Abstract
In this paper, we expound on the hypergeometric series solutions for the second-order non-homogeneous k-hypergeometric differential equation with the polynomial term. The general solutions of this equation are obtained in the form of k-hypergeometric series based on the Frobenius method. Lastly, we employ the result of the theorem to find the solutions of several non-homogeneous k-hypergeometric differential equations.
Highlights
It is well known that many phenomena in physical and technical applications are governed by a variety of differential equations
In 2014, Mubeen et al [38,39] solved the k-hypergeometric differential equation by using the Frobenius method and gave its solution in the form of the so-called k-hypergeometric series 2 F1,k introduced by Díaz et al [30]
The method of Frobenius enables us to gain a power series solution of the differential equation defined by Equation (8), provided that both Y0 (z) and Y1 (z) are themselves analytic at z0 or they are analytic elsewhere and their limits exist at z0
Summary
It is well known that many phenomena in physical and technical applications are governed by a variety of differential equations. In the case of k = 1, if the function f (z) vanishes identically, Equation (1) degrades into a linear homogeneous hypergeometric ordinary differential equation presented by Euler [1] in 1769, which has the following normalized form:. Such an equation has been extensively studied. In 2014, Mubeen et al [38,39] solved the k-hypergeometric differential equation by using the Frobenius method and gave its solution in the form of the so-called k-hypergeometric series 2 F1,k introduced by Díaz et al [30]. Throughout this paper, we let C, R, R+ , and N+ stand for the set of complex numbers, the set of real numbers, the set of positive real numbers, and the set of positive integers, respectively
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