Wright Type Hypergeometric Function and Its Properties

Snehal B. Rao, Jyotindra C. Prajapati, Ajay K. Shukla

Department of Applied Mathematics and Humenities, S.V. National Institute of Technology, Surat, India.
Department of Applied Mathematics, The M.S. University of Baroda, Vadodara, India.
Department of Mathematical Sciences, Faculty of Applied Sciences, Charotar University of Science and Technology, Anand, India.

DOI: 10.4236/apm.2013.33048                

Abstract

Let s and z be complex variables, Γ(s) be the Gamma function, and for any complex v be the generalized Pochhammer symbol. Wright Type Hypergeometric Function is defined (Virchenko et al. [1]), as: where which is a direct generalization of classical Gauss Hypergeometric Function ₂F₁(a,b;c;z). The principal aim of this paper is to study the various properties of this Wright type hypergeometric function ₂R₁(a,b;c;τ;z); which includes differentiation and integration, representation in terms of _pF_q and in terms of Mellin-Barnes type integral. Euler (Beta) transforms, Laplace transform, Mellin transform, Whittaker transform have also been obtained; along with its relationship with Fox H-function and Wright hypergeometric function.

Keywords

Euler Transform; Fox H-Function; Wright Type Hypergeometric Function; Laplace Transform; Mellin Transform; Whittaker Transform; Wright Hypergeometric Function

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	N. Virchenko, S. L. Kalla and A. Al-Zamel, “Some Results on a Generalized Hypergeometric Function,” Integral Transforms and Special Functions, Vol. 12, No. 1, 2001, pp. 89-100. doi:10.1080/10652460108819336
[2]	E. D. Rainville, “Special Functions,” The Macmillan Company, New York, 1960.
[3]	A. Erdelyi, et al., “Higher Transcendental Functions,” McGaw-Hill, New York, 1953-1954.
[4]	E. M. Wright, “On the Coefficient of Power Series Having Exponential Singularities,” Journal London Mathematical Society, Vol. s1-8, No. 1, 1933, pp. 71-79. doi:10.1112/jlms/s1-8.1.71
[5]	M. Dotsenko, “On Some Applications of Wright’s Hypergeometric Function,” Comptes Rendus de l’Académie Bulgare des Sciences, Vol. 44, 1991, pp. 13-16.
[6]	V. Malovichko, “On a Generalized Hypergeometric Function and Some Integral Operators,” Mathematical Physics, Vol. 19, 1976, pp. 99-103.
[7]	L. Galue, A. Al-Zamel and S. L. Kalla, “Further Results on Generalized Hypergeometric Functions,” Applied Mathematics and Computation, Vol. 136, No. 1, 2003, pp. 17-25. doi:10.1016/S0096-3003(02)00014-0
[8]	I. N. Sneddon, “The Use of Integral Transforms,” Tata McGraw-Hill Publication Co. Ltd., New Delhi, 1979.
[9]	H. M. Srivastava and H. L. Manocha, “A Treatise on Generating Functions,” John Wiley and Sons/Ellis Horwood, New York/Chichester, 1984.
[10]	A. M. Mathai, R. K. Saxena and H. J. Haubold, “The H-Function,” Springer, Berlin, 2010. doi:10.1007/978-1-4419-0916-9