Some congruences involving central q-binomial coefficients

TL;DR

This paper proves new congruences involving central q-binomial coefficients, extending recent identities and providing q-analogues of classical number theory results, with several conjectures for future research.

Contribution

It introduces variations of the Green-Krammer identity involving q-binomial coefficients and establishes new congruences, including q-analogues of known number theory results.

Findings

Proved a q-analogue of a Green-Krammer identity involving Legendre symbols.

Derived congruences for sums of q-binomial coefficients modulo cyclotomic polynomials.

Proposed several related conjectures for further exploration.

Abstract

Motivated by recent works of Sun and Tauraso, we prove some variations on the Green-Krammer identity involving central q-binomial coefficients, such as $$ \sum_{k=0}^{n-1}(-1)^kq^{-{k+1\choose 2}}{2k\brack k}_q \equiv (\frac{n}{5}) q^{-\lfloor n^4/5\rfloor} \pmod{\Phi_n(q)}, $$ where $\big(\frac{n}{p}\big)$ is the Legendre symbol and $\Phi_n(q)$ is the $n$th cyclotomic polynomial. As consequences, we deduce that $$ \sum_{k=0}^{3^a m-1} q^{k}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{3^a})/(1-q)}, \sum_{k=0}^{5^a m-1}(-1)^kq^{-{k+1\choose 2}}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{5^a})/(1-q)}, $$ for $a,m\geq 1$, the first one being a partial q-analogue of the Strauss-Shallit-Zagier congruence modulo powers of 3. Several related conjectures are proposed.