Extension of exton's hypergeometric function \(K_{16}\)

Ahmed Ali Atash; Maisoon Ahmed Kulib

doi:10.4067/S0719-06462021000300489

DOI:

https://doi.org/10.4067/S0719-06462021000300489

Abstract

The purpose of this article is to introduce an extension of Exton's hypergeometric function \(K_{16}\) by using the extended beta function given by OÌˆzergin et al. [11]. Some integral representations, generating functions, recurrence relations, transformation formulas, derivative formula and summation formulas are obtained for this extended function. Some special cases of the main results of this paper are also considered.

Keywords

Extended beta function , Extended Exton‘s function , Integral representations , Generating functions , Recurrence relation , Transformation formula , Derivative formula , Summation formula

Ahmed Ali Atash Department of Mathematics, Faculty of Education Shabwah, Aden University, Aden, Yemen.
Maisoon Ahmed Kulib Department of Mathematics, Faculty of Engineering, Aden University, Aden, Yemen.

Pages: 489–501
Date Published: 2021-12-01
Vol. 23 No. 3 (2021)

P. Agarwal, J. Choi and S. Jain, “Extended hypergeometric functions of two and three variables”, Commun. Korean Math. Soc., vol. 30, no. 4, pp. 403–414, 2015.

R. P. Agarwal, M. J. Luo and P. Agarwal, “On the extended Appell-Lauricella hypergeometric functions and their applications”, Filomat, vol. 31, no. 12, pp. 3693–3713, 2017.

A. Çetinkaya, I. O. KÄ±ymaz, P. Agarwal and R. Agarwal, “A comparative study on generating function relations for generalized hypergeometric functions via generalized fractional operators”, Adv. Difference Equ., vol. 2018, paper no. 156, pp. 1–11, 2018.

R. C. Singh Chandel and A. Tiwari, “Generating relations involving hypergeometric functions of four variables”, Pure Appl. Math. Sci., vol. 36, no. 1-2, pp. 15–25, 1991.

M. A. Chaudhry, A. Qadir, M. Rafique and S. M. Zubair, “Extension of Euler‘s beta function”, J. Comp. Appl. Math., vol. 78, no. 1, pp. 19–32, 1997.

M. A. Chaudhry, A. Qadir, H. M. Srivastava and R. B. Paris, “Extended hypergeometric and confluent hypergeometric functions”, Appl. Math. Comp., vol. 159, no. 2, pp. 589–602, 2004.

H. Exton, Multiple hypergeometric functions and applications, New York: Halsted Press, 1976.

H. Liu, “Some generating relations for extended Appell‘s and Lauricella‘s hypergeometric functions”, Rocky Mountain J. Math., vol. 44, no. 6, pp. 1987–2007, 2014.

Y. L. Luke, The special functions and their approximations, New York: Academic Press, 1969.

M. A. Özarslan and E. Özergin, “Some generating relations for extended hypergeometric functions via generalized fractional derivative operator”, Math. Comput. Modelling, vol. 52, no. 9-10, pp. 1825–1833, 2010.

E. Özergin, M. A. Özarslan, and A. Altin, “Extension of gamma, beta and hypergeometric functions”, J. Comp. Appl. Math., vol. 235, no. 16, pp. 4601–4610, 2011.

H. M. Srivastava and P. W. Karlsson, Multiple Gaussian hypergeometric Series, New York: Halsted Press, 1985.

H. M. Srivastava and H. L. Manocha, A treatise on generating functions, New York: Halsted Press, 1984.

X. Wang, “Recursion formulas for Appell functions”, Integral Transforms Spec. Funct., vol. 23, no. 6, pp. 421–433, 2012.