From the hyper-Bessel operators of Dimovski to the generalized fractional calculus

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Virginia Kiryakova
Keyword(s):  
Fractional Calculus ◽  
Differential Operators ◽  
Integral Transform ◽  
Special Functions ◽  
Operational Calculus ◽  
Integral Transforms ◽  
Bessel Operators ◽  
Generalized Fractional Calculus ◽  
Basic Facts ◽  
Fox’S H Function

AbstractIn 1966 Ivan Dimovski introduced and started detailed studies on the Bessel type differential operators B of arbitrary (integer) order m ≥ 1. He also suggested a variant of the Obrechkoff integral transform (arising in a paper of 1958 by another Bulgarian mathematician Nikola Obrechkoff) as a Laplace-type transform basis of a corresponding operational calculus for B and for its linear right inverse integral operator L. Later, the developments on these linear singular differential operators appearing in many problems of mathematical physics, have been continued by the author of this survey who called them hyper-Bessel differential operators, in relation to the notion of hyper-Bessel functions of Delerue (1953), shown to form a fundamental system of solutions of the IVPs for By(t) = λy(t). We have been able to extend Dimovski’s results on the hyper-Bessel operators and on the Obrechkoff transform due to the happy hint to attract the tools of the special functions as Meijer’s G-function and Fox’s H-function to handle successfully these matters. These author’s studies have lead to the introduction and development of a theory of generalized fractional calculus (GFC) in her monograph (1994) and subsequent papers, and to various applications of this GFC in other topics of analysis, differential equations, special functions and integral transforms.Here we try briefly to expose the ideas leading to this GFC, its basic facts and some of the mentioned applications.

The mellin integral transform in fractional calculus

2013 ◽  
Vol 16 (2) ◽  
Author(s):  
Yuri Luchko ◽  
Virginia Kiryakova
Keyword(s):  
Fractional Calculus ◽  
Fourier Transforms ◽  
Integral Transform ◽  
Special Functions ◽  
Integral Transforms ◽  
Fourier Integral ◽  
Monotone Functions ◽  
Completely Monotone ◽  
Mellin Convolution ◽  
New Applications

AbstractIn Fractional Calculus (FC), the Laplace and the Fourier integral transforms are traditionally employed for solving different problems. In this paper, we demonstrate the role of the Mellin integral transform in FC. We note that the Laplace integral transform, the sin- and cos-Fourier transforms, and the FC operators can all be represented as Mellin convolution type integral transforms. Moreover, the special functions of FC are all particular cases of the Fox H-function that is defined as an inverse Mellin transform of a quotient of some products of the Gamma functions.In this paper, several known and some new applications of the Mellin integral transform to different problems in FC are exemplarily presented. The Mellin integral transform is employed to derive the inversion formulas for the FC operators and to evaluate some FC related integrals and in particular, the Laplace transforms and the sin- and cos-Fourier transforms of some special functions of FC. We show how to use the Mellin integral transform to prove the Post-Widder formula (and to obtain its new modi-fication), to derive some new Leibniz type rules for the FC operators, and to get new completely monotone functions from the known ones.

scholarly journals Certain Integral Transform and Fractional Integral Formulas for the Generalized Gauss Hypergeometric Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Junesang Choi ◽  
Praveen Agarwal
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Integral Transform ◽  
Special Functions ◽  
Integral Transforms ◽  
Integral Formulas ◽  
Special Cases

A remarkably large number of integral transforms and fractional integral formulas involving various special functions have been investigated by many authors. Very recently, Agarwal gave some integral transforms and fractional integral formulas involving theFp(α,β)(·). In this sequel, using the same technique, we establish certain integral transforms and fractional integral formulas for the generalized Gauss hypergeometric functionsFp(α,β,m)(·). Some interesting special cases of our main results are also considered.

The obrechkoff integral transform: properties and relation to a generalized fractional calculus

2000 ◽  
Vol 21 (1-2) ◽  
pp. 121-144 ◽  
Author(s):  
Dimovski Ivan ◽  
Kiryakova Virginia
Keyword(s):  
Fractional Calculus ◽  
Integral Transform ◽  
Generalized Fractional Calculus

scholarly journals Certain generalized fractional calculus formulas and integral transforms involving $(p,q)$-Mathieu-type series

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Ravi P. Agarwal ◽  
Adem Kılıçman ◽  
Rakesh K. Parmar ◽  
Arjun K. Rathie
Keyword(s):  
Fractional Calculus ◽  
Integral Transforms ◽  
Type Series ◽  
Generalized Fractional Calculus

