利用差商终止零平衡超几何级数

中文摘要 AI 辅助

差商与超几何级数之间存在密切联系，近期研究表明差商可有效用于推导终止超几何恒等式。本文将此方法应用于终止零平衡超几何级数。利用由差商产生的显式乘积求值，我们给出了零平衡 ${}_3F_2(1)$ 和 ${}_4F_3(1)$ 求和公式的替代证明，并将论证推广至一般终止 ${}_{r+1}F_r(1)$ 情形。我们还结合拉格朗日表示与差商的莱布尼茨法则，推导出终止 ${}_{r+2}F_{r+1}(1)$ 级数的有限卷积变换。因此，差商方法不仅重现了已知的零平衡求和公式，还产生了终止超几何级数的有限卷积变换及其零平衡特例。

英文摘要

There is a close relationship between divided differences and hypergeometric series, and recent studies have shown that divided differences can be used effectively to derive terminating hypergeometric identities. In this paper, we apply this approach to terminating zero-balanced hypergeometric series. Using explicit product evaluations arising from divided differences, we give alternative proofs of zero-balanced ${}_3F_2(1)$ and ${}_4F_3(1)$ summation formulas and then extend the argument to the general terminating ${}_{r+1}F_r(1)$ case. We also combine the Lagrange representation with the Leibniz rule for divided differences to derive a finite convolution transformation for a terminating ${}_{r+2}F_{r+1}(1)$ series. Thus, the divided-difference approach not only reproduces the known zero-balanced summation formulas but also yields a finite convolution transformation for terminating hypergeometric series, together with its zero-balanced specialization.

发表机构

• Istanbul Technical University（伊斯坦布尔理工大学）

机构由 AI 辅助整理，请以论文原文为准。