaker function by introducing an extra parameter in the extended con uent hypergeometric function dened by Parmar [15]. We also investigate some integral representations, some integral transforms, differential formulas and recurrence relations of these new generalizations

scholarly journals Note on the Hurwitz–Lerch Zeta Function of Two Variables

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1431
Author(s):  
Junesang Choi ◽  
Recep Şahin ◽  
Oğuz Yağcı ◽  
Dojin Kim
Keyword(s):  
Zeta Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Zeta Functions ◽  
Integral Representations ◽  
Lerch Zeta Function ◽  
Derivative Formulas ◽  
The One ◽  
Function Of Two Variables

A number of generalized Hurwitz–Lerch zeta functions have been presented and investigated. In this study, by choosing a known extended Hurwitz–Lerch zeta function of two variables, which has been very recently presented, in a systematic way, we propose to establish certain formulas and representations for this extended Hurwitz–Lerch zeta function such as integral representations, generating functions, derivative formulas and recurrence relations. We also point out that the results presented here can be reduced to yield corresponding results for several less generalized Hurwitz–Lerch zeta functions than the extended Hurwitz–Lerch zeta function considered here. For further investigation, among possibly various more generalized Hurwitz–Lerch zeta functions than the one considered here, two more generalized settings are provided.

The Neumann problem in plane deformations of a micropolar elastic solid with micropolar surface effects

2020 ◽  
Vol 26 (1) ◽  
pp. 30-44
Author(s):  
Alireza Gharahi ◽  
Peter Schiavone
Keyword(s):  
Neumann Problem ◽  
Existence Theorems ◽  
Surface Elasticity ◽  
Elastic Solid ◽  
Singular Integral Equations ◽  
Surface Effects ◽  
Integral Representations ◽  
Matrix Functions ◽  
Micropolar Elasticity ◽  
The Neumann Problem

We consider the Neumann problem in a theory of plane micropolar elasticity incorporating micropolar surface effects. The incorporation of surface elasticity utilizes the Eremeyev–Lebedev–Altenbach shell model, leading to a set of second-order boundary conditions describing the separate micropolar elasticity of the surface. The Neumann problem is of particular interest, since the question of solvability is complicated by the fact that the corresponding systems of homogeneous singular integral equations admit nontrivial solutions that affect the solvability of both the interior and exterior Neumann boundary value problems. We overcome this difficulty by constructing integral representations of the solutions based on specifically constructed auxiliary matrix functions leading to uniqueness and existence theorems in appropriate classes of smooth matrix functions.

scholarly journals On new extensions of the generalized Hermite matrix polynomials

2019 ◽  
Vol 22 (2) ◽  
pp. 203-222
Author(s):  
Ayman Shehata
Keyword(s):  
Integral Representations ◽  
Matrix Functions ◽  
Matrix Polynomials ◽  
Summation Formulae ◽  
Summation Formulas ◽  
Fractional Calculus Operators ◽  
Generating Matrix Functions ◽  
Hermite Matrix ◽  
Generating Matrix ◽  
Matrix Recurrence Relations

Various families of generating matrix functions have been established in diverse ways. The objective of the present paper is to investigate these generalized Hermite matrix polynomials, and derive some important results for them, such as, the generating matrix functions, matrix recurrence relations, an expansion of xnI, finite summation formulas, addition theorems, integral representations, fractional calculus operators, and certain other implicit summation formulae.

scholarly journals Some new results for Jacobi matrix polynomials

Filomat ◽  
2013 ◽  
Vol 27 (4) ◽  
pp. 713-719 ◽  
Author(s):  
Bayram Çekim ◽  
Abdullah Altin ◽  
Rabia Aktaş
Keyword(s):  
Matrix Function ◽  
Recurrence Relations ◽  
Jacobi Matrix ◽  
Integral Representations ◽  
Matrix Polynomials ◽  
Generating Matrix

The main aim of this paper is to obtain some recurrence relations and generating matrix function for Jacobi matrix polynomials (JMP). Also, various integral representations satisfied by JMP are derived.

Computation of Fourier transform representations involving the generalized Bessel matrix polynomials

2021 ◽  
Author(s):  
mohamed abdalla
Keyword(s):  
Fourier Transform ◽  
Orthogonal Polynomials ◽  
Integral Transforms ◽  
General Character ◽  
Matrix Functions ◽  
Fourier Cosine ◽  
Matrix Polynomials ◽  
The Matrix ◽  
Special Matrix ◽  
Special Matrix Functions

Abstract Motivated by the recent studies and developments of the integral transforms with various special matrix functions, including the matrix orthogonal polynomials as kernels. In this article, we derive the formulas for Fourier cosine transforms and Fourier sine transforms of matrix functions involving generalized Bessel matrix polynomials. With the help of these transforms a number of results are considered which are extensions of the corresponding results in the standard cases. The results given here are of general character and can yield a number of (known and new) results in modern integral transforms.

scholarly journals Some Incomplete Hypergeometric Functions and Incomplete Riemann-Liouville Fractional Integral Operators

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 483 ◽  
Author(s):  
Mehmet Ali Özarslan ◽  
Ceren Ustaoğlu
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Recurrence Relations ◽  
Beta Function ◽  
Integral Operators ◽  
Integral Representations ◽  
Fractional Integral Operators ◽  
Derivative Formulas ◽  
Generating Relations ◽  
Transformation Formulas

Very recently, the incomplete Pochhammer ratios were defined in terms of the incomplete beta function B y ( x , z ) . With the help of these incomplete Pochhammer ratios, we introduce new incomplete Gauss, confluent hypergeometric, and Appell’s functions and investigate several properties of them such as integral representations, derivative formulas, transformation formulas, and recurrence relations. Furthermore, incomplete Riemann-Liouville fractional integral operators are introduced. This definition helps us to obtain linear and bilinear generating relations for the new incomplete Gauss hypergeometric functions.

scholarly journals Investigation of the k-Analogue of Gauss Hypergeometric Functions Constructed by the Hadamard Product

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 714
Author(s):  
Mohamed Abdalla ◽  
Muajebah Hidan
Keyword(s):  
Hypergeometric Functions ◽  
Hadamard Product ◽  
Special Function ◽  
Function Theory ◽  
Special Functions ◽  
Integral Transforms ◽  
Integral Representations ◽  
Convergence Properties ◽  
Pochhammer Symbol ◽  
Definition Of

Traditionally, the special function theory has many applications in various areas of mathematical physics, economics, statistics, engineering, and many other branches of science. Inspired by certain recent extensions of the k-analogue of gamma, the Pochhammer symbol, and hypergeometric functions, this work is devoted to the study of the k-analogue of Gauss hypergeometric functions by the Hadamard product. We give a definition of the Hadamard product of k-Gauss hypergeometric functions (HPkGHF) associated with the fourth numerator and two denominator parameters. In addition, convergence properties are derived from this function. We also discuss interesting properties such as derivative formulae, integral representations, and integral transforms including beta transform and Laplace transform. Furthermore, we investigate some contiguous function relations and differential equations connecting the HPkGHF. The current results are more general than previous ones. Moreover, the proposed results are useful in the theory of k-special functions where the hypergeometric function naturally occurs.

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