论文标题
涉及超几何函数的系数的复发关系与应用
Recurrence relations of coefficients involving hypergeometric function with an application
论文作者
论文摘要
对于$ a，b，p \ in \ mathbb {r} $，$ -c \ notin \ mathbb {n \ cup} \ left \ {0 \ right \} $和$θ\ in \ in \ in \ left [-1,1 \ right] 1-θx\ right）^{p} f \ left（a，b; c; x \ right）= \ sum_ {n = 0}^{\ infty} u_ {n} \ left（θ\ prount（θ\ right）x^{n}。 \ end {equation*}％在本文中，我们证明系数$ u_ {n} \ left（θ\ right）$ for $ n \ geq 0 $满足3阶复发关系。特别是，$ u_ {n} \ left（1 \右）$满足2阶复发关系。这些提供了一种研究高几何功能的新方法。例如，我们提供了必要和充分的条件，使得超几何平均值是schur m-power凸面或在$ \ mathbb {r} _ {+}^{2} $上的凹入。
For $a,b,p\in \mathbb{R}$, $-c\notin \mathbb{N\cup }\left\{ 0\right\} $ and $ θ\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( θ\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 $\mathbb{R}_{+}^{2}$.
