Properties and Applications of a New Extended Gamma Function Involving Confluent Hypergeometric Function

In this paper, a new confluent hypergeometric gamma function and an associated confluent hypergeometric Pochhammer symbol are introduced. We discuss some properties, for instance, their different integral representations, derivative formulas, and generating function relations. Different special cases are also considered.

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On the integral representations for the confluent hypergeometric function

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2016.0421 ◽

2016 ◽

Vol 472 (2193) ◽

pp. 20160421 ◽

Cited By ~ 1

Author(s):

Bujar Xh. Fejzullahu

Keyword(s):

Integral Representation ◽

Hypergeometric Function ◽

Hypergeometric Functions ◽

Confluent Hypergeometric Function ◽

Integral Representations ◽

Contour Integral ◽

Confluent Hypergeometric Functions ◽

Special Cases

In this paper, we derive a new contour integral representation for the confluent hypergeometric function as well as for its various special cases. Consequently, we derive expansions of the confluent hypergeometric function in terms of functions of the same kind. Furthermore, we obtain a new identity involving integrals and sums of confluent hypergeometric functions. Our results generalized several well-known results in the literature.

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A new extension of Srivastava’s triple hypergeometric functions and their associated properties

Analysis ◽

10.1515/anly-2020-0036 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Abdus Saboor ◽

Gauhar Rahman ◽

Zunaira Anjum ◽

Kottakkaran Sooppy Nisar ◽

Serkan Araci

Keyword(s):

Hypergeometric Functions ◽

Recurrence Relations ◽

Integral Representations ◽

Pochhammer Symbol ◽

Basic Properties ◽

Special Cases ◽

Derivative Formulas

AbstractIn this paper, we define a new extension of Srivastava’s triple hypergeometric functions by using a new extension of Pochhammer’s symbol that was recently proposed by Srivastava, Rahman and Nisar [H. M. Srivastava, G. Rahman and K. S. Nisar, Some extensions of the Pochhammer symbol and the associated hypergeometric functions, Iran. J. Sci. Technol. Trans. A Sci. 43 2019, 5, 2601–2606]. We present their certain basic properties such as integral representations, derivative formulas, and recurrence relations. Also, certain new special cases have been identified and some known results are recovered from main results.

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A Note on the New Extended Beta Function with Its Application

Journal of Mathematics ◽

10.1155/2021/8353719 ◽

2021 ◽

Vol 2021 ◽

pp. 1-12

Author(s):

Vandana Palsaniya ◽

Ekta Mittal ◽

Sunil Joshi ◽

D. L. Suthar

Keyword(s):

Hypergeometric Function ◽

Beta Function ◽

Moment Generating Function ◽

Confluent Hypergeometric Function ◽

Integral Representations ◽

Summation Formulas ◽

Derivative Formulas ◽

New Type ◽

Extended Beta Function ◽

Transformation Formulas

The purpose of this research is to provide a systematic review of a new type of extended beta function and hypergeometric function using a confluent hypergeometric function, as well as to examine various belongings and formulas of the new type of extended beta function, such as integral representations, derivative formulas, transformation formulas, and summation formulas. In addition, we also investigate extended Riemann–Liouville (R-L) fractional integral operator with associated properties. Furthermore, we develop new beta distribution and present graphically the relation between moment generating function and ℓ .

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A study on certain properties of generalized special functions defined by Fox-Wright function

Applied Mathematics and Nonlinear Sciences ◽

10.2478/amns.2020.1.00014 ◽

2020 ◽

Vol 5 (1) ◽

pp. 147-162

Author(s):

Enes Ata ◽

İ. Onur Kıymaz

Keyword(s):

Hypergeometric Function ◽

Special Functions ◽

Confluent Hypergeometric Function ◽

Integral Representations ◽

Gauss Hypergeometric Function ◽

Wright Function ◽

Summation Formulas ◽

Frequent Use ◽

Derivative Formulas ◽

Transformation Formulas

AbstractIn this study, motivated by the frequent use of Fox-Wright function in the theory of special functions, we first introduced new generalizations of gamma and beta functions with the help of Fox-Wright function. Then by using these functions, we defined generalized Gauss hypergeometric function and generalized confluent hypergeometric function. For all the generalized functions we have defined, we obtained their integral representations, summation formulas, transformation formulas, derivative formulas and difference formulas. Also, we calculated the Mellin transformations of these functions.

