Some congruences with $q-$binomial sums

Authors

NEŞE ÖMÜR
ZEHRA BETÜL GÜR
SİBEL KOPARAL
LAİD ELKHIRI

DOI

10.55730/1300-0098.3478

Abstract

In this paper, using some combinatorial identities and congruences involving $q-$harmonic numbers, we establish congruences that for any odd prime $p$ and any positive integer $\alpha$,% \begin{equation*} \text{ }\sum\limits_{k=1 \pmod{2}}^{p-1}(-1)^{nk}\frac{% q^{-\alpha npk+ n\tbinom{k+1}{2}+2k}}{[k]_{q}}{\alpha p-1 \brack k}_{q}^{n} \pmod{[p]_{q}^{2}} , \end{equation*}% and \begin{equation*} \sum\limits_{k=1 \pmod{2}}^{p-1}(-1)^{nk}q^{-\alpha npk+ n\tbinom{k+1}{2}+k}{\alpha p-1 \brack k}_{q}^{n}% \widetilde{H}_{k}(q)\pmod{[p]_{q}^{2}} ,\text{ } \end{equation*}% where $n$ is any integer.

Keywords

Congruence, $q-$analog, $q-$harmonic number

First Page

2006

Last Page

2027

Recommended Citation

ÖMÜR, NEŞE; GÜR, ZEHRA BETÜL; KOPARAL, SİBEL; and ELKHIRI, LAİD (2023) "Some congruences with $q-$binomial sums," Turkish Journal of Mathematics: Vol. 47: No. 7, Article 12. https://doi.org/10.55730/1300-0098.3478
Available at: https://journals.tubitak.gov.tr/math/vol47/iss7/12