H_A^(τ_1,τ_2,τ_3 ) A Srivastava Hypergeometric Function

Year 2018, Volume: 6 Issue: 2, 1 - 9, 31.10.2018

Recep Şahin , Oğuz Yağcı

Abstract

Formulas and identities involving many well known special functions (such as the Gamma and Beta functions, Gauss hypergeometric function, and so on) play important roles in themselves and their diverse applications. In this paper, we will add τ1, τ2, τ3 parameters to the HA Srivastava hypergeometric function and we introduce new H τ1,τ2,τ3 A Srivastava’s triple τ -hypergeometric function. Then, we present some properties of the H τ1,τ2,τ3 A Srivastava’s triple τ -hypergeometric function In this paper, N, Z −, and C denote the sets of positive integers, negative integers, complex numbers, respectively. Also, N0 and Z  represent the sets of positive integers and complex numbers by excluding origin, respectively. N0 := N ∪ {0} and Z − 0:= Z − ∪ {0} . The classical Gamma function Γ(x) is defined by [13, 16, 17, 19, 20

Keywords

Srivastava hypergeometric funtion, integral representation, derivative formula

References

• [1] Agarwal, P., Certain properties of the generalized Gauss hypergeometric functions, Applied Mathematics & Information Sciences, 8(2014), no:5, 2315.
• [2] Appell Kampé de Fériet, P.J., Fonctions Hypergéométriques et Hypersphériques, Gauthier-Villars, Paris, 1926.
• [3] Al-Shammery, A.H. and Kalla, S.L., An extension of some hypergeometric functions of two variables, Rev. Acad. Canaria Cienc., 12 (2000), no. 1-2, 189-196.
• [4] Choi, J. and Agarwal, P., Certain generating functions involving Appell series, Far East Journal of Mathematical Sciences, 84 (2014), no:1, 25-32.
• [5] Choi, J., Parmar, R.K. and Chopra, P., The incomplete generalized 􀀀hypergeometric and second 􀀀Appell functions, Honam Math. J. 36 (2014), no. 3, 531-542.
• [6] Çetinkaya, A., Ya˘gbasan, M.B. and Kıymaz, ˙I.O., The extended Srivastava’s hypergeometric function and their integral representation, J. Nonlinear sci. Appl., 9 (2016), no:6, 4860-4866.
• [7] Exton, H., On Srivastava’s symmetrical triple hypergeometric function HB, J. Indian Acad. Math., 25 (2003), 17-22.
• [8] Goswami, A., Jain, S., Agarwal, P. and Araci, S., A Note on the New Extended Beta and Gauss Hypergeometric Functions, Appl. Math, 12 (2018), no:1, 139-144.
• [9] Hasanov, A., Srivastava H. M. and Turaev, M., Integral representation of Srivastava’s triple hypergeometric function, Taiwanese Journal of Mathematics, 15(2011), no. 6 , 2751-2762.
• [10] Luo, M. J., Milovanovic, G. V. and Agarwal, P., Some results on the extended beta and extended hypergeometric functions. Applied Mathematics and Computation, 248 (2014), 631-651.
• [11] Opps, S.B., Saad, N. and Srivastava, H.M., Some reduction and transformation formulas for the Appell hypergeometric function F2, J. Math. Anal. Appl., 302 (2005), no:1, 180-195.
• [12] Opps, S.B., Saad, N. and Srivastava, H.M., Recursion formulas for Appell’s hypergeometric function F2 with some applications to radiation field problem, Appl. Math. Comput., 207 (2009), no:2, 545-558.
• [13] Ruzhansky, M., Cho, Y. J.,Agarwal, P. and Area, I., Advances in Real and Complex Analysis with Applications, Springer Singapore, 2017.
• [14] Parmar, R. K., Extended hypergeometric functions and associated properties, C. R. Math. Acad. Sci. Paris, 353 (2015), no:5, 421-426.
• [15] Parmar, R. K. and Saxena, R. K., The incomplete generalized 􀀀-hypergeometric and second 􀀀-Appell functions, J. Korean Math. Soc. 53 (2016), no. 2, 363-379.
• [16] Rainville, E.D., Special functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
• [17] Slater, L.J. Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966.
• [18] Srivastava, H.M., Hypergeometric functions of three variables, Ganita 15 (1964), no:2, 97-108.
• [19] Srivastava, H.M. and Karlsson, P.W., Multiple Gaussian Hpergeometric Series, Halsted Press (John Wiley and Sons), New York, 1985.
• [20] Srivastava, H.M. and Monacha, H.L., A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester),Wiley, New York, Chichester, Brisbane and Toronto, 1984.
• [21] Srivastava, H.M., Some integrals representing triple hypergeometric functions , Rend. Circ. Mat. Palermo, 16 (1967), no:2, 99-115.
• [22] Srivastava, H.M., Relations between functions contiguous to certain hypergeometric functions of three variables, Proc. Nat. Acad. Sci. India Sect. A, 36 (1966), 377-385.
• [23] Şahin, R., Recursion Formulas for Srivastava Hypergeometric Functions. Mathematica Slovaca, 65 (2015), no:6, 1345-1360.
• [24] Virchenko, N., Kalla, S. L. and Al-Zamel, A., Some results on a generalized hypergeometric function, Integral Transforms and Special Functions 12 (2001), no:1, 89-100.
• [25] Virchenko, N., On some generalizations of the functions of hypergeometric type, Fract. Calc. Appl. Anal., 2 (1999), no. 3, 233-244.