On a family of the incomplete H-functions and associated integral transforms

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Manish Kumar Bansal ◽  
Devendra Kumar
Keyword(s):  
Integral Transform ◽  
Special Functions ◽  
Natural Extension ◽  
Integral Transforms ◽  
Legendre Transform ◽  
Physical Sciences ◽  
Special Cases ◽  
Improper Integrals ◽  
Jacobi Transform ◽  
Math Phys

Abstract Recently, Srivastava, Saxena and Parmar [H. M. Srivastava, R. K. Saxena and R. K. Parmar, Some families of the incomplete H-functions and the incomplete H ¯ {\overline{H}} -functions and associated integral transforms and operators of fractional calculus with applications, Russ. J. Math. Phys. 25 2018, 1, 116–138] suggested incomplete H-functions (IHF) that paved the way to a natural extension and decomposition of H-function and other connected functions as well as to some important closed-form portrayals of definite and improper integrals of different kinds of special functions of physical sciences. In this article, our key aim is to present some new integral transform (Jacobi transform, Gegenbauer transform, Legendre transform and 𝖯 δ {\mathsf{P}_{\delta}} -transform) of this family of incomplete H-functions. Further, we give several interesting new and known results which are special cases our key results.

scholarly journals Unified Approach to Fractional Calculus Images of Special Functions—A Survey

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2260 ◽  
Author(s):  
Virginia Kiryakova
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Function ◽  
Special Functions ◽  
Great Part ◽  
General Setting ◽  
Generalized Hypergeometric Function ◽  
Unified Approach ◽  
Generalized Fractional Calculus ◽  
Special Cases ◽  
Google Search

Evaluation of images of special functions under operators of fractional calculus has become a hot topic with hundreds of recently published papers. These are growing daily and we are able to comment here only on a few of them, including also some of the latest of 2019–2020, just for the purpose of illustrating our unified approach. Many authors are producing a flood of results for various operators of fractional order integration and differentiation and their generalizations of different special (and elementary) functions. This effect is natural because there are great varieties of special functions, respectively, of operators of (classical and generalized) fractional calculus, and thus, their combinations amount to a large number. As examples, we mentioned only two such operators from thousands of results found by a Google search. Most of the mentioned works use the same formal and standard procedures. Furthermore, in such results, often the originals and the images are special functions of different kinds, or the images are not recognized as known special functions, and thus are not easy to use. In this survey we present a unified approach to fulfill the mentioned task at once in a general setting and in a well visible form: for the operators of generalized fractional calculus (including also the classical operators of fractional calculus); and for all generalized hypergeometric functions such as pΨq and pFq, Fox H- and Meijer G-functions, thus incorporating wide classes of special functions. In this way, a great part of the results in the mentioned publications are well predicted and appear as very special cases of ours. The proposed general scheme is based on a few basic classical results (from the Bateman Project and works by Askey, Lavoie–Osler–Tremblay, etc.) combined with ideas and developments from more than 30 years of author’s research, and reflected in the cited recent works. The main idea is as follows: From one side, the operators considered by other authors are cases of generalized fractional calculus and so, are shown to be (m-times) compositions of weighted Riemann–Lioville, i.e., Erdélyi–Kober operators. On the other side, from each generalized hypergeometric function pΨq or pFq (p≤q or p=q+1) we can reach, from the final number of applications of such operators, one of the simplest cases where the classical results are known, for example: to 0Fq−p (hyper-Bessel functions, in particular trigonometric functions of order (q−p)), 0F0 (exponential function), or 1F0 (beta-distribution of form (1−z)αzβ). The final result, written explicitly, is that any GFC operator (of multiplicity m≥1) transforms a generalized hypergeometric function into the same kind of special function with indices p and q increased by m.

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.

A family of the incomplete hypergeometric functions and associated integral transform and fractional derivative formulas

Filomat ◽  
2017 ◽  
Vol 31 (1) ◽  
pp. 125-140 ◽  
Author(s):  
Rekha Srivastava ◽  
Ritu Agarwal ◽  
Sonal Jain
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Derivatives ◽  
Integral Transform ◽  
Special Functions ◽  
Applied Mathematics ◽  
Integral Transforms ◽  
Mathematical Physics ◽  
Natural Generalization ◽  
Derivative Formulas ◽  
Derivatives Of

Recently, Srivastava et al. [Integral Transforms Spec. Funct. 23 (2012), 659-683] introduced the incomplete Pochhammer symbols that led to a natural generalization and decomposition of a class of hypergeometric and other related functions as well as to certain potentially useful closed-form representations of definite and improper integrals of various special functions of applied mathematics and mathematical physics. In the present paper, our aim is to establish several formulas involving integral transforms and fractional derivatives of this family of incomplete hypergeometric functions. As corollaries and consequences, many interesting results are shown to follow from our main results.

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