Kiryakova, Virginia S. Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus. (English) Zbl 0966.33011 J. Comput. Appl. Math. 118, No. 1-2, 241-259 (2000). The Mittag-Leffler functions were studied by Agarwal, Humbert, Dzrbashjan, and others in the 1950s, but not a lot of attention has been given to them since. The functions under consideration here are the generalizations \[ E_{(1/ \rho_i), (\mu_i)}(z)= \sum^\infty_{k=0} {z^k\over\Gamma (\mu_1+k/ \rho_1) \cdots \Gamma(\mu_m+k/ \rho_m)}, \] which are interesting cases of Wright’s generalization of the hypergeometric function and of Fox’s \(H\)-function. Examples of special cases are displayed. Connections are obtained with various generalized fractional integral operators and with the solutions of differential and integral equations. A convolution relation is derived and the appropriate Borel-Laplace integral transform and its inverse is developed. The details of some of the proofs, the connection with the \(m\)-dimensional Laplace transformation, other representations, and additional results are promised in another paper. Reviewer: Robert G.Buschman (Langlois) Cited in 5 ReviewsCited in 68 Documents MSC: 33E12 Mittag-Leffler functions and generalizations 26A33 Fractional derivatives and integrals 33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions) 33E30 Other functions coming from differential, difference and integral equations 44A15 Special integral transforms (Legendre, Hilbert, etc.) 44A30 Multiple integral transforms Keywords:Mittag-Leffler functions; fractional integrals; integral transforms; \(H\)-functions PDFBibTeX XMLCite \textit{V. S. Kiryakova}, J. Comput. Appl. Math. 118, No. 1--2, 241--259 (2000; Zbl 0966.33011) Full Text: DOI References: [1] Agarwal, R. P., A propos d’une note de M. Pierre Humbert, C.R. Acad. Sci. Paris, 236, 2031-2032 (1953) · Zbl 0051.30801 [2] Dimovski, I., Operational calculus for a differential operator, C.R. Acad. Bulg. Sci., 27, 1, 513-516 (1966) · Zbl 0188.43002 [6] Dzrbashjan, M. M., On the integral representation and uniqueness of some classes of entire functions (in Russian), Dokl. AN SSSR, 85, 1, 29-32 (1952) [7] Dzrbashjan, M. M., On the integral transformations generated by the generalized Mittag-Leffler function (in Russian), Izv. AN Arm. SSR, 13, 3, 21-63 (1960) [10] Gelfond, A. O.; Leontiev, A. F., On a generalization of the Fourier series (in Russian), Mat. Sbornik, 29, 71, 477-500 (1951) [11] Gorenflo, R.; Kilbas, A. A.; Rogozin, S., On the generalized Mittag-Leffler type functions, Integral Transforms Special Functions, 7, 3-4, 215-224 (1998) · Zbl 0935.33012 [12] Gorenflo, R.; Mainardi, F., Fractional calculus: integral and differential equations of fractional order, (Carpinteri, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer: Springer Wien and New York), 223-276 · Zbl 1438.26010 [13] Kiryakova, V., Generalized \(H_{m,m}^{m,0\) · Zbl 0713.33006 [15] Kiryakova, V., All the special functions are fractional differintegrals of elementary functions, J. Phys. A, 30, 5083-5103 (1997) · Zbl 0928.33010 [16] Kiryakova, V.; Al-Saqabi, B. N., Transmutation method for solving Erdélyi-Kober fractional differintegral equations, J. Math. Anal. Appl., 211, 347-364 (1997) · Zbl 0879.45005 [17] Luchko, Yu.; Yakubovich, S., Operational calculi for the generalized fractional differential operator and applications, Math. Balkanica, 4, 2, 119-130 (1990) · Zbl 0830.44004 [21] Obrechkoff, N., On some integral representations of real functions on the real half-line (in Bulg.), Izv. Mat. Inst. (Sofia), 1, 2, 3-33 (1958) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.