On the divisibility of some truncated hypergeometric series
Volume 195 / 2020
Acta Arithmetica 195 (2020), 199-206
MSC: Primary 11B65; Secondary 05A10, 11A07, 11T06, 33C05.
DOI: 10.4064/aa190511-2-1
Published online: 11 May 2020
Abstract
Let $p$ be an odd prime and $r\geq 1$. Suppose that $\alpha $ is a $p$-adic integer with $\alpha \equiv 2a\pmod p$ for some $1\leq a \lt (p+r)/(2r+1)$. We confirm a conjecture of Sun and prove that $${}_{2r+1}F_{2r}\biggl [\begin {matrix}\alpha &\alpha &\ldots &\alpha \\ &1&\ldots &1\end {matrix}\,\bigg |\,1\biggr ]_{p-1}\equiv 0\pmod {p^2},$$ where the truncated hypergeometric series is defined by $$ {}_{m+1}F_{m}\biggl [\begin {matrix}x_0&x_1&\ldots &x_{m}\\ &y_1&\ldots &y_m\end {matrix}\,\bigg |\,z\biggr ]_{n}:=\sum _{k=0}^n\frac {(x_0)_k(x_1)_k\cdots (x_m)_k}{(y_1)_k (y_m)_k}\frac {z^k}{k!}. $$