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

Mathematics ◽

10.3390/math9222944 ◽

2021 ◽

Vol 9 (22) ◽

pp. 2944

Author(s):

Shilpi Jain ◽

Rahul Goyal ◽

Praveen Agarwal ◽

Antonella Lupica ◽

Clemente Cesarano

Keyword(s):

Hypergeometric Function ◽

Beta Function ◽

Confluent Hypergeometric Function ◽

Integral Representations ◽

Gauss Hypergeometric Function ◽

Logarithmic Convexity ◽

Functional Relations ◽

Derivative Formulas ◽

Extended Beta Function ◽

Transformation Formulas

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

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A New Generalization of Pochhammer Symbol and Its Applications

Applied Mathematics and Nonlinear Sciences ◽

10.2478/amns.2020.1.00023 ◽

2020 ◽

Vol 5 (1) ◽

pp. 255-266

Author(s):

Recep Şahin ◽

Oğuz Yağcı

Keyword(s):

Gamma Function ◽

Integral Representations ◽

Pochhammer Symbol ◽

Several Variables ◽

Derivative Formulas ◽

Metric Functions

AbstractIn this paper, we introduce a new generalization of the Pochhammer symbol by means of the generalization of extended gamma function (4). Using the generalization of Pochhammer symbol, we give a generalization of the extended hypergeo-metric functions one or several variables. Also, we obtain various integral representations, derivative formulas and certain properties of these functions.

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A (p,v)-Extension of Hurwitz-Lerch Zeta Function and its Properties

10.20944/preprints201803.0008.v1 ◽

2018 ◽

Author(s):

Gauhar Rahman ◽

KS Nisar ◽

Shahid Mubeen

Keyword(s):

Zeta Function ◽

Beta Function ◽

Integral Representations ◽

Basic Properties ◽

Mellin Transformation ◽

Special Cases ◽

Lerch Zeta Function ◽

Derivative Formulas ◽

Generating Relations

In this paper, we define a (p,v)-extension of Hurwitz-Lerch Zeta function by considering an extension of beta function defined by Parmar et al. [J. Classical Anal. 11 (2017) 81–106]. We obtain its basic properties which include integral representations, Mellin transformation, derivative formulas and certain generating relations. Also, we establish the special cases of the main results.

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DKP equation with smooth barrier

International Journal of Modern Physics A ◽

10.1142/s0217751x21501876 ◽

2021 ◽

Author(s):

O. Langueur ◽

M. Merad ◽

A. Rassoul

Keyword(s):

Hypergeometric Function ◽

Numerical Study ◽

Confluent Hypergeometric Function ◽

Reflection Coefficients ◽

Step Potential ◽

Dkp Equation ◽

Transmission And Reflection Coefficients ◽

Special Cases ◽

Rectangular Barrier ◽

Transmission And Reflection

In this paper, we study the Duffin–Kemmer–Petiau (DKP) equation in the presence of a smooth barrier in dimensions space–time (1+1) dimensions. The eigenfunctions are determined in terms of the confluent hypergeometric function [Formula: see text]. The transmission and reflection coefficients are calculated, special cases as a rectangular barrier and step potential are analyzed. A numerical study is presented for the transmission and reflection coefficients graphs for some values of the parameters [Formula: see text] are plotted.

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Further Extension of the Generalized Hurwitz-Lerch Zeta Function of Two Variables

Mathematics ◽

10.3390/math7010048 ◽

2019 ◽

Vol 7 (1) ◽

pp. 48

Author(s):

Kottakkaran Sooppy Nisar

Keyword(s):

Zeta Function ◽

Hypergeometric Function ◽

Summation Formula ◽

Integral Representations ◽

Generalized Hypergeometric Function ◽

Special Cases ◽

Lerch Zeta Function ◽

Function Of Two Variables

The main aim of this paper is to provide a new generalization of Hurwitz-Lerch Zeta function of two variables. We also investigate several interesting properties such as integral representations, summation formula, and a connection with the generalized hypergeometric function. To strengthen the main results we also consider some important special cases.

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Extended Mittag-Leffler function and associated fractional calculus operators

Georgian Mathematical Journal ◽

10.1515/gmj-2019-2030 ◽

2020 ◽

Vol 27 (2) ◽

pp. 199-209 ◽

Cited By ~ 2

Author(s):

Junesang Choi ◽

Rakesh K. Parmar ◽

Purnima Chopra

Keyword(s):

Fractional Calculus ◽

Hypergeometric Function ◽

Hypergeometric Functions ◽

Fractional Integrals ◽

Generalized Hypergeometric Function ◽

Pochhammer Symbol ◽

Special Cases ◽

Fractional Calculus Operators ◽

Fractional Integrals And Derivatives ◽

Appl Math

AbstractMotivated mainly by certain interesting recent extensions of the generalized hypergeometric function [H. M. Srivastava, A. Çetinkaya and I. Onur Kıymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 2014, 484–491] by means of the generalized Pochhammer symbol, we introduce here a new extension of the generalized Mittag-Leffler function. We then systematically investigate several properties of the extended Mittag-Leffler function including some basic properties, Mellin, Euler-Beta, Laplace and Whittaker transforms. Furthermore, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag-Leffler function are also investigated. Some interesting special cases of our main results are pointed out.

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