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Mubeen [5] has recently introduced the second order linear differential equations of k-hypergeometric and con uent k-hypergeometric functions. The main objective of this present paper is to develop these k-hypergeometric and con uent k-hypergeometric differential equations by adopting the same method, used by Rainville in [6]. Also, we find out the solutions of the form w =2 F1,k ((α, k); (β, k); (γ, k); z) and w =1 F1,k ((α, k); (β, k); z) of k- hypergeometric and con uent k-hypergeometric differential equations respectively, in the same way. [ABSTRACT FROM AUTHOR]

In this research work, our aim is to determine the contiguous function relations for fc-hypergeometric functions with one parameter corresponding to Gauss fifteen contiguous function relations for hypergeometric functions and also obtain contiguous function relations for two parameters. Throughout in this research paper, we find out the contiguous function relations for both the cases in terms of a new parameter k > 0. Obviously if k → 1, then the contiguous function relations for fc-hypergeometric functions are Gauss contiguous function relations. [ABSTRACT FROM AUTHOR]