scholarly journals Certain Recurrence Relations of Two Parametric Mittag-Leffler Function and Their Application in Fractional Calculus

2021 ◽  
Vol 5 (4) ◽  
pp. 215
Author(s):  
Dheerandra Shanker Sachan ◽  
Shailesh Jaloree ◽  
Junesang Choi
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Recurrence Relations ◽  
Differential Operators ◽  
Elementary Functions ◽  
Generalized Hypergeometric Functions

The purpose of this paper is to develop some new recurrence relations for the two parametric Mittag-Leffler function. Then, we consider some applications of those recurrence relations. Firstly, we express many of the two parametric Mittag-Leffler functions in terms of elementary functions by combining suitable pairings of certain specific instances of those recurrence relations. Secondly, by applying Riemann–Liouville fractional integral and differential operators to one of those recurrence relations, we establish four new relations among the Fox–Wright functions, certain particular cases of which exhibit four relations among the generalized hypergeometric functions. Finally, we raise several relevant issues for further research.

scholarly journals Fractional calculus and integral transforms of the product of a general class of polynomial and incomplete Fox–Wright functions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
K. Jangid ◽  
R. K. Parmar ◽  
R. Agarwal ◽  
Sunil D. Purohit
Keyword(s):  
Fractional Calculus ◽  
General Class ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Differential Operators ◽  
Integral Transforms ◽  
Parameter Selection ◽  
Fractional Operators ◽  
Polynomial Class ◽  
Math Phys

Abstract Motivated by a recent study on certain families of the incomplete H-functions (Srivastava et al. in Russ. J. Math. Phys. 25(1):116–138, 2018), we aim to investigate and develop several interesting properties related to product of a more general polynomial class together with incomplete Fox–Wright hypergeometric functions ${}_{p}\Psi _{q}^{(\gamma )}(\mathfrak{t})$ Ψ q ( γ ) p ( t ) and ${}_{p}\Psi _{q}^{(\Gamma )}(\mathfrak{t})$ Ψ q ( Γ ) p ( t ) including Marichev–Saigo–Maeda (M–S–M) fractional integral and differential operators, which contain Saigo hypergeometric, Riemann–Liouville, and Erdélyi–Kober fractional operators as particular cases regarding different parameter selection. Furthermore, we derive several integral transforms such as Jacobi, Gegenbauer (or ultraspherical), Legendre, Laplace, Mellin, Hankel, and Euler’s beta transforms.

scholarly journals Some characterization and distortion theorems involving fractional calculus, generalized hypergeometric functions, Hadamard products, linear operators, and certain subclasses of analytic functions

1987 ◽  
Vol 106 ◽  
pp. 1-28 ◽  
Author(s):  
H. M. Srivastava ◽  
Shigeyoshi Owa
Keyword(s):  
Fractional Calculus ◽  
Analytic Functions ◽  
Hypergeometric Functions ◽  
Linear Operators ◽  
Coefficient Estimates ◽  
Hadamard Products ◽  
Generalized Hypergeometric Functions ◽  
Hypergeometric Type ◽  
Distortion Theorems ◽  
Characterization Theorems

By using a certain linear operator defined by a Hadamard product or convolution, several interesting subclasses of analytic functions in the unit disk are introduced and studied systematically. The various results presented here include, for example, a number of coefficient estimates and distortion theorems for functions belonging to these subclasses, some interesting relationships between these subclasses, and a wide variety of characterization theorems involving a certain functional, some general functions of hypergeometric type, and operators of fractional calculus. Some of the coefficient estimates obtained here are fruitfully applied in the investigation of certain subclasses of analytic functions with fixed finitely many coefficients.

Multi-parametric mittag-leffler functions and their extension

2013 ◽  
Vol 16 (2) ◽  
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.

Fractional integral operators characterized by some new hypergeometric summation formulas

2017 ◽  
Vol 20 (2) ◽  
Author(s):  
Min-Jie Luo ◽  
Ravinder Krishna Raina
Keyword(s):  
Special Class ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Generalized Hypergeometric Functions ◽  
Fractional Integral Operators ◽  
Summation Formulas ◽  
Generalized Fractional Integral ◽  
Generalized Fractional Integral Operators

AbstractThe purpose of this paper is to study generalized fractional integral operators whose kernels involve a very special class of generalized hypergeometric functions

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

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 483 ◽  
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.

scholarly journals The (k, s)-Fractional Calculus of Class of a Function

2017 ◽  
Author(s):  
Gauhar Rahman ◽  
Abdul Ghaffar ◽  
K. S. Nisar ◽  
Azeema Azeema
Keyword(s):  
Fractional Calculus ◽  
Fractional Integral ◽  
Differential Operators

In this present paper, we deal with the generalized (k, s)-fractional integral and differential operators recently defined by Nisar et al. and obtain some generalized (k, s)-fractional integral and differential formulas involving the class of a function as its kernels. Also, we investigate a certain number of their consequences containing the said function in their kernels.

scholarly journals Generalized Mittag-Leffler Function Associated with Weyl Fractional Calculus Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Ahmad Faraj ◽  
Tariq Salim ◽  
Safaa Sadek ◽  
Jamal Ismail
Keyword(s):  
Integral Operator ◽  
Fractional Calculus ◽  
Fractional Integral ◽  
Differential Operators ◽  
Fractional Integral Operator ◽  
Fractional Calculus Operators ◽  
Weyl Fractional Integral ◽  
Weyl Fractional Calculus

This paper is devoted to study further properties of generalized Mittag-Leffler functionEα,β,pγ,δ,qassociated with Weyl fractional integral and differential operators. A new integral operatorℰα,β,p,w,∞γ,δ,qdepending on Weyl fractional integral operator and containingEα,β,pγ,δ,q(z)in its kernel is defined and studied, namely, its boundedness. Also, composition of Weyl fractional integral and differential operators with the new operatorℰα,β,p,w,∞γ,δ,qis established.

scholarly journals A Guide to Special Functions in Fractional Calculus

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 106
Author(s):  
Virginia Kiryakova
Keyword(s):  
Applied Sciences ◽  
Fractional Calculus ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Generalized Hypergeometric Functions ◽  
Social Events ◽  
Transcendental Functions ◽  
Meijer G Function ◽  
Wright Generalized Hypergeometric Functions

Dedicated to the memory of Professor Richard Askey (1933–2019) and to pay tribute to the Bateman Project. Harry Bateman planned his “shoe-boxes” project (accomplished after his death as Higher Transcendental Functions, Vols. 1–3, 1953–1955, under the editorship by A. Erdélyi) as a “Guide to the Functions”. This inspired the author to use the modified title of the present survey. Most of the standard (classical) Special Functions are representable in terms of the Meijer G-function and, specially, of the generalized hypergeometric functions pFq. These appeared as solutions of differential equations in mathematical physics and other applied sciences that are of integer order, usually of second order. However, recently, mathematical models of fractional order are preferred because they reflect more adequately the nature and various social events, and these needs attracted attention to “new” classes of special functions as their solutions, the so-called Special Functions of Fractional Calculus (SF of FC). Generally, under this notion, we have in mind the Fox H-functions, their most widely used cases of the Wright generalized hypergeometric functions pΨq and, in particular, the Mittag–Leffler type functions, among them the “Queen function of fractional calculus”, the Mittag–Leffler function. These fractional indices/parameters extensions of the classical special functions became an unavoidable tool when fractalized models of phenomena and events are treated. Here, we try to review some of the basic results on the theory of the SF of FC, obtained in the author’s works for more than 30 years, and support the wide spreading and important role of these functions by several examples.

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