Combinatorial investigation of q-analogs of two recursions for Catalan numbers with alternating signs

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Recently Zeilberger conjectured that if the integers $\{A_n\}_{n\ge 1}$ are inductively defined via the recursion for Catalan numbers with alternating signs, then the integers $\{A_n\}_{n\ge 2}$ are increasing and positive. Motivated by his conjecture and Andrews' open questions on Koshy's identity, we are primarily concerned with $q$-analogs of two recursions for Catalan numbers with alternating signs. Here, the $q$-analog of a sequence of numbers resp.\ an identity is an extension of those by parameter $q$ that naturally reduces to the original in the limit $q\rightarrow 1$. Our main objective is to generalize Zeilberger's conjecture on sequence $\{A_n\}_{n\ge 1}$ into a combinatorial and probabilistic study of $q$-analogs of $\{A_n\}_{n\ge 1}$. By further analyzing the $q$-analogs of a similar recursion for Fuss-Catalan numbers, we aim toshow the universal positivity of the coefficients of the $q$-analogs. Much more are beyond Zeilberger's conjecture. Indeed our investigation will contribute to approach the intrinsic connection of two seemingly unrelated $q$-series with positive coefficients. One is the $q$-analog of $\{A_n\}_{n\ge 1}$, the other one comes from the $q$-analog of Koshy's identity due to Andrews. Along our way we also expect to achieve some intermediate results of relevance, including completely solving Andrew's open problems on the $q$-analog of Koshy's identity in the context of the $q$-hypergeometric series, a combinatorial proof of the $q$-analog of Koshy's identity in terms of Narayana polynomials, providing a new proof of Zeilberger's conjecture by employing analytic combinatorial tools and answering Lassalle's open questions on the positivity of Schur functions associated to the complete functions generalized from the exponential generating function of Catalan numbers.