Some q-congruences arising from certain identities

Abstract

In this paper, by constructing some new q-identities, we prove some q-congruences. For example, for any odd integer $n>1$, we show that

$$\begin{aligned} \sum _{k=0}^{n-1}\frac{(q^{-1};q^2)_k}{(q;q)_k}q^k\equiv & {} (-1)^{(n+1)/2}q^{(n^2-1)/4}-(1+q)[n]\pmod {\Phi _n(q)^2},\\ \sum _{k=0}^{n-1}\frac{(q^3;q^2)_k}{(q;q)_k}q^k\equiv & {} (-1)^{(n+1)/2}q^{(n^2-9)/4}+\frac{1+q}{q^2}[n]\pmod {\Phi _n(q)^2}, \end{aligned}$$

where the q-Pochhammer symbol is defined by $(x;q)_0=1$ and $(x;q)_k=(1-x)(1-xq)\cdots (1-xq^{k-1})$ for $k\ge 1$, the q-integer is defined by $[n]=1+q+\cdots +q^{n-1}$ and $\Phi _n(q)$ is the n-th cyclotomic polynomial. The q-congruences above confirm some recent conjectures of Gu and Guo.