A New Type Multivariable Multiple Hypergeometric Functions

Year 2021, Volume: 13 Issue: 2, 359 - 372, 31.12.2021

Duriye Korkmaz-duzgun

https://doi.org/10.47000/tjmcs.954676

Abstract

We define a new type of multivariable multiple hypergeometric functions in this paper, which is inspired by Exton's multiple hypergeometric functions given by in [13]. Then, for these functions, we obtain some certain type linear generating functions. After that, we derive a variety classes of multilinear and multilateral generating functions for a family of the multivariable multiple hypergeometric functions. In addition, by employing the Erkus-Srivastava polynomials (see [11]) and the fourth type multivariable Horn functions (see [13]), we have also provided some of its conclusions.

Keywords

Multiple hypergeometric functions, Multivariable Horn Functions, Lauricella functions, Appell functions, Generating functions, Multilinear and Multilateral generating functions, Pochhammer symbol

Supporting Institution

No

Project Number

No

References

• [1] Abreu, S., Britto, R., Duhr, C., Gardi, E., Matthew, J., From positive geometries to a coaction on hypergeometric functions, Journal of High Energy Physics, 2(2020), 1–45.
• [2] Agarwal, R.P., Luo, M.J., Agarwal, P., On the extended Appell Lauricella hypergeometric functions and their applications, Filomat, 31(2017), 3693–3713.
• [3] Altin, A., Cekim, B., Sahin, R., On the matrix versions of Appell hypergeometric functions, Quaestiones Mathematicae, 37(2014), 31–38.
• [4] Bezrodnykh, S.I., Analytic continuation of Lauricella’s functions, FA(N), FB(N) and FD(N), Integral Transforms and Special Functions, 31(2020), 921–940.
• [5] Bezrodnykh, S.I., Horn’s hypergeometric functions with three variables, Integral Transforms and Special Functions, 32(2021), 207–223.
• [6] Brychkov, Y.A., Saad, N., On some formulas for the Appell function F 2 (a, b, b’; c, c’; w; z), Integral Transforms and Special Functions, 25(2014), 111–123.
• [7] Brychkov, Y.A., Savischenko, N.V., On some formulas for the Horn functions H 4 (a, b; c, c’; w, z) and H 7 (c)(a; c, c’; w, z), Integral Transforms and Special Functions, 32(2021), 1–19.
• [8] Choi, J., A generalization of Gottlieb polynomials in several variables, Applied Mathematics Letters, 25(2012), 43–46.
• [9] Dwivedi, R., Sahai, V., A note on the Appell matrix functions, Quaestiones Mathematicae, 43(2020), 321–334.
• [10] Ergashev, T.G., Fundamental solutions of the generalized Helmholtz equation with several singular coefficients and confluent hypergeometric functions of many variables, Lobachevskii Journal of Mathematics, 41(2020), 15–26.
• [11] Erkuş, E., Srivastava, H.M., A unified presentation of some families of multivariable polynomials, Integral Transform. Spec. Funct., 17(2006), 267–273.
• [12] Ernst, T., Some results for q functions of many variables, Rendiconti del Seminario Matematico della Universit´a di Padova, 112(2004), 199-235.
• [13] Exton, H., Multiple Hypergeometric Functions and Applications, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, 1976.
• [14] Gürel Yılmaz, Ö., Aktaş, R., Taşdelen, F., On some formulas for the k-analogue of Appell functions and generating relations via k-fractional derivative, Fractal and Fractional, 4(2020), 48.
• [15] Hidan, M., Abdalla, M., A note on the Appell hypergeometric matrix function F2, Mathematical Problems in Engineering, (2020).
• [16] Horn, J., Hypergeometrische funktionen zweier Ver¨anderlichen, Math. Ann., 105(1931), 381–407.
• [17] Jaeger, J.C., Hulme, H.R., The internal conversion of Gamma rays with the production of electrons and positrons, Proceedings of the Royal Society of London A, 148(1935), 708–728.
• [18] Kalla, S.L., Parmar, R.K., Purohit, S.D., Some extensions of Lauricella Functions of sevearal variables, Communications of the Korean Mathematical Society, 30(2015), 239–252.
• [19] Korkmaz Duzgun, D., Erkus Duman, E., Extended multivariable fourth type Horn functions, Gazi University Journal of Science, 32(2019), 225–240.
• [20] Korkmaz Duzgun, D., Erkuş Duman, E., Generating functions for the extended multivariable fourth type Horn functions, International Journal of Applied Physics and Mathematics, 10(2020), 65–72.
• [21] Lauricella, G., Sulle funzioni ipergeometriche a piu variabili, Rendiconti del Circolo Matematico di Palermo, 7(1983), 111–158.
• [22] Liu, S.J., Chyan, C.J., Lu, H.C., Srivastava, H.M., Bilateral generating functions for the Chan-Chyan-Srivastava polynomials and the generalized Lauricella functions, Integral Transforms and Special Functions, 23(2012), 539–549.
• [23] Ma, H., Some properties for Appell series F2 over finite fields, Integral Transforms and Special Functions, 30(2019), 992–1003.
• [24] Olsson P.O.M., A hypergeometric function of two variables of importance in perturbation theory I and II, Arkiv för Fysik; 30(1965), 187–191, ibid. 29(1965), 459–465.
• [25] Ozmen, N., Erkuş Duman, E., Some generating functions for a class of hypergeometric polynomials, Gazi University Journal of Science, 31(2018), 1179–1190.
• [26] Özmen, N., Some new properties of generalized Bessel polynomials, Applicationes Mathematicae, 46(2019), 85–98.
• [27] Rainville, E.D., Special Functions, The Macmillan Company, New York, 1960.
• [28] Scarpello, G.M., Ritelli, D., On computing some special values of multivariate hypergeometric functions, Journal of Mathematical Analysis and Applications, 420(2014), 1693–1718.
• [29] Srivastava, H.M., Manocha, H.L., A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, 1984.
• [30] Upadhyaya, L.M., Remarks on Horn’s double hypergeometric functions of matrix arguments, Bulletin of Pure and Applied Sciences-Mathematics, 1(2011), 11–18.