Some $q$-congruences involving central $q$-binomial coefficients

Abstract

Suppose that $p$ is an odd prime and $m$ is an integer not divisible by $p$. Sun and Tauraso [Adv. in Appl. Math., 45(2010), 125--148] gave ${\sum }_{k=0}^{n-1}\left(\genfrac{}{}{0ex}{}{2k}{k+d}\right)/m^k$ and ${\sum }_{k=0}^{n-1}\left(\genfrac{}{}{0ex}{}{2k}{k+d}\right)/(k{m}^k)$ modulo $p$ for all $d=0,1,\dots n$ and $n=p^a$, where $a$ is a positive integer. In this paper, we present some $q$-analogues of these congruences in the cases $m=2,4$ for any positive integer $n$.

13 pages