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 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 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 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 On the integral representations for the confluent hypergeometric function

2016 ◽  
Vol 472 (2193) ◽  
pp. 20160421 ◽  
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.

scholarly journals A New Extension of Beta and Hypergeometric Functions

2018 ◽  
Author(s):  
Gauhar Rahman ◽  
Gulnaz Kanwal ◽  
Kottakkaran Sooppy Nisar ◽  
Abdul Ghaffar
Keyword(s):  
Beta Distribution ◽  
Hypergeometric Functions ◽  
Mellin Transform ◽  
Beta Function ◽  
Moment Generating Function ◽  
Integral Representations ◽  
Confluent Hypergeometric Functions ◽  
Summation Formulas ◽  
Differentiation Formulas ◽  
Mean Variance

The main objective of this paper is to introduce a further extension of extended (p, q)-beta function by considering two Mittag-Leffler function in the kernel. We investigate various properties of this newly defined beta function such as integral representations, summation formulas and Mellin transform. We define extended beta distribution and its mean, variance and moment generating function with the help of extension of beta function. Also, we establish an extension of extended (p, q)-hypergeometric and (p, q)-confluent hypergeometric functions by using the extension of beta function. Various properties of newly defined extended hypergeometric and confluent hypergeometric functions such as integral representations, Mellin transformations, differentiation formulas, transformation and summation formulas are investigated.

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 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 A (p,ν)-extension of Srivastava’s triple hypergeometric function HC

2020 ◽  
Vol 108 (122) ◽  
pp. 33-45
Author(s):  
S.A. Dar ◽  
R.B. Paris
Keyword(s):  
Hypergeometric Function ◽  
Mellin Transform ◽  
Beta Function ◽  
Integral Representations ◽  
Recursion Formulas ◽  
Extended Beta Function ◽  
Extended Function

We obtain a (p,?)-extension of Srivastava?s triple hypergeometric function HC(?) by employing the extended Beta function Bp,?(x, y) introduced in Parmar et al. [J. Class. Anal. 11 (2017), 91-106]. We give some of the main properties of this extended function, which include several integral representations, the Mellin transform, a differential formula, recursion formulas and a bounded inequality.

scholarly journals Further Generalized Beta Function with Three Parameters Mittag- Leffler Function

2018 ◽  
pp. 41-49
Author(s):  
Salem Saleh Barahmah
Keyword(s):  
Mellin Transform ◽  
Beta Function ◽  
Integral Representations ◽  
Summation Formulas ◽  
Extended Beta Function

The purpose of the present paper is to introduce a new extension of extended Beta function by product of two Mittag-Leffler functions. Further, we present certain results including summation formulas, integral representations and Mellin transform.

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.

Sign in / Sign up

Export Citation Format

Share Document