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

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.

scholarly journals A Note on the New Extended Beta Function with Its Application

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 .

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

Mathematics ◽  
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.

scholarly journals Certain investigations in the field of special functions

2020 ◽  
Vol 1 (1) ◽  
pp. 87-98
Author(s):  
Maisoon A. Kulib ◽  
Ahmed A. Al-Gonah ◽  
Salem S. Barahmah
Keyword(s):  
Hypergeometric Function ◽  
Special Functions ◽  
Beta Function ◽  
Confluent Hypergeometric Function ◽  
Integral Representations ◽  
Confluent Hypergeometric Functions ◽  
Research Fields ◽  
Summation Formulas ◽  
Physical Problems ◽  
Wide Range

Motivated mainly by a variety of applications of Euler's Beta, hypergeometric, and confluent hypergeometric functions together with their extensions in a wide range of research fields such asengineering, chemical, and physical problems. In this paper, we introduce modified forms of some extended special functions such as Gamma function, Beta function, hypergeometric function and confluent hypergeometric function by making use of the idea given in reference \cite{9}. Also, certain investigations including summation formulas, integral representations and Mellin transform of these modified functions are derived. Further, many known results are obtained asspecial cases of our main results.

scholarly journals An Extension of Wright Function and Its Properties

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Moustafa El-Shahed ◽  
Ahmed Salem
Keyword(s):  
Hypergeometric Function ◽  
Confluent Hypergeometric Function ◽  
Integral Representations ◽  
Wright Function ◽  
Auxiliary Functions

The paper is devoted to the study of the functionWα,βγ,δ(z), which is an extension of the classical Wright function and Kummer confluent hypergeometric function. The properties ofWα,βγ,δ(z)including its auxiliary functions and the integral representations are proven.

scholarly journals MORE SPECIAL FUNCTIONS TRAPPED

2013 ◽  
Vol 24 (02) ◽  
pp. 1350004
Author(s):  
CHARLES SCHWARTZ
Keyword(s):  
Hypergeometric Function ◽  
Bessel Functions ◽  
Special Functions ◽  
Mathematical Physics ◽  
Confluent Hypergeometric Function ◽  
Trapezoidal Rule ◽  
Integral Representations ◽  
Gamma Functions ◽  
Incomplete Gamma Functions

We extend the technique of using the trapezoidal rule for efficient evaluation of the special functions of mathematical physics given by integral representations. This technique was recently used for Bessel functions, and here we treat incomplete gamma functions and the general confluent hypergeometric function.

scholarly journals ON A NEW GENERALIZED BETA FUNCTION DEFINED BY THE GENERALIZED WRIGHT FUNCTION AND ITS APPLICATIONS

2021 ◽  
Vol 6 (2) ◽  
pp. 852
Author(s):  
UMAR MUHAMMAD ABUBAKAR ◽  
Soraj Patel
Keyword(s):  
Hypergeometric Functions ◽  
Special Functions ◽  
Beta Function ◽  
Integral Representations ◽  
Wright Function ◽  
Confluent Hypergeometric Functions ◽  
Summation Formulas ◽  
Generalized Wright Function ◽  
Beta Error ◽  
Extended Beta Function

Various extensions of classical gamma, beta, Gauss hypergeometric and confluent hypergeometric functions have been proposed recently by many researchers. In this paper, we further generalized extended beta function with some of its properties such as symmetric properties, summation formulas, integral representations, connection with some other special functions such as classical beta, error, Mittag – Leffler, incomplete gamma, hypergeometric, classical Wright, Fox – Wright, Fox H and Meijer G – functions. Furthermore, the generalized beta function is used to generalize classical and other extended Gauss hypergeometric, confluent hypergeometric, Appell’s and Lauricella’s functions.

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

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Abdus Saboor ◽  
Gauhar Rahman ◽  
Hazrat Ali ◽  
Kottakkaran Sooppy Nisar ◽  
Thabet Abdeljawad
Keyword(s):  
Hypergeometric Function ◽  
Generating Function ◽  
Gamma Function ◽  
Confluent Hypergeometric Function ◽  
Integral Representations ◽  
Pochhammer Symbol ◽  
Special Cases ◽  
Derivative Formulas

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.

INTEGRAL TRANSFORM OF K4-FUNCTION

2021 ◽  
Vol 21 (2) ◽  
pp. 429-436
Author(s):  
SEEMA KABRA ◽  
HARISH NAGAR
Keyword(s):  
Laplace Transform ◽  
Hypergeometric Function ◽  
Integral Transform ◽  
Integral Transforms ◽  
Gauss Hypergeometric Function ◽  
Wright Function ◽  
Generalized Wright Function ◽  
Special Cases ◽  
Euler Transform

In this present work we derived integral transforms such as Euler transform, Laplace transform, and Whittaker transform of K4-function. The results are given in generalized Wright function. Some special cases of the main result are also presented here with new and interesting results. We further extended integral transforms derived here in terms of Gauss Hypergeometric function.

scholarly journals On the Volterra-Type Fractional Integro-Differential Equations Pertaining to Special Functions

2020 ◽  
Vol 4 (3) ◽  
pp. 33
Author(s):  
Yudhveer Singh ◽  
Vinod Gill ◽  
Jagdev Singh ◽  
Devendra Kumar ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Hypergeometric Function ◽  
Fractional Order ◽  
Integral Transform ◽  
Special Functions ◽  
Confluent Hypergeometric Function ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Volterra Type

In this article, we apply an integral transform-based technique to solve the fractional order Volterra-type integro-differential equation (FVIDE) involving the generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function in terms of several complex variables in the kernel. We also investigate and introduce the Elazki transform of Hilfer-derivative, generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function. In this article, we have established three results that are present in the form of lemmas, which give us new results on the above mentioned three functions, and by using these results we have derived our main results that are given in the form of theorems. Our main results are very general in nature, which gives us some new and known results as a particular case of results established here.

scholarly journals A New Extension of the τ-Gauss Hypergeometric Function and Its Associated Properties

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 996 ◽  
Author(s):  
Hari Mohan Srivastava ◽  
Asifa Tassaddiq ◽  
Gauhar Rahman ◽  
Kottakkaran Sooppy Nisar ◽  
Ilyas Khan
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Function ◽  
Mellin Transform ◽  
Gauss Hypergeometric Function ◽  
Extended Version ◽  
Pochhammer Symbol ◽  
Basic Properties ◽  
Derivative Formulas

In this article, we define an extended version of the Pochhammer symbol and then introduce the corresponding extension of the τ-Gauss hypergeometric function. The basic properties of the extended τ-Gauss hypergeometric function, including integral and derivative formulas involving the Mellin transform and the operators of fractional calculus, are derived. We also consider some new and known results as consequences of our proposed extension of the τ-Gauss hypergeometric function.

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