Recurrence relations of coefficients involving hypergeometric function with an application

Abstract

For $a,b,p\in \mathbb{R}$, $-c\notin \mathbb{N}\cup \left\{0\right\}$ and $\theta \in \left[-1,1\right]$, let \begin{equation*} U_{θ}\left( x\right) =\left( 1-θx\right) ^{p}F\left( a,b;c;x\right) =\sum_{n=0}^{\infty }u_{n}\left( θ\right) x^{n}. \end{equation*}% In this paper, we prove that the coefficients $u_n\left(\theta \right)$ for $n\ge 0$ satisfies a 3-order recurrence relation. In particular, $u_n\left(1\right)$ satisfies a 2-order recurrence relation. These offer a new way to study for hypergeometric function. As an example, we present the necessary and sufficient conditions such that a hypergeometric mean value is Schur m-power convex or concave on ${\mathbb{R}}_{+}^2$.

17 pages