Some generalizations of a congruence by Sun and Tauraso

Abstract

In 2010, Sun and Tauraso proved the combinatorial congruence

$$\begin{aligned} \sum _{k=0}^{p^r-1}\frac{1}{2^k}{2k\atopwithdelims ()k} \equiv (-1)^{\frac{p^r-1}{2}} \pmod {p}. \end{aligned}$$

Later, Guo and Zeng gave a q-analogue of this congruence: for any positive odd integer n,

$$\begin{aligned} \sum _{k=0}^{n-1}\frac{(q;q^2)_k}{(q;q)_k}q^{k} \equiv (-1)^{\frac{(n-1)}{2}}q^{\frac{n^2-1}{4}} \pmod {\Phi _n(q)}, \end{aligned}$$

where $\Phi _n(q)$ represents the n-th cyclotomic polynomial in q and $(a,q)_n$ is the q-shifted factorial. In this paper, we shall provide some generalizations of Guo and Zeng’s q-congruences.