New proofs of two $q$-analogues of Koshy's formula

Abstract

In this paper we prove a $q$-analogue of Koshy's formula in terms of the Narayana polynomial due to Lassalle and a $q$-analogue of Koshy's formula in terms of $q$-hypergeometric series due to Andrews by applying the inclusion-exclusion principle on Dyck paths and on partitions. We generalize these two $q$-analogues of Koshy's formula for $q$-Catalan numbers to that for $q$-Ballot numbers. This work also answers an open question by Lassalle and two questions raised by Andrews in 2010. We conjecture that if $n$ is odd, then for $m\ge n\ge 1$, the polynomial $(1+q^n){\left[\genfrac{}{}{0ex}{}{m}{n-1}\right]}_q$ is unimodal. If $n$ is even, for any even $j\ne 0$ and $m\ge n\ge 1$, the polynomial $(1+q^n)[j{]}_q{\left[\genfrac{}{}{0ex}{}{m}{n-1}\right]}_q$ is unimodal. This implies the answer to the second problem posed by Andrews.

15 pages, 1 figures

References in corpus (1)

• Some remarks about q-Chebyshev polynomials and q-Catalan numbers and related results