Asymptotics of the special functions of fractional calculus

Basic Theory ◽  
2019 ◽  
pp. 297-326 ◽  
Author(s):  
Richard B. Paris
Keyword(s):  
Fractional Calculus ◽  
Special Functions

scholarly journals New fractional approaches for n-polynomial P-convexity with applications in special function theory

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shu-Bo Chen ◽  
Saima Rashid ◽  
Muhammad Aslam Noor ◽  
Zakia Hammouch ◽  
Yu-Ming Chu
Keyword(s):  
Fractional Calculus ◽  
Convex Functions ◽  
Special Functions ◽  
Real Life ◽  
Fractional Integrals ◽  
Digamma Function ◽  
Novel Approach ◽  
Inequality Theory ◽  
New Strategy ◽  
Significant Mechanism

Abstract Inequality theory provides a significant mechanism for managing symmetrical aspects in real-life circumstances. The renowned distinguishing feature of integral inequalities and fractional calculus has a solid possibility to regulate continuous issues with high proficiency. This manuscript contributes to a captivating association of fractional calculus, special functions and convex functions. The authors develop a novel approach for investigating a new class of convex functions which is known as an n-polynomial $\mathcal{P}$ P -convex function. Meanwhile, considering two identities via generalized fractional integrals, provide several generalizations of the Hermite–Hadamard and Ostrowski type inequalities by employing the better approaches of Hölder and power-mean inequalities. By this new strategy, using the concept of n-polynomial $\mathcal{P}$ P -convexity we can evaluate several other classes of n-polynomial harmonically convex, n-polynomial convex, classical harmonically convex and classical convex functions as particular cases. In order to investigate the efficiency and supremacy of the suggested scheme regarding the fractional calculus, special functions and n-polynomial $\mathcal{P}$ P -convexity, we present two applications for the modified Bessel function and $\mathfrak{q}$ q -digamma function. Finally, these outcomes can evaluate the possible symmetric roles of the criterion that express the real phenomena of the problem.

scholarly journals On new applications of fractional calculus

2017 ◽  
Vol 37 (3) ◽  
pp. 113-118 ◽  
Author(s):  
Shilpi Jain ◽  
Praveen Agarwal
Keyword(s):  
Fractional Calculus ◽  
Special Functions ◽  
Leibniz Rule ◽  
The One ◽  
New Applications ◽  
Paper Author

In the present paper author derive a number of integrals concerning various special functions which are applications of the one of Osler result. Osler provided extensions to the familiar Leibniz rule for the nth derivative of product of two functions.

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.

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.

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.

scholarly journals Special Functions of Fractional Calculus in the Form of Convolution Series and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2132
Author(s):  
Yuri Luchko
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Analytic Solutions ◽  
Fractional Integrals ◽  
Function Type ◽  
Fractional Differential ◽  
Derivatives Of

In this paper, we first discuss the convolution series that are generated by Sonine kernels from a class of functions continuous on a real positive semi-axis that have an integrable singularity of power function type at point zero. These convolution series are closely related to the general fractional integrals and derivatives with Sonine kernels and represent a new class of special functions of fractional calculus. The Mittag-Leffler functions as solutions to the fractional differential equations with the fractional derivatives of both Riemann-Liouville and Caputo types are particular cases of the convolution series generated by the Sonine kernel κ(t)=tα−1/Γ(α),0<α<1. The main result of the paper is the derivation of analytic solutions to the single- and multi-term fractional differential equations with the general fractional derivatives of the Riemann-Liouville type that have not yet been studied in the fractional calculus literature.

scholarly journals AN ELEMENTARY AND INTRODUCTORY APPROACH TO FRACTIONAL CALCULUS AND ITS APPLICATIONS

2017 ◽  
Vol 29 (1-2) ◽  
Author(s):  
Hari M. Srivastava
Keyword(s):  
Fractional Calculus ◽  
Integral Equations ◽  
Special Functions ◽  
Mathematical Physics ◽  
Science And Engineering ◽  
The Past ◽  
Fractional Differintegral ◽  
The Subject ◽  
Derivatives Of ◽  
Differential And Integral Equations

A b s t r a c t: The subject of fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. It does indeed provide several potentially useful tools for solving differential and integral equations, and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables. The main object of this paper* is to present a brief elementary and introductory approach to the theory of fractional calculus and its applications especially in developing solutions of certain interesting families of ordinary and partial fractional differintegral equations. Relevant connections of some of the results presented in this lecture with those obtained in many other earlier works on this subject will also be indicated.

scholarly journals A Tutorial on the Basic Special Functions of Fractional Calculus

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Probability Theory ◽  
Integral Equation ◽  
Fractional Calculus ◽  
Diffusion Equation ◽  
Special Function ◽  
Special Functions ◽  
Fractional Diffusion ◽  
Basic Properties ◽  
Fractional Relaxation ◽  
Time Fractional Diffusion Equation

In this tutorial survey we recall the basic properties of the special function of the Mittag-Leffler and Wright type that are known to be relevant in processes dealt with the fractional calculus. We outline the major applications of these functions. For the Mittag-Leffler functions we analyze the Abel integral equation of the second kind and the fractional relaxation and oscillation phenomena. For the Wright functions we distinguish them in two kinds. We mainly stress the relevance of the Wright functions of the second kind in probability theory with particular regard to the so-called M-Wright functions that generalizes the Gaussian and is related with the time-fractional diffusion equation.

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