On a Unification of Generalized Mittag-Leffler Function and Family of Bessel Functions

Affiliation(s)

Department of Mathematical Sciences, Faculty of Applied Sciences, Charotar University of Science and Technology, Changa, India.

Department of Mathematics, Faculty of Science, The Maharaja Sayajirao University of Baroda, Vadodara, India.

Department of Mathematical Sciences, Faculty of Applied Sciences, Charotar University of Science and Technology, Changa, India.

Department of Mathematics, Faculty of Science, The Maharaja Sayajirao University of Baroda, Vadodara, India.

ABSTRACT

In the present work, a unification of certain functions of mathematical physics is proposed and its properties are studied. The proposed function unifies Lommel function, Struve function, the Bessel-Maitland function and its generalization, Dotsenko function, generalized Mittag-Leffler function etc. The properties include absolute and uniform convergence, differential recurrence relation, integral representations in the form of Euler-Beta transform, Mellin-Barnes transform, Laplace transform and Whittaker transform. The special cases namely the generalized hypergeometric function, generalized Laguerre polynomial, Fox H-function etc. are also obtained.

Cite this paper

J. Prajapati, B. Dave and B. Nathwani, "On a Unification of Generalized Mittag-Leffler Function and Family of Bessel Functions,"*Advances in Pure Mathematics*, Vol. 3 No. 1, 2013, pp. 127-137. doi: 10.4236/apm.2013.31017.

J. Prajapati, B. Dave and B. Nathwani, "On a Unification of Generalized Mittag-Leffler Function and Family of Bessel Functions,"

References

[1] A. K. Shukla and J. C. Prajapti, “On a Generlization of Mittag-Leffler Functions 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

[2] G. Mittag-Leffler, “Sur la Nouvelle Fonction Eα(x),” Comptes Rendus de l’Academie des Sciences Paris, Vol. 137, 1903, pp. 554-558.

[3] A. Wiman, “über die Nullstellen der Funktionen Eα(x),” Acta Mathematica, Vol. 29, No. 1, 1905, pp. 217-234. doi:10.1007/BF02403204

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

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

[6] 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

[7] T. O. Salim, “Some Properties Relating to the Generalized Mittag-Leffler Function,” Advances in Applied Mathematical Analysis, Vol. 4, No. 1, 2009, pp. 21-30.

[8] H. J. Haubold, A. M. Mathai and R. K. Saxena, “The H-Function: Theory and Applications,” Publication No. 37 of Centre for Mathematical Sciences, Pala Campus, 2008.

[9] Y. L. Luke, “The Special Functions and their approximations,” Academic Press, New York, London, 1969.

[10] I. N. Sneddon, “The Use of Integral Transforms,” McGraw-Hill Book Company, New York, 1972.

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

[12] H. M. Srivastava and H. L. Manocha, “A Treatise on Generating Functions,” Ellis Horwood Ltd., Chichester, 1984.

[1] A. K. Shukla and J. C. Prajapti, “On a Generlization of Mittag-Leffler Functions 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

[2] G. Mittag-Leffler, “Sur la Nouvelle Fonction Eα(x),” Comptes Rendus de l’Academie des Sciences Paris, Vol. 137, 1903, pp. 554-558.

[3] A. Wiman, “über die Nullstellen der Funktionen Eα(x),” Acta Mathematica, Vol. 29, No. 1, 1905, pp. 217-234. doi:10.1007/BF02403204

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

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

[6] 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

[7] T. O. Salim, “Some Properties Relating to the Generalized Mittag-Leffler Function,” Advances in Applied Mathematical Analysis, Vol. 4, No. 1, 2009, pp. 21-30.

[8] H. J. Haubold, A. M. Mathai and R. K. Saxena, “The H-Function: Theory and Applications,” Publication No. 37 of Centre for Mathematical Sciences, Pala Campus, 2008.

[9] Y. L. Luke, “The Special Functions and their approximations,” Academic Press, New York, London, 1969.

[10] I. N. Sneddon, “The Use of Integral Transforms,” McGraw-Hill Book Company, New York, 1972.

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

[12] H. M. Srivastava and H. L. Manocha, “A Treatise on Generating Functions,” Ellis Horwood Ltd., Chichester, 1984.