- Hacettepe Journal of Mathematics and Statistics
- Vol: 52 Issue: 2
- Some congruences with $q-$binomial coefficients and $q-$harmonic numbers
Some congruences with $q-$binomial coefficients and $q-$harmonic numbers
Authors : Sibel Koparal, Neşe Ömür, Laid Elkhiri
Pages : 445-458
Doi:10.15672/hujms.1076409
View : 15 | Download : 7
Publication Date : 2023-03-31
Article Type : Research
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}}. \\]Keywords : congruences, q−binomial coefficients, q−harmonic numbers, Abel sum