English

Abstract

We give several different q-analogues of the following two congruences of Z.-W. Sun: $$\sum\limits_{k = 0}^{({p^r} - 1)/2} {\frac{1}{{{8^k}}}\left( {\begin{array}{*{20}{c}} {2k} \\ k \end{array}} \right) \equiv \left( {\frac{2}{{{p^r}}}} \right)(\bmod {p^2})\;\text{and}\;} \sum\limits_{k = 0}^{({p^r} - 1)/2} {\frac{1}{{{{16}^k}}}\left( {\begin{array}{*{20}{c}} {2k} \\ k \end{array}} \right) \equiv \left( {\frac{3}{{{p^r}}}} \right)(\bmod {p^2})} $$ is the Jacobi symbol. The proofs of them require the use of some curious q-series identities, two of which are related to Franklin’s involution on partitions into distinct parts. We also confirm a conjecture of the latter author and Zeng in 2012.