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

TL;DR

This paper establishes new $q$-analogues of Koshy's formula using combinatorial methods, extends them to $q$-Ballot numbers, and addresses open questions about unimodality of related polynomials.

Contribution

It introduces novel $q$-analogues of Koshy's formula, generalizes these to $q$-Ballot numbers, and resolves open questions on polynomial unimodality.

Findings

Proved $q$-analogues of Koshy's formula using Dyck paths and partitions.

Extended $q$-analogues to $q$-Ballot numbers.

Confirmed conjectures on polynomial unimodality for specific cases.

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 $(1+q^n){m\brack n-1}_q$ is unimodal. If $n$ is even, for any even $j\ne 0$ and $m\ge n\ge 1$, the polynomial $(1+q^n)[j]_q{m\brack n-1}_q$ is unimodal. This implies the answer to the second problem posed by Andrews.