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.
[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.