Decomposition of Generalized Mittag-Leffler Function and Its Properties

ABSTRACT

The principal aim of the paper is devoted to the study of some special properties of the E_{α,β}^{γ,q}(Z) function for α =1/n . Authors defined the decomposition of the function E_{α,β}^{γ,q}(Z) in the form of truncated power series as Equations (1.7), (1.8) and their various properties including Integral representation, Derivative, Inequalities and their several special cases are obtained. Some new results are also established for the function E_{α,β}^{γ,q}(Z).

The principal aim of the paper is devoted to the study of some special properties of the E

KEYWORDS

Generalized Mittag-Leffler Function; Beta Function; Gamma Function; Integral Representation

Generalized Mittag-Leffler Function; Beta Function; Gamma Function; Integral Representation

Cite this paper

J. Prajapati and A. Shukla, "Decomposition of Generalized Mittag-Leffler Function and Its Properties,"*Advances in Pure Mathematics*, Vol. 2 No. 1, 2012, pp. 8-14. doi: 10.4236/apm.2012.21003.

J. Prajapati and A. Shukla, "Decomposition of Generalized Mittag-Leffler Function and Its Properties,"

References

[1] R. Gorenflo, A. A. Kilbas and S. V. Rogosin, “On the Generalised Mittag-Leffler Type Function,” Integral Transforms and Special Functions, Vol. 7, No. 3-4, 1998, pp. 215- 224. doi:10.1080/10652469808819200

[2] A. Wiman, “Uber de Fundamental Satz in der Theorie der Funktionen ,” Acta Mathematica, Vol. 29, No. 1, 1905, pp. 191-201. doi:10.1007/BF02403202

[3] T. R. Prabhakar, “A Singular Integral Equation with a Generalized Mittag-Leffler Function in the Kernel,” Yokohama Mathematical Journal, Vol. 19, 1971, pp. 7-15.

[4] R. Gorenflo and F. Mainardi, “On Mittag-Leffler Function in Fractional Evaluation Processes,” Journal of Computational and Applied Mathematics, Vol. 118, No. 1-2, 2000, pp. 283-299. doi:10.1016/S0377-0427(00)00294-6

[5] A. A. Kilbas and M. Saigo, “On Mittag-Leffler Type Function, Fractional Calculus Operators and Solution of Integral Equations,” Integral Transforms and Special Functions, Vol. 4, No. 4, 1996, pp. 355-370. doi:10.1080/10652469608819121

[6] A. A. Kilbas, M. Saigo and R. K. Saxena, “Generalised Mittag-Leffler Function and Generalised Fractional Calculus Operators,” Integral Transforms and Special Functions, Vol. 15, No. 1, 2004, pp. 31-49. doi:10.1080/10652460310001600717

[7] K. S. Miller, “The Mittag-Leffler and Related Functions,” Integral Transforms and Special Functions, Vol. 1, No. 1, 1993, pp. 41-49. doi:10.1080/10652469308819007

[8] M. Saigo and A. A. Kilbas, “On Mittag-Leffler Type Function and Applications,” Integral Transforms and Special Functions, Vol. 7, No. 1-2, 1998, pp. 97-112. doi:10.1080/10652469808819189

[9] A. K. Shukla and J. C. Prajapati, “On a Generalization of Mittag-Leffler Function and Its Properties,” Journal of Mathematical Analysis and Applications, Vol. 336, No. 2, 2007, pp. 797-811. doi:10.1016/j.jmaa.2007.03.018

[10] M. Ikehata and S. Siltanen, “Electrical Impedance Tomography and Mittag-Leffler Function,” Inverse Problems, Vol. 20, No. 4, 2004, pp. 1325-1348. doi:10.1088/0266-5611/20/4/019

[11] E. D. Rainville, “Special Functions,” The Macmillan Company, New York, 1960.

[12] I. N. Sneddon, “The Use of Integral Transforms,” Tata McGraw-Hill Publication Co. Ltd., New Delhi, 1979.

[13] A. K. Shukla and J. C. Prajapati, “On Generalized Mittag-Leffler Type Function and Generated Integral Operator,” Mathematical Sciences Research Journal, Vol. 12, No. 12, 2008, pp. 283-290.

[1] R. Gorenflo, A. A. Kilbas and S. V. Rogosin, “On the Generalised Mittag-Leffler Type Function,” Integral Transforms and Special Functions, Vol. 7, No. 3-4, 1998, pp. 215- 224. doi:10.1080/10652469808819200

[2] A. Wiman, “Uber de Fundamental Satz in der Theorie der Funktionen ,” Acta Mathematica, Vol. 29, No. 1, 1905, pp. 191-201. doi:10.1007/BF02403202

[3] T. R. Prabhakar, “A Singular Integral Equation with a Generalized Mittag-Leffler Function in the Kernel,” Yokohama Mathematical Journal, Vol. 19, 1971, pp. 7-15.

[4] R. Gorenflo and F. Mainardi, “On Mittag-Leffler Function in Fractional Evaluation Processes,” Journal of Computational and Applied Mathematics, Vol. 118, No. 1-2, 2000, pp. 283-299. doi:10.1016/S0377-0427(00)00294-6

[5] A. A. Kilbas and M. Saigo, “On Mittag-Leffler Type Function, Fractional Calculus Operators and Solution of Integral Equations,” Integral Transforms and Special Functions, Vol. 4, No. 4, 1996, pp. 355-370. doi:10.1080/10652469608819121

[6] A. A. Kilbas, M. Saigo and R. K. Saxena, “Generalised Mittag-Leffler Function and Generalised Fractional Calculus Operators,” Integral Transforms and Special Functions, Vol. 15, No. 1, 2004, pp. 31-49. doi:10.1080/10652460310001600717

[7] K. S. Miller, “The Mittag-Leffler and Related Functions,” Integral Transforms and Special Functions, Vol. 1, No. 1, 1993, pp. 41-49. doi:10.1080/10652469308819007

[8] M. Saigo and A. A. Kilbas, “On Mittag-Leffler Type Function and Applications,” Integral Transforms and Special Functions, Vol. 7, No. 1-2, 1998, pp. 97-112. doi:10.1080/10652469808819189

[9] A. K. Shukla and J. C. Prajapati, “On a Generalization of Mittag-Leffler Function and Its Properties,” Journal of Mathematical Analysis and Applications, Vol. 336, No. 2, 2007, pp. 797-811. doi:10.1016/j.jmaa.2007.03.018

[10] M. Ikehata and S. Siltanen, “Electrical Impedance Tomography and Mittag-Leffler Function,” Inverse Problems, Vol. 20, No. 4, 2004, pp. 1325-1348. doi:10.1088/0266-5611/20/4/019

[11] E. D. Rainville, “Special Functions,” The Macmillan Company, New York, 1960.

[12] I. N. Sneddon, “The Use of Integral Transforms,” Tata McGraw-Hill Publication Co. Ltd., New Delhi, 1979.

[13] A. K. Shukla and J. C. Prajapati, “On Generalized Mittag-Leffler Type Function and Generated Integral Operator,” Mathematical Sciences Research Journal, Vol. 12, No. 12, 2008, pp. 283-290.