zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 pΨq, 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.
33E10Lamé, Mathieu, and spheroidal wave functions
26A33Fractional derivatives and integrals (real functions)
33-02Research monographs (special functions)
[1], Higher transcendental functions 1–3 (1953)
[2]Mathai, A. M.; Saxena, R. K.: The H-function with applications in statistics and other disciplines, (1978) · Zbl 0382.33001
[3]Srivastava, H. M.; Kashyap, B. R. K.: Special functions in queuing theory and related stohastic processes, (1981)
[4]Srivastava, H. M.; Gupta, K. C.; Goyal, S. P.: The H-functions of one and two variables with applications, (1982) · Zbl 0506.33007
[5]Marichev, O. I.: Handbook of integral transforms of higher transcendental functions: theory and algorithmic tables, (1983) · Zbl 0494.33001
[6]Prudnikov, A.; Brychkov, Yu.; Marichev, O.: Integrals and series, vol. 3: some more special functions, (1992)
[7]Kiryakova, V.: Generalized fractional calculus and applications, (1994)
[8]Podlubny, I.: Fractional differential equations, (1999)
[9], Applications of fractional calculus in physics (2000)
[10]Kilbas, A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006)
[11]Samko, S.; Kilbas, A.; Marichev, O.: Fractional integrals and derivatives: theory and applications, (1993) · 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 · doi:10.1137/1018042
[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 · doi:10.1088/0305-4470/30/14/019
[14]Gorenflo, R.; Mainardi, F.: Fractional calculus: integral and differential equations of fractional order, Fractals and fractional calculus in continuum mechanics, 223-278 (1997)
[15]Dzrbashjan, M.: Integral transforms and representations of functions in the complex domain, (1966)
[16]Mainardi, F.: Applications of fractional calculus in mechanics, Transform methods and special functions, varna’96, 309-334 (1996) · 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 · doi:10.1006/jmaa.1997.5469
[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 · doi:10.1080/10652469808819200
[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 · doi:10.1080/10652460310001600717
[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 · doi:10.1016/S0377-0427(00)00292-2
[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, Algebras, groups and geometries 20, 363-386 (2003) · 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 · doi:10.1016/S0022-247X(02)00012-4
[26]Dzrbashjan, M.: On the integral transformations generated by the generalized Mittag–Leffler function, Izv. akad. Nauk armen. SSR 13, 21-63 (1960)
[27]Mainardi, F.; Tomirotti, M.: On a special function arising in the fractional diffusion-wave equation, Transform methods and special functions, sofia’94, 171-183 (1994) · 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 · doi:10.1080/01630560008816944
[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)
[34]Al-Musallam, F.; Kiryakova, V.; Tuan, Vu Kim: A multi-index Borel–dzrbashjan transform, Rocky mountain J. Math. 32, 409-428 (2002) · Zbl 1035.44002 · doi:10.1216/rmjm/1030539678 · doi:http://math.la.asu.edu/~rmmc/rmj/Vol32-2/CONT32-2/CONT32-2.html
[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 · doi:10.1016/j.amc.2007.01.035
[37]Kalla, S. L.; Galue, L.: Generalized fractional calculus based upon composition of some basic operators, Recent advances in fractional calculus, 145-178 (1993) · Zbl 0790.26004
[38]Caputo, M.; Mainardi, F.: A new dissipation model based on memory mechanism, Pure appl. Geophys. 91, 134-147 (1971)
[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 Eα,β 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 · doi:10.1016/j.cma.2004.06.006
[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 · doi:10.1080/10652460600725341
[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)