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 $\alpha \in\mathbb{Z}^{+},$
\[
\sum\limits_{k=1}^{p-1}\left( -1\right) ^{k}q^{-\alpha pk+\binom{k+1}{2}
+k}\left[ k\right] _{q}{\alpha p-1 \brack k}_{q}
\]
\[
\equiv\frac{q^{1-\alpha p}}{(1-q^{2})^{2}}\left( q^{\alpha p+2}\left(
q^{p}-2\right) +q^{\alpha p}-q^{p}+q^{2}\right) \left[ p-1\right] _{q} %
\pmod{\left[ p\right] _{q}^{3}}.
\]
Primary Language | English |
---|---|
Subjects | Mathematical Sciences |
Journal Section | Mathematics |
Authors | |
Publication Date | March 31, 2023 |
Published in Issue | Year 2023 Volume: 52 Issue: 2 |