×

zbMATH — the first resource for mathematics

Generalized Mittag-Leffler function and generalized fractional calculus operators. (English) Zbl 1047.33011
The paper studies the function \[ E^\gamma_{\rho,\mu}(z)= \sum^\infty_{k=0} {(\gamma)_k\over \Gamma(\rho k+\mu)k!}, \] where \(\rho\), \(\mu\), and \(\gamma\) are complex parameters with \(\text{Re}(\rho)> 0\). This is a generalization of the classical Mittag-Leffler function \(E_{\rho,\mu}(z)\) as well as a generalization of the Kummer confluent hypergeometric function \(\Phi(\gamma,\mu,z)\).
Reviewer: Kehe Zhu (Albany)

MSC:
33E12 Mittag-Leffler functions and generalizations
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
26A33 Fractional derivatives and integrals
47B38 Linear operators on function spaces (general)
47G10 Integral operators
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Al-Bassam M. A., J. Fract. Calc. 7 pp 69– (1995)
[2] Dzherbashyan M. M., Integral Transforms and Representations of Functions in Complex Domain. (1966)
[3] Dzherbashyan M. M., Harmonic Analysis and Boundary Value Problems in the Complex Domain. (1993)
[4] Erdélyi A., Higher Transcendental Functions (1953) · Zbl 0051.30303
[5] Erdélyi A., Higher Transcendental Functions (1953) · Zbl 0051.30303
[6] Erdélyi A., Higher Transcendental Functions (1955) · Zbl 0064.06302
[7] Gorenflo R., Dokl. Nats. Akad. Nauk Belarusi 42 pp 34– (1998)
[8] DOI: 10.1080/10652469808819200 · Zbl 0935.33012
[9] DOI: 10.1080/10652469708819125 · Zbl 0887.44003
[10] Gorenflo R., Int. J. Math. Stat. Sci. 6 pp 179– (1997)
[11] DOI: 10.1016/S0377-0427(00)00288-0 · Zbl 0973.35012
[12] Hadid S. B., Panamer. Math. J. 6 pp 57– (1996)
[13] Kilbas A. A., Differential and Integral Equations 8 pp 993– (1995) · Zbl 0970.45003
[14] DOI: 10.1080/10652469608819121 · Zbl 0876.26007
[15] Kilbas A. A., Differ. Uravn. 33 pp 195– (1997)
[16] Konhauser J. D. E., Pacific. J. Math. 21 pp 303– (1967)
[17] Luchko Yu. F., Acta Math. Vietnam 24 pp 207– (1999)
[18] DOI: 10.1016/0898-1221(95)00031-S · Zbl 0824.44011
[19] Luchko Yu. F., Math. Balkanica., (N.S.) 4 pp 119– (1990)
[20] Luchko Yu. F., Differ. Uravn. 30 pp 269– (1994)
[21] Mathai A. M., The H-Function with Applications in Statistics and Other Disciplines. (1978) · Zbl 0382.33001
[22] Mittag-Leffler G. M., C.R. Acad. Sci. Paris 137 pp 554– (1903)
[23] DOI: 10.1007/BF02403200 · JFM 36.0469.02
[24] Prabhakar T. R., Yokohama. Math. J. 19 pp 7– (1971)
[25] Prudnikov A. P., Integrals and Series, Vol. 3 More Special Functions (1990) · Zbl 0967.00503
[26] DOI: 10.1080/10652469808819189 · Zbl 0933.26001
[27] Saigo M., Differ. Uravn. 36 pp 168– (2000)
[28] Samko S. G., Fractional Integrals and Derivatives. Theory and Applications (1993)
[29] Srivastava H. M., The H-Function of One and Two Variables with Applications. (1982)
[30] Titchmarsh E. C., Introduction to the Theory of Fourier Integrals (1986)
[31] DOI: 10.1007/BF02403202 · JFM 36.0471.01
[32] DOI: 10.1112/jlms/s1-10.40.286 · Zbl 0013.02104
[33] DOI: 10.1112/plms/s2-46.1.389 · Zbl 0025.40402
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.