Some q-supercongruences from Watson’s $$_8\phi _7$$ Transformation Formula

Abstract

Recently, Jana and Kalita proved the following supercongruences involving rising factorials $$(\frac{1}{d})_k^3$$: $$\begin{aligned}&\sum _{k=0}^{N} (-1)^k (2dk+1)\frac{(\frac{1}{d})_k^3}{k!^3}\\&\quad \equiv {\left\{ \begin{array}{ll} p^r&{}\pmod {p^{r+2}},\quad \text {if } r \text { is even};\\ (-1)^{\frac{(p-d+1)r}{d}}(d-1)p^r&{}\pmod {p^{r+2}},\quad \text {if } r \text { is odd}; \end{array}\right. }\\&\sum _{k=0}^{N} (-1)^k (2dk+1)^3\frac{(\frac{1}{d})_k^3}{k!^3}\\&\quad \equiv {\left\{ \begin{array}{ll} -3p^r&{}\pmod {p^{r+2}},\quad \text {if } r \text { is even};\\ (-1)^{\frac{(p+1)r}{d}}3(d-1)p^r&{}\pmod {p^{r+2}},\quad \text {if } r \text { is odd}, \end{array}\right. } \end{aligned}$$where $$N={\left\{ \begin{array}{ll} \frac{p^r-1}{d}, \quad &{}\text {if } r \text { is even};\\ \frac{(d-1)p^r-1}{d},\quad &{}\text {if } r \text { is odd}.\end{array}\right. }$$ From Watson’s $$_8\phi _7$$ transformation formula, we give q-analogues of the above supercongruences, generalizing some previous conjectural results of Van Hamme. Our proof uses the ‘creative microscoping’ method which was introduced by Guo and Zudilin.