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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. 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Nebel","submitted_at":"2013-09-04T09:07:15Z","abstract_excerpt":"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. 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