Wright Type Hypergeometric Function and Its Properties

Affiliation(s)

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.

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.

Department of Applied Mathematics and Humenities, S.V. National Institute of Technology, Surat, India.

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 _{2}*F*_{1}(*a*,*b*;*c*;*z*). The principal aim of this paper is to study the various properties of this Wright type hypergeometric function _{2}*R*_{1}(*a*,*b*;*c*;*τ*;*z*); which includes differentiation and integration, representation in terms of * _{p}F_{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.

Cite this paper

S. Rao, J. Prajapati and A. Shukla, "Wright Type Hypergeometric Function and Its Properties,"*Advances in Pure Mathematics*, Vol. 3 No. 3, 2013, pp. 335-342. doi: 10.4236/apm.2013.33048.

S. Rao, J. Prajapati and A. Shukla, "Wright Type Hypergeometric Function and Its Properties,"

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

[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