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 $\alpha \in\mathbb{Z}^{+},$
\[
\sum\limits_{k=1}^{p-1}\left( -1\right) ^{k}q^{-\alpha pk+\binom{k+1}{2}
+k}\left[ k\right] _{q}{\alpha p-1 \brack k}_{q}
\]
\[
\equiv\frac{q^{1-\alpha p}}{(1-q^{2})^{2}}\left( q^{\alpha p+2}\left(
q^{p}-2\right) +q^{\alpha p}-q^{p}+q^{2}\right) \left[ p-1\right] _{q} %
\pmod{\left[ p\right] _{q}^{3}}.
\]
congruences q−binomial coefficients q−harmonic numbers Abel sum
Birincil Dil | İngilizce |
---|---|
Konular | Matematik |
Bölüm | Matematik |
Yazarlar | |
Yayımlanma Tarihi | 31 Mart 2023 |
Yayımlandığı Sayı | Yıl 2023 Cilt: 52 Sayı: 2 |