The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus. (English) Zbl 1189.33034

Summary: This paper is a short description of our recent results on an important class of the so-called “Special Functions of Fractional Calculus” (SF of FC), which became important as solutions of fractional order (or multi-order) differential and integral equations, control systems and refined mathematical models of various physical, chemical, economical, management, bioengineering phenomena. Basically, under “SF of FC” we mean the Wright generalized hypergeometric function \(p\Psi q\), as a special case of the Fox \(H\)-function. We have introduced and studied the multi-index Mittag-Leffler functions as their typical representatives, including many interesting special cases that have already proven their usefulness in FC and its applications. Some new results are also presented and open problems are discussed.


33E10 Lamé, Mathieu, and spheroidal wave functions
26A33 Fractional derivatives and integrals
33-02 Research exposition (monographs, survey articles) pertaining to special functions
Full Text: DOI


[1] ()
[2] Mathai, A.M.; Saxena, R.K., The \(H\)-function with applications in statistics and other disciplines, (1978), Wiley East. Ltd. New Delhi · Zbl 0382.33001
[3] Srivastava, H.M.; Kashyap, B.R.K., Special functions in queuing theory and related stohastic processes, (1981), Acad. Press New York · Zbl 0492.60089
[4] Srivastava, H.M.; Gupta, K.C.; Goyal, S.P., The \(H\)-functions of one and two variables with applications, (1982), South Asian Publs New Delhi · Zbl 0506.33007
[5] Marichev, O.I., Handbook of integral transforms of higher transcendental functions: theory and algorithmic tables, (1983), Ellis Horwood & Halsted Press New York · Zbl 0494.33001
[6] Prudnikov, A.; Brychkov, Yu.; Marichev, O., Integrals and series, vol. 3: some more special functions, (1992), Gordon & Breach New York · Zbl 0786.44003
[7] Kiryakova, V., Generalized fractional calculus and applications, (1994), Longman & J. Wiley Harlow, New York · Zbl 0882.26003
[8] Podlubny, I., Fractional differential equations, (1999), Acad. Press San Diego · Zbl 0918.34010
[9] ()
[10] Kilbas, A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier Amsterdam etc. · Zbl 1092.45003
[11] Samko, S.; Kilbas, A.; Marichev, O., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach Switzerland etc. · Zbl 0818.26003
[12] Lavoie, J.L.; Osler, T.J.; Tremblay, R., Fractional derivatives and special functions, SIAM rev., 18, 240-268, (1976) · Zbl 0324.44002
[13] Kiryakova, V., All the special functions are fractional differintegrals of elementary functions, J. phys. A: math. gen., 30, 5085-5103, (1997) · Zbl 0928.33010
[14] Gorenflo, R.; Mainardi, F., Fractional calculus: integral and differential equations of fractional order, (), 223-278
[15] Dzrbashjan, M., Integral transforms and representations of functions in the complex domain, (1966), Nauka Moscow, (in Russian)
[16] Mainardi, F., Applications of fractional calculus in mechanics, (), 309-334 · Zbl 1113.26303
[17] Al-Saqabi, B.; Kiryakova, V., Transmutation method for solving erdélyi – kober fractional differintegral equations, J. math. anal. appl., 211, 347-364, (1997) · Zbl 0879.45005
[18] Gorenflo, R.; Kilbas, A.A.; Rogozin, S.V., On the generalized mittag – leffler type function, Integr. transform. spec. funct., 7, 215-224, (1998) · Zbl 0935.33012
[19] Kilbas, A.A.; Saigo, M.; Saxena, R.K., Generalizad mittag – leffler functions and generalized fractional calculus operators, Integr. transform. spec. funct., 15, 31-49, (2004) · Zbl 1047.33011
[20] Kiryakova, V., Multiindex mittag – leffler functions, related gelfond – leontiev operators and Laplace type integral transforms, Fract. calc. appl. anal., 2, 445-462, (1999) · Zbl 1111.33300
[21] Kiryakova, V., Multiple (multiindex) mittag – leffler functions and relations to generalized fractional calculus, J. comput. appl. math., 118, 214-259, (2000) · Zbl 0966.33011
