A parametric congruence motivated by Orr's identity

Abstract

For any $m,n\in \mathbb{N}=\{0,1,2\dots \}$, the truncated hypergeometric series ${}_{m+1}{F}_m$ is defined by $${}_{m+1}{F}_m\left[\begin{array}{cccc}{x}_0& x_1& \dots & x_m\\ & y_1& \dots & y_m\end{array}\right|z{]}_n=\sum _{k=0}^n\frac{(x_0{)}_k(x_1{)}_k\cdots (x_m{)}_k}{(y_1{)}_k\cdots (y_m{)}_k}\cdot \frac{z^k}{k!},$$ where $(x{)}_k=x(x+1)\cdots (x+k-1)$ is the Pochhammer symbol. Let $p$ be an odd prime. For $\alpha ,z\in {\mathbb{Z}}_p$ with $⟨-\alpha {⟩}_p\equiv 0(\mathrm{mod}2)$, where $⟨x{⟩}_p$ denotes the least nonnegative residue of $x$ modulo $p$ for any $x\in {\mathbb{Z}}_p$, we mainly prove the following congruence motivated by Orr's identity: $${}_2{F}_1\left[\begin{array}{cc}\frac{1}{2}\alpha & \frac{3}{2}-\frac{1}{2}\alpha \\ & 1\end{array}\right|z{]}_{p-1}{}_2{F}_1\left[\begin{array}{cc}\frac{1}{2}\alpha & \frac{1}{2}-\frac{1}{2}\alpha \\ & 1\end{array}\right|z{]}_{p-1}\equiv {}_3{F}_2\left[\begin{array}{ccc}\alpha & 2-\alpha & \frac{1}{2}\\ & 1& 1\end{array}\right|z{]}_{p-1}(\mathrm{mod}{p}^2).$$ As a corollary, for any positive integer $b$ with $p\equiv \pm 1(\mathrm{mod}b)$ and $⟨-1/b{⟩}_p\equiv 0(\mathrm{mod}2)$, we deduce that $$\sum _{k=0}^{p-1}(b^{2}k+b-1)\frac{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}{4^k}\left(\genfrac{}{}{0ex}{}{-1/b}{k}\right)\left(\genfrac{}{}{0ex}{}{1/b-1}{k}\right)\equiv 0(\mathrm{mod}{p}^2).$$ This confirms a conjectural congruence of the second author.

8 pages, accepted by Journal of Difference Equations and Applications

References in corpus (3)

• $p$-adic analogues of hypergeometric identities
• Supercongruences concerning truncated hypergeometric series
• $p$-adic analogues of hypergeometric identities and their applications