Some congruences involving central q-binomial coefficients

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 $$\sum _{k=0}^{n-1}(-1{)}^k{q}^{-\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)}{\left[\genfrac{}{}{0ex}{}{2k}{k}\right]}_q\equiv (\frac{n}{5})q^{-\lfloor n^4/5\rfloor }(\mathrm{mod}{\Phi }_n(q)),$$ where $\left(\frac{n}{p}\right)$ 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 $a,m\ge 1$, the first one being a partial q-analogue of the Strauss-Shallit-Zagier congruence modulo powers of 3. Several related conjectures are proposed.

arXiv comment16 pages, detailed proofs of Theorems 4.1 and 4.3 are added, to appear in Adv. Appl. Math

References in corpus (1)

• Multiple extensions of a finite Euler's pentagonal number theorem and the Lucas formulas

Cited by in corpus (1)

• Some q-analogues of supercongruences of Rodriguez-Villegas