[22] Luchko, Yu., Operational method in fractional calculus, Fract. calc. appl. anal., 2, 463-488, (1999) · Zbl 1030.26009
[23] Delerue, P., Sur le calcul symbolique à \(n\) variables et fonctions hyperbesseliennes (II), Ann. soc. sci. bruxelles, ser. 1, 3, 229-274, (1953) · Zbl 0053.37201
[24] Saxena, R.K.; Kalla, S.L.; Kiryakova, V., Relations connecting multiindex mittag – leffler functions and riemann – liouville fractional calculus, (), 363-386 · Zbl 1050.33016
[25] Ali, I.; Kiryakova, V.; Kalla, S.L., Solutions of fractional multi-order integral and differential equations using a Poisson-type transform, J. math. anal. appl., 269, 172-199, (2002) · Zbl 1026.45009
[26] Dzrbashjan, M., On the integral transformations generated by the generalized mittag – leffler function, Izv. akad. nauk armen. SSR, 13, 21-63, (1960), (in Russian)
[27] Mainardi, F.; Tomirotti, M., On a special function arising in the fractional diffusion-wave equation, (), 171-183 · Zbl 0921.33010
[28] de Oteiza, M.M.M.; Kalla, S.L.; Conde, S., Un estudio sobre la función lommel – maitalnd, Rev. tecn. fac. ingr. univ. del zulia, 9, 33-40, (1986)
[29] Paneva-Konovska, J., Index-asymptotic formulae for wright’s generalized Bessel functions, Math. sci. res. J., 11, 424-431, (2007) · Zbl 1122.30001
[30] Paneva-Konovska, J., Cauchy – hadamard, Abel and tauber type theorems for series in generalized bessel – maitland functions, C. R. acad. bulgare sci., 61, 9-14, (2008) · Zbl 1164.30001
[31] Dimovski, I.; Kiryakova, V., The obrechkoff integral transform: properties and relation to a generalized fractional calculus, Numer. funct. anal. optim., 21, 121-144, (2000) · Zbl 0956.44002
[32] S. Spirova, V. Hernandez, Explicit solutions to \(n\)-th order Bessel-Clifford integral and differential equations, in: Proc. XXIV Summer School Application of Mathematics in Engineering, Sozopol, 1998, pp. 124-128
[33] Gelfond, A.O.; Leontiev, A.F., On a generalization of the Fourier series, Mat. sbornik, 29, 477-500, (1951), (in Russian)
[34] Al-Musallam, F.; Kiryakova, V.; Kim Tuan, Vu, A multi-index borel – dzrbashjan transform, Rocky mountain J. math., 32, 409-428, (2002) · Zbl 1035.44002
[35] Ishteva, M.; Boyadjiev, L., On the C-Laguerre functions, Compt. rend. acad. bulgare sci., 58, 1019-1024, (2005) · Zbl 1093.33005
[36] Mirevski, S.P.; Boyadjiev, L.; Scherer, R., On the riemann – liouville fractional calculus, \(g\)-Jacobi functions and F-Gauss functions, Appl. math. comput., 187, 315-325, (2007) · Zbl 1117.33010
[37] Kalla, S.L.; Galue, L., Generalized fractional calculus based upon composition of some basic operators, (), 145-178 · Zbl 0790.26004
[38] Caputo, M.; Mainardi, F., A new dissipation model based on memory mechanism, Pure appl. geophys., 91, 134-147, (1971), Repr. in: Fract. Calc. Appl. Anal. 10 (2007) 309-324
[39] I. Podlubny, M. Kacenak, Mittag-Leffler function, Matlab Central File Exchange, File ID: #8738 (17 Oct. 2005). Available at http://www.mathworks.com/matlabcentral/fileexchange/8738
[40] Gorenflo, R.; Loutchko, J.; Luchko, Yu., Computation of the mittag – leffler function \(E_{\alpha, \beta}\) and its derivatives, Fract. calc. appl. anal., 5, 491-518, (2002) · Zbl 1027.33016
[41] Diethelm, K.; Ford, N.; Freed, A.; Luchko, Yu., Algorithms for the fractional calculus: A selection of numerical methods, Comput. methods appl. mech. eng., 194, 743-773, (2005) · Zbl 1119.65352
[42] Hilfer, R.; Seybold, H.J., Computation of the generalized mittag – leffler function and its inverse in the complex plane, Integr. transform. spec. funct., 17, 637-652, (2006) · Zbl 1096.65024
[43] Luchko, Yu., Algorithms for evaluation of the wright function for the real arguments’ values, Fract. calc. appl. anal., 11, 57-75, (2008) · Zbl 1145.33005
[44] V. Kiryakova, The special functions of fractional calculus as generalized fractional calculus operators of some basic functions, Comput. Math. Appl. (2009), in press (doi:10.1016/j.camwa.2009.05.014) · Zbl 1189.26007
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.