  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_{\theta }\left(
x\right) =\left( 1-\theta x\right) ^{p}F\left( a,b;c;x\right)
=\sum_{n=0}^{\infty }u_{n}\left( \theta \right) x^{n}. \end{equation*}% In this
paper, we prove that the coefficients $u_{n}\left( \theta \right) $ for $n\geq
0$ 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 $\mathbb{R}_{+}^{2}$.
