Curious q-Analogues of Two Supercongrunces Modulo the Third Power of a Prime

Abstract

Let $$(a)_{n}=a(a+1)\cdots (a+n-1)$$ be the Pochhammer symbol. Recently, Jana and Kalita proved the following two supercongruences: $$\begin{aligned} \sum _{k=0}^{(p-1)/3} (6k+1)\frac{(\frac{1}{3})_k^4 (2k)!}{k!^4 (\frac{2}{3})_{2k}}&\equiv p\pmod {p^3}\quad \text {for primes }\,{ p\equiv 1\pmod {3},} \\ \sum _{k=0}^{(p+1)/3} (6k-1)\frac{(-\frac{1}{3})_k^4 (2k)!}{k!^4 (-\frac{2}{3})_{2k}}&\equiv p\pmod {p^3} \quad \text {for primes }\,{p\equiv 2\pmod {3},} \end{aligned}$$ which were originally conjectured by the second author and Schlosser. In this note, employing the creative microscoping method introduced by the second author and Zudilin, we give curious q-analogues of the above two supercongruences.