Abstract—In mathematics, generating functions are important way to transform formal power series into functions and to analyze asymptotic properties of sequences. On the other hand, hypergeometric function is a special function represented by the hypergeometric series. Gauss, Confluent, Appell, Lauricella, and Horn functions are as an example of hypergeometric functions. In this article, we will introduce k-hypergeometric functions which are extensions of the Gauss hypergeometric functions including k- Pochhammer symbol. We first give an identity for k-Pochhammer symbol and certain linear generating functions for k-hypergeometric functions. Then we derive a family of multilinear and multilateral generating functions for these functions. In the main theorems, specially, bilateral generating function relations are obtained by applying extended multivariable hypergeometric functions and Cesáro Polynomials. Additionally we present bilinear generating functions for k-hypergeometric functions.

Index Terms—Hypergeometric functions, generating functions, multilinear and multilateral generating functions, k-hypergeometric functions, k-Pochhammer symbol.

Duriye Korkmaz-Duzgun is with Kafkas University, Faculty of Economics and Administrative Sciences, Department of Business Administration, Central Campus TR-36100 Kars, Turkey (email: dkduzgun@kafkas.edu.tr).
Esra Erkus-Duman is with Gazi University, Faculty of Science, Department of Mathematics, 06500 Teknikokullar, Ankara, Turkey.

Cite: Duriye Korkmaz-Duzgun, Esra Erkus-Duman, "Generating Functions for k-Hypergeometric Functions," International Journal of Applied Physics and Mathematics vol. 9, no. 3, pp. 119-126, 2019.