MULTI-INDEX MITTAG-LEFFLER FUNCTIONS, GENERALIZED FRACTIONAL CALCULUS AND LAPLACE TYPE TRANSFORM

2006 ◽  
Vol 39 (11) ◽  
pp. 118-123
Author(s):  
Virginia Kiryakova
Keyword(s):  
Fractional Calculus ◽  
Multi Index ◽  
Generalized Fractional Calculus ◽  
Laplace Type

Differential and integral relations in the class of multi-index Mittag-Leffler functions

2018 ◽  
Vol 21 (1) ◽  
pp. 254-265 ◽  
Author(s):  
Jordanka Paneva-Konovska
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Fractional Integration ◽  
Multi Index ◽  
Generalized Fractional Calculus ◽  
Special Cases ◽  
Integral Relations ◽  
Derivatives Of ◽  
Prabhakar Function

Abstract As recently observed by Bazhlekova and Dimovski [1], the n-th derivative of the 2-parametric Mittag-Leffler function gives a 3-parametric Mittag-Leffler function, known as the Prabhakar function. Following this analogy, the n-th derivative of the (2m-index) multi-index Mittag-Leffler functions [6] is obtained, and it turns out that it is expressed in terms of the (3m-index) Mittag-Leffler functions [10, 11]. Further, some special cases of the fractional order Riemann-Liouville and Erdélyi-Kober integrals of the Mittag-Leffler functions are calculated and interesting relations are proved. Analogous relations happen to connect the 3m-Mittag-Leffler functions with the integrals and derivatives of 2m-Mittag-Leffler functions. Finally, multiple Erdélyi-Kober fractional integration operators, as operators of the generalized fractional calculus [5], are shown to relate the 2m- and 3m-parametric Mittag-Leffler functions.

On the multi-index (3m-parametric) Mittag-Leffler functions, fractional calculus relations and series convergence

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Jordanka Paneva-Konovska
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Entire Functions ◽  
Asymptotic Estimates ◽  
Multi Index ◽  
Contour Integrals ◽  
Convergence Of Series ◽  
Generalized Fractional Calculus ◽  
Series Convergence ◽  
Order And Type

AbstractIn this paper we consider a family of 3m-indices generalizations of the classical Mittag-Leffler function, called multi-index (3m-parametric) Mittag-Leffler functions. We survey the basic properties of these entire functions, find their order and type, and new representations by means of Mellin-Barnes type contour integrals, Wright pΨq-functions and Fox H-functions, asymptotic estimates. Formulas for integer and fractional order integration and differentiations are found, and these are extended also for the operators of the generalized fractional calculus (multiple Erdélyi-Kober operators). Some interesting particular cases of the multi-index Mittag-Leffler functions are discussed. The convergence of series of such type functions in the complex plane is considered, and analogues of the Cauchy-Hadamard, Abel, Tauber and Littlewood theorems are provided.

scholarly journals Some Fractional Calculus Results Pertaining To Mittag-Leffler Type Functions

2017 ◽  
Vol 13 (1) ◽  
pp. 31-48
Author(s):  
Anupama Choudhary ◽  
Devendra Kumar ◽  
Jagdev Singh
Keyword(s):  
Fractional Calculus ◽  
Wright Function ◽  
Fractional Operators ◽  
Multi Index ◽  
Generalized Wright Function ◽  
Engineering And Technology

Abstract In this paper, we study the generalized fractional operators pertaining to the generalized Mittag-Leffler function and multi-index Mittag-Leffler function. Some applications of the established results associated with generalized Wright function are also deduced as corollaries. The results are useful in solving the problems of science, engineering and technology where the Mittag-Leffler function occurs naturally.

Some generalized fractional calculus operators and their applications in integral equations

2012 ◽  
Vol 15 (4) ◽  
Author(s):  
Om Agrawal
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Integral Equations ◽  
Systematic Approach ◽  
Fractional Integrals ◽  
Generalized Fractional Calculus ◽  
Fractional Calculus Operators ◽  
Fractional Integrals And Derivatives ◽  
Generalized Fractional Calculus Operators

AbstractIn this paper, we survey some generalizations of fractional integrals and derivatives and present some of their properties. Using these properties, we show that many integral equations can be solved in a much elegant way. We believe that this will blur the distinction between the integral and differential equations, and provide a systematic approach for the two of these classes.

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