Certain formulas involving a multi-index Mittag-Leffler function
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Since Mittag-Leffler introduced the so-called Mittag-Leffler function, a number of its extensions have been investigated due mainly to their applications in a variety of research subjects. Shukla and Prajapati presented a lot of formulas involving a generalized Mittag-Leffler function in a systematic manner. Motivated mainly by Shukla and Prajapati’s work, we aim to investigate a generalized multi-index Mittag-Leffler func- tion and, among possible numerous formulas, choose to present several for- mulas involving this generalized multi-index Mittag-Leffler function such as a recurrence formula, derivative formula, three integral transformation formulas. The results presented here, being general, are pointed out to reduce to yield relatively simple formulas including known ones.
Abstract
Since Mittag-Leffler introduced the so-called Mittag-Leffler function, a number of its extensions have been investigated due mainly to their applications in a variety of research subjects. Shukla and Prajapati presented a lot of formulas involving a generalized Mittag-Leffler function in a systematic manner. Motivated mainly by Shukla and Prajapati’s work, we aim to investigate a generalized multi-index Mittag-Leffler func- tion and, among possible numerous formulas, choose to present several for- mulas involving this generalized multi-index Mittag-Leffler function such as a recurrence formula, derivative formula, three integral transformation formulas. The results presented here, being general, are pointed out to reduce to yield relatively simple formulas including known ones.
참고문헌 (22)
- C. Fox, "The asymptotic expansion of generalized hypergeometric functions", Proc. London Math. Soc, 27(2), pp.389~400 (1928)
- A. A. Kilbas, "Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, Vol. 204" (2006), Elsevier (North-Holland) Science Publishers
- M. Mathai, "Mittag-Leffler functions to pathway model to Tsallis statistics", Integral Transforms Spec. Funct, 21(11), pp.867~875 (2010)
- G. M. Mittag-Leffler, "Sur la representation analytique d'une branche uniforme d'une fonction monogene", Acta Math, 29, pp.101~181 (1905)
- G. M. Mittag-Leffler, "Sur la nouvelle fonction Eα(x)", C. R. Acad. Sci. Paris, 137, pp.554~558 (1903)
- R. N. Pillai, "On Mittag-Leffler functions and related distributions", Ann. Inst. Statist. Math, 42(1), pp.157~161 (1990)
- T. R. Prabhakar, "A singular integral equation with a generalized Mittag-Leffler function in the kernel", Yokohama Math. J, 19, pp.7~15 (1971)
- E. D. Rainville, "Special Functions" (1971), Chelsea Publishing Company
- R. K. Saxena, "Further results on generalized Mittag-Leffler functions of fractional calculus", J. Fract. Calc, 39, pp.29~41 (2010)
- R. K. Saxena, "N-fractional calculus of generalized Mittag-Leffler functions", J. Fract. Calc, 37, pp.43~52 (2010)
- R. K. Saxena, "Dirichlet averages of generalized multi-index Mittag-Leffler functions", Armen. J. Math, 3(4), pp.174~187 (2010)
- A. K. Shukla, "On a generalization of Mittag-Leffler functions and its properties", J. Math. Anal. Appl, 336(2), pp.797~811 (2007)
- H. M. Srivastava, "Zeta and q-Zeta Functions and Associated Series and Integrals" (2012), Elsevier Science Publishers
- H. M. Srivastava, "Seires Associated with the Zeta and Related Functions" (2001), Kluwer Academic Publishers
- H. M. Srivastava, "A general fractional differential equation associated with an integral operator with the H-function in the kernel", Russian J. Math. Phys, 22(1), pp.112~126 (2015)
- H. M. Srivastava, "Multiple Gaussian Hypergeometric Series" (1985), Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons
- H. M. Srivastava, "Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel", Appl. Math. Comput, 211(1), pp.198~210 (2009)
- E. T. Whittaker., "A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; With an Account of the Principal Transcendental Functions, Fourth edition" (1963), Cambridge University Press
- A. Wiman, "Uber den fundamentalsatz in der theorie der funktionen Eα(x)", Acta Math, 29, pp.191~201 (1905)
- E. M. Wright, "The asymptotic expansion of the generalized hypergeometric functions", J. London Math. Soc, 10, pp.286~293 (1935)
- E. M. Wright, "The asymptotic expansion of integral functions defined by Taylor series", Philos. Trans. Roy. Soc. London A, 238, pp.423~451 (1940)
- E. M. Wright, "The asymptotic expansion of the generalized hypergeometric function II", Proc. London Math. Soc, 46(2), pp.389~408 (1940)