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

最近Zeilberger猜想，如果整数$A_n_{n\ge1}$是通过交错符号加泰罗尼亚数的递归归纳定义的，则整数$_n_{n\ge2}$是递增的且为正的。受他的猜想和安德鲁斯关于Koshy恒等式的公开问题的启发，我们主要关注$Q$-具有交替符号的加泰罗尼亚数的两个递归的类似。在这里，数字序列的$q$-模拟分别是通过参数$q$对这些恒等式的扩展，该参数在极限$q\right tarrow 1$中自然缩减为原始的。我们的主要目的是将Zeilberger关于序列A_n_1}的猜想推广到关于$A_n_1}的$Q_-类似的组合概率研究中。通过进一步分析Fuss-Catalan数的类似递归的$Q$-模拟，我们的目的是证明$Q$-模拟的系数的普遍正性。更多的情况超出了泽尔伯格的猜测。事实上，我们的研究将有助于探讨两个看似不相关的正系数$Q$级数的内在联系。一个是$Q$-模拟$A_n_{n\ge1}$，另一个来自于安德鲁斯对Koshy身份的$Q$模拟。在我们的道路上，我们还期望取得一些相关的中间结果，包括完全解决Andrew关于$q$的公开问题-在$q$-超几何级数的背景下模拟Koshy的恒等式，用Narayana多项式证明$q$-类似Koshy的恒等式的组合证明，通过使用分析组合工具提供Zeilberger猜想的新证明，并回答Lassalle关于Schur函数的正性的公开问题，该问题与从加泰罗尼亚数的指数生成函数推广的完备函数相关。

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.