Some congruences with $q-$binomial coefficients and $q-$harmonic numbers

Abstract

In this paper, considering $q-$analogues and $q-$combinatorial identities, we gave some congruences including $q-$binomial coefficients and $q-$ harmonic numbers. For example, for any prime number $p$ and $\alpha \in\mathbb{Z}^{+},$ \[ \sum\limits_{k=1}^{p-1}\left( -1\right) ^{k}q^{-\alpha pk+\binom{k+1}{2} +k}\left[ k\right] _{q}{\alpha p-1 \brack k}_{q} \] \[ \equiv\frac{q^{1-\alpha p}}{(1-q^{2})^{2}}\left( q^{\alpha p+2}\left( q^{p}-2\right) +q^{\alpha p}-q^{p}+q^{2}\right) \left[ p-1\right] _{q} % \pmod{\left[ p\right] _{q}^{3}}. \]