Generalizations of Fractional Calculus, Special Functions and Integral Transforms, and Their Mutual Relationships
In this survey talk we aim to clarify the close relationships between the operators of the generalized fractional calculus (GFC), some classes of generalized hypergeometric functions and generalizations of the classical integral transforms. The GFC developed in 13 is based on the essential use of the Special Functions (SF). The generalized (multiple) fractional integrals and derivatives are defined by single (differ-)integrals with Meijers G- and Fox H-functions as their kernels, but represent also products of classical Erdélyi-Kober operators. This theory is widely illustrated by numerous special cases and applications in different areas of Applied Analysis: SF, Integral Transforms (IT), operational calculus, differential and integral equations, hyper-Bessel and Gelfond-Leontiev operators, geometric function theory, etc. Here we focus our attention to the first two topics of applications, SF and IT. First of all, the GFC operators are defined by means of IT whose kernels are SF. Next, we provide a new sight on the SF (meant as all generalized hypergeometric functions pFq and pΨq) as operators of GFC of 3 simplest elementary functions, depending on either p < q, p = q, or p = q+1 (13, 14, 18). On this GFC base, a new classification of the SF is proposed together with many new integral and differential representations (generalizations of Poisson and Euler integrals and Rodrigues formulas). Besides, the GFC operators can be interpreted as Gelfond-Leontiev operators generated by the introduced multi-index Mittag-Leffler (M-L) functions. The basic properties of this new class of SF of FC are studied and their applications in mathematical models of fractional order are emphasized, 16, 19. Also, by means of the GFC operators in the role of transmutation operators, we introduce and study some important generalizations of the Laplace integral transform. The Laplace, Borel- Dzrbashjan, Meijer, Kratzel, Obrechkoff, multi-index Borel-Dzrbashjan and other useful integral transforms are shown to be special cases. FC and SF techniques allow to develop the theory of these generalized IT, including operational rules, tables of images, convolutions, real and complex inversion formulas, Abelian type theorems, etc; 13, 6, 1. Vice versa, we need to emphasize the role of the Obrechko?integral transform and to the related hyper-Bessel operators 6, 13, as giving initial hint to develop the GFC 13.
generalized fractional calculus special functions integral transforms
Virginia Kiryakova
Institute of Mathematics and Informatics - Bulgarian Academy of Sciences, Sofia 1113, BULGARIA
国际会议
The Fifth Symposium on Fractional Differentiation and Its Applications(第五届国际自动控制联合会分数阶导数及其应用会议)
南京
英文
1-9
2012-05-14（万方平台首次上网日期，不代表论文的发表时间）