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Rahman Gauhar,Mubeen Shahid,Nisar Kottakkaran Sooppy,Choi Junesang 경남대학교 수학교육과 2019 Nonlinear Functional Analysis and Applications Vol.24 No.1
Various extensions of the Euler’s beta function have, recently, been presented and investigated. Here, choosing to use a fully extended beta function, we introduce an extended hypergeometric function, an extended confluent hypergeometric function, and an extension of the Appell function F1. We, also, use the fully extended beta function to introduce an extended Riemann-Liouville type integral operator and investigate its associated formulas and generating relations. The results presented here, being very general, can be specialized to yield some known and new results.
Pathway Fractional Integral Formulas Involving Extended Mittag-Leffler Functions in the Kernel
Rahman, Gauhar,Nisar, Kottakkaran Sooppy,Choi, Junesang,Mubeen, Shahid,Arshad, Muhammad Department of Mathematics 2019 Kyungpook mathematical journal Vol.59 No.1
Since the Mittag-Leffler function was introduced in 1903, a variety of extensions and generalizations with diverse applications have been presented and investigated. In this paper, we aim to introduce some presumably new and remarkably different extensions of the Mittag-Leffler function, and use these to present the pathway fractional integral formulas. We point out relevant connections of some particular cases of our main results with known results.
ON A CERTAIN EXTENSION OF THE RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE OPERATOR
Nisar, Kottakkaran Sooppy,Rahman, Gauhar,Tomovski, Zivorad Korean Mathematical Society 2019 대한수학회논문집 Vol.34 No.2
The main aim of this present paper is to present a new extension of the fractional derivative operator by using the extension of beta function recently defined by Shadab et al. [19]. Moreover, we establish some results related to the newly defined modified fractional derivative operator such as Mellin transform and relations to extended hypergeometric and Appell's function via generating functions.
Nisar, Kottakkaran Sooppy,Rahman, Gauhar,Choi, Junesang,Mubeen, Shahid,Arshad, Muhammad The Youngnam Mathematical Society 2018 East Asian mathematical journal Vol.34 No.3
We aim to establish certain Gronwall type inequalities associated with Riemann-Liouville k- and Hadamard k-fractional derivatives. The results presented here are sure to be new and potentially useful, in particular, in analyzing dependence solutions of certain k-fractional differential equations of arbitrary real order with initial conditions. Some interesting special cases of our main results are also considered.
AN EXTENSION OF THE WHITTAKER FUNCTION
Choi, Junesang,Nisar, Kottakkaran Sooppy,Rahman, Gauhar Korean Mathematical Society 2021 대한수학회논문집 Vol.36 No.4
The Whittaker function and its diverse extensions have been actively investigated. Here we aim to introduce an extension of the Whittaker function by using the known extended confluent hypergeometric function 𝚽<sub>p,v</sub> and investigate some of its formulas such as integral representations, a transformation formula, Mellin transform, and a differential formula. Some special cases of our results are also considered.
INEQUALITIES OF EXTENDED (p, q)-BETA AND CONFLUENT HYPERGEOMETRIC FUNCTIONS
Mubeen, Shahid,Nisar, Kottakkaran Sooppy,Rahman, Gauhar,Arshad, Muhammad The Honam Mathematical Society 2019 호남수학학술지 Vol.41 No.4
In this paper, we establish the log convexity and Turán type inequalities of extended (p, q)-beta functions. Likewise, we present the log-convexity, the monotonicity and Turán type inequalities for extended (p, q)-confluent hypergeometric function by utilizing the inequalities of extended (p, q)-beta functions.
Inequalities of extended $(p,q)$-beta and confluent hypergeometric functions
Shahid Mubeen,Kottakkaran Sooppy Nisar,Gauhar Rahman,Muhammad Arshad 호남수학회 2019 호남수학학술지 Vol.41 No.4
In this paper, we establish the log convexity and Tur\'{a}n type inequalities of extended $(p,q)$-beta functions. Likewise, we present the log-convexity, the monotonicity and Tur\'{a}n type inequalities for extended $(p,q)$-confluent hypergeometric function by utilizing the inequalities of extended $(p,q)$-beta functions.
EXTENDED WRIGHT-BESSEL FUNCTION AND ITS PROPERTIES
Arshad, Muhammad,Mubeen, Shahid,Nisar, Kottakkaran Sooppy,Rahman, Gauhar Korean Mathematical Society 2018 대한수학회논문집 Vol.33 No.1
In this present paper, our aim is to introduce an extended Wright-Bessel function $J^{{\lambda},{\gamma},c}_{{\alpha},q}(z)$ which is established with the help of the extended beta function. Also, we investigate certain integral transforms and generalized integration formulas for the newly defined extended Wright-Bessel function $J^{{\lambda},{\gamma},c}_{{\alpha},q}(z)$ and the obtained results are expressed in terms of Fox-Wright function. Some interesting special cases involving an extended Mittag-Leffler functions are deduced.
A NEW EXTENSION OF THE MITTAG-LEFFLER FUNCTION
Arshad, Muhammad,Choi, Junesang,Mubeen, Shahid,Nisar, Kottakkaran Sooppy,Rahman, Gauhar Korean Mathematical Society 2018 대한수학회논문집 Vol.33 No.2
Since Mittag-Leffler introduced the so-called Mittag-Leffler function in 1903, due to its usefulness and diverse applications, a variety and large number of its extensions (and generalizations) and variants have been presented and investigated. In this sequel, we aim to introduce a new extension of the Mittag-Leffler function by using a known extended beta function. Then we investigate ceratin useful properties and formulas associated with the extended Mittag-Leffler function such as integral representation, Mellin transform, recurrence relation, and derivative formulas. We also introduce an extended Riemann-Liouville fractional derivative to present a fractional derivative formula for a known extended Mittag-Leffler function, the result of which is expressed in terms of the new extended Mittag-Leffler functions.