zbMATH — the first resource for mathematics

Certain integrals involving generalized Mittag-Leffler functions. (English) Zbl 1325.44003
Summary: In this paper, we aim to establish certain new integrals involving the generalized Mittag-Leffler functions which are associated with the Laguerre polynomials. All the results derived here are of general character and can yield a number of known and new results in the theory of Mittage-Leffler functions. Furthermore, we also established the expansion formulas for the generalized Mittag-Leffler functions and the Laguerre polynomials.

44A15 Special integral transforms (Legendre, Hilbert, etc.)
65A05 Tables in numerical analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
Full Text: DOI
[1] Agarwal, RP, A propos dúne note de M. pierre Humbert, CR Acad Sci Paris, 236, 2031-2032, (1953) · Zbl 0051.30801
[2] Humbert, P, Quelques resultants retifs a la fonction de Mittag-Leffler, CR Acad Sci Paris, 236, 1467-1468, (1953) · Zbl 0050.10404
[3] Humbert, P; Agarwal, RP, Sur la fonction de Mittag-Leffler et quelques unes de ses generalizations, Bull Sci Math (Ser. II), 77, 180-185, (1953) · Zbl 0052.06402
[4] Hilfer R (ed) (2000) Applications of fractional calculus in physics. World Scientific, Singapore · Zbl 0998.26002
[5] Lang KR (1999a) Astrophysical formulae, vol. 1: radiation, gas processes and high-energy astrophysics, 3rd edition, revised edition. Springer, New York · Zbl 0957.85001
[6] Carlitz, L, Some explanation and convolution formula related to mac mohan’s master theorems, SIAMS J Math Anal, 8, 320-336, (1977) · Zbl 0359.05003
[7] Mittag-Leffler, GM, Une generalisation de lintegrale de Laplace-Abel, CR Acad Sci Paris (Ser. II), 137, 537-539, (1903) · JFM 34.0434.02
[8] Mittag-Leffler, GM, Sur la representation analytiqie dúne fonction monogene (cinquieme note), Acta Math, 29, 101-181, (1905) · JFM 36.0469.02
[9] Prabhakar, TR, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math J, 19, 7-15, (1971) · Zbl 0221.45003
[10] Saxena, RK; Mathai, AM; Haubold, HJ, On fractional kinetic equations, Astrophys Space Sci, 282, 281-287, (2002)
[11] Srivastava, HM; Tomovski, Z, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl Math Comput, 211, 198-210, (2009) · Zbl 1432.30022
[12] Wiman, A, Über den fundamental satz in der theorie der funcktionen, \(E_{α }(x)\), Acta Math, 29, 191-201, (1905) · JFM 36.0471.01
[13] Ahmed, A, On the generalized fractional integrals of the generalized Mittag-Leffler function, SpringerPlus, 3, 198, (2014)
[14] Özarslan MA, Yilmaz B (2014) The extended Mittag-Leffer function and its properties. J Inequal Appl. doi:10.1186/1029-242X-2014-85 · Zbl 1432.30022
[15] Erdélyi A, Magnus W, Oberhettinger F, Tricomi FG (1955) Higher transcendental functions, vol 3. McGraw-Hill, New York · Zbl 0064.06302
[16] Dzherbashyan MM (1966) Integral transforms and representations of functions in the complex plane. Nauka, Moscow (in Russian)
[17] Shukla, AK; Prajapati, JC, On a generalization of Mittag-Leffler function and its properties, J Math Anal Appl, 336, 797-811, (2007) · Zbl 1122.33017
[18] Gautam S (2008) Investigations in fractional differential operators of arbitrary order and their applications to special functions of one and several variables. Ph. D. Thesis, University of Kota, Kota, India · JFM 34.0434.02
[19] Saxena, RK; Kalla, SL; Saxena, R, Multivariable analogue of generalized Mittag-Leffler function, Integral Transforms Spec Funct, 22, 533-548, (2011) · Zbl 1275.33030
[20] Srivastava HM, Karlsson PW (1985) Multiple Gaussian hypergeometric Series. Ellis Horwood series: mathematics and its applications. Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York · Zbl 0052.06402
[21] Prabhakar, TR; Suman, R, Some results on the polynomials \(L^{α ,β }_n(x)\), Rocky Mt J Math, 8, 751-754, (1978) · Zbl 0398.33009
[22] Rainville ED (1960) Special functions. Macmillan, New York · Zbl 0092.06503
[23] Srivastava, HM, A multilinear generating function for the konhauser sets of bi-orthogonal polynomials suggested by the Laguerre polynomials, Pac J Math, 117, 183-191, (1985) · Zbl 0535.33003
[24] Shukla, AK; Prajapati, JC; Salehbhai, IA, On a set of polynomials suggested by the family of konhauser polynomial, Int J Math Anal, 3, 637-643, (2009) · Zbl 1198.33006
[25] Khan, MA, On some properties of the generalized Mittag-Leffler function, SpringerPlus, 2, 337, (2013)
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.