Turkish Journal of Mathematics
DOI
10.3906/mat-1905-88
Abstract
In this paper, we examine some congruences with $q$-analogues of ballot numbers. For example, for $n>1$ and $% d=0,1,...,n-1$, \begin{eqnarray*} &&\sum\limits_{k=1}^{n-d}q^{k}B_{k,d}^{q}\equiv -2+\left( -1\right) ^{n-d}\left( \frac{n-d+1}{3}\right) q^{-\frac{1}{3}\binom{n-d}{2}}-\left( \frac{n-d-1% }{3}\right) q^{d+1-\frac{1}{3}\binom{n-d-2}{2}} \pmod{\Phi _{n}\left( q\right)} , \end{eqnarray*}% with the Legendre symbol $\left( \frac{.}{3}\right) ,~$the$~q$-analogue of ballot number $B_{n,d}^{q}$, and the $n$th cyclotomic polynomial $% \Phi _{n}\left( q\right) $.
Keywords
Congruence, $q$-binomial coefficient, cyclotomic polynomial, ballot number
First Page
2642
Last Page
2648
Recommended Citation
KOPARAL, SİBEL
(2019)
"On congruences for $q$-analogues of ballot numbers,"
Turkish Journal of Mathematics: Vol. 43:
No.
5, Article 45.
https://doi.org/10.3906/mat-1905-88
Available at:
https://journals.tubitak.gov.tr/math/vol43/iss5/45
