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

TL;DR

This paper develops $q$-analogues of existing prime modulus congruences involving central $q$-binomial coefficients, specifically for cases $m=2$ and $m=4$, extending previous results to a broader $q$-series context.

Contribution

It introduces new $q$-congruences related to central $q$-binomial coefficients for specific cases, expanding the scope of prior prime modulus congruences.

Findings

Established $q$-congruences for $m=2$ and $m=4$ cases.

Extended classical congruences to $q$-analogues.

Provided formulas valid for any positive integer $n$.

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 $\sum_{k=0}^{n-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=0}^{n-1}\binom{2k}{k+d}/(km^k)$ modulo $p$ for all $d=0,1, \ldots n$ 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 $n$.