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

Authors : Sibel KOPARAL, Neşe ÖMÜR, Laid ELKHİRİ

Pages : 445-458

Doi:10.15672/hujms.1076409

View : 24 | Download : 12

Publication Date : 2023-03-31

Article Type : Research Paper

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}\\leftinsert ignore into journalissuearticles values( -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}}{insert ignore into journalissuearticles values(1-q^{2});^{2}}\\leftinsert ignore into journalissuearticles values( q^{\\alpha p+2}\\leftinsert ignore into journalissuearticles values( 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