Degenerate Analogues of Euler Zeta, Digamma, and Polygamma Functions

In recent years, much attention has been paid to the role of degenerate versions of special functions and polynomials in mathematical physics and engineering. In the present paper, we introduce a degenerate Euler zeta function, a degenerate digamma function, and a degenerate polygamma function. We present several properties, recurrence relations, infinite series, and integral representations for these functions. Furthermore, we establish identities involving hypergeometric functions in terms of degenerate digamma function.

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On the matrix versions of q-zeta function, q-digamma function and q-polygamma function

Asian-European Journal of Mathematics ◽

10.1142/s1793557120501429 ◽

2019 ◽

Vol 13 (08) ◽

pp. 2050142

Author(s):

Ravi Dwivedi ◽

Vivek Sahai

Keyword(s):

Zeta Function ◽

Matrix Function ◽

Integral Representations ◽

Digamma Function ◽

Polygamma Function ◽

The Matrix ◽

Matrix Series ◽

Zeta Matrix

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.

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A procedure for generating infinite series identities

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s016117120440310x ◽

2004 ◽

Vol 2004 (67) ◽

pp. 3653-3662

Author(s):

Anthony A. Ruffa

Keyword(s):

Zeta Function ◽

Hypergeometric Function ◽

Infinite Series ◽

Special Functions ◽

Hurwitz Zeta Function ◽

Gauss Hypergeometric Function ◽

Polygamma Function ◽

Polygamma Functions ◽

Infinite Sums ◽

Lerch Transcendent

A procedure for generating infinite series identities makes use of the generalized method of exhaustion by analytically evaluating the inner series of the resulting double summation. Identities are generated involving both elementary and special functions. Infinite sums of special functions include those of the gamma and polygamma functions, the Hurwitz Zeta function, the polygamma function, the Gauss hypergeometric function, and the Lerch transcendent. The procedure can be automated withMathematica(or equivalent software).

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Note on the Hurwitz–Lerch Zeta Function of Two Variables

Symmetry ◽

10.3390/sym12091431 ◽

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.

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Multi-parametric mittag-leffler functions and their extension

Fractional Calculus and Applied Analysis ◽

10.2478/s13540-013-0024-9 ◽

2013 ◽

Vol 16 (2) ◽

Cited By ~ 28

Author(s):

Anatoly Kilbas ◽

Anna Koroleva ◽

Sergei Rogosin

Keyword(s):

Fractional Calculus ◽

Historical Development ◽

Hypergeometric Functions ◽

Special Functions ◽

Integral Representations ◽

Generalized Hypergeometric Functions ◽

Late Professor ◽

Wright Generalized Hypergeometric Functions

AbstractThis paper surveys one of the last contributions by the late Professor Anatoly Kilbas (1948–2010) and research made under his advisorship. We briefly describe the historical development of the theory of the discussed multi-parametric Mittag-Leffler functions as a class of the Wright generalized hypergeometric functions. The method of the Mellin-Barnes integral representations allows us to extend the considered functions to the case of arbitrary values of parameters. Thus, the extended Mittag-Leffler-type functions appear. The properties of these special functions and their relations to the fractional calculus are considered. Our results are based mainly on the properties of the Fox H-functions, as one of the widest class of special functions.

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

Mathematics ◽

10.3390/math7050483 ◽

2019 ◽

Vol 7 (5) ◽

pp. 483 ◽

Cited By ~ 6

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.

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Extended k-Gamma and k-Beta Functions of Matrix Arguments

Mathematics ◽

10.3390/math8101715 ◽

2020 ◽

Vol 8 (10) ◽

pp. 1715

Author(s):

Ghazi S. Khammash ◽

Praveen Agarwal ◽

Junesang Choi

Keyword(s):

Gamma Function ◽

Hypergeometric Functions ◽

Special Functions ◽

Integral Formula ◽

Beta Function ◽

Integral Representations ◽

Matrix Argument ◽

Functional Relations

Various k-special functions such as k-gamma function, k-beta function and k-hypergeometric functions have been introduced and investigated. Recently, the k-gamma function of a matrix argument and k-beta function of matrix arguments have been presented and studied. In this paper, we aim to introduce an extended k-gamma function of a matrix argument and an extended k-beta function of matrix arguments and investigate some of their properties such as functional relations, inequality, integral formula, and integral representations. Also an application of the extended k-beta function of matrix arguments to statistics is considered.

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

Symmetry ◽

10.3390/sym13040714 ◽

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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NUMERICAL CALCULATION OF BESSEL FUNCTIONS

International Journal of Modern Physics C ◽

10.1142/s0129183112500842 ◽

2012 ◽

Vol 23 (12) ◽

pp. 1250084 ◽

Cited By ~ 2

Author(s):

CHARLES SCHWARTZ

Keyword(s):

Numerical Calculation ◽

Bessel Functions ◽

Special Functions ◽

Computational Procedure ◽

Mathematical Physics ◽

Trapezoidal Rule ◽

Integral Representations ◽

The Many

A new computational procedure is offered to provide simple, accurate and flexible methods for using modern computers to give numerical evaluations of the various Bessel functions. The trapezoidal rule, applied to suitable integral representations, may become the method of choice for evaluation of the many special functions of mathematical physics.

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A family of the incomplete hypergeometric functions and associated integral transform and fractional derivative formulas

Filomat ◽

10.2298/fil1701125s ◽

2017 ◽

Vol 31 (1) ◽

pp. 125-140 ◽

Cited By ~ 10

Author(s):

Rekha Srivastava ◽

Ritu Agarwal ◽

Sonal Jain

Keyword(s):

Hypergeometric Functions ◽

Fractional Derivatives ◽

Integral Transform ◽

Special Functions ◽

Applied Mathematics ◽

Integral Transforms ◽

Mathematical Physics ◽

Natural Generalization ◽

Derivative Formulas ◽

Derivatives Of

Recently, Srivastava et al. [Integral Transforms Spec. Funct. 23 (2012), 659-683] introduced the incomplete Pochhammer symbols that led to a natural generalization and decomposition of a class of hypergeometric and other related functions as well as to certain potentially useful closed-form representations of definite and improper integrals of various special functions of applied mathematics and mathematical physics. In the present paper, our aim is to establish several formulas involving integral transforms and fractional derivatives of this family of incomplete hypergeometric functions. As corollaries and consequences, many interesting results are shown to follow from our main results.

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A NEW EXTENSION OF THE HURWITZ- LERCH ZETA FUNCTION AND PROPERTIES USING THE EXTENDED BETA FUNCTION \(B_{p,q}^{(ρ,σ,τ)}(x,y)\)

Electronic Journal of University of Aden for Basic and Applied Sciences ◽

10.47372/ejua-ba.2020.3.33 ◽

2020 ◽

Vol 1 (3) ◽

pp. 120-127

Author(s):

Salem Saleh Barahmah

Keyword(s):

Zeta Function ◽

Recurrence Relations ◽

Beta Function ◽

Integral Representations ◽

Lerch Zeta Function ◽

Generating Relations ◽

Extended Beta Function

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.

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