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

最近Zeilberger推测，如果整数$\{A_n\}_{n\ge 1}$是通过交替符号的加泰罗尼亚数的递归归纳定义的，那么整数$\{A_n\}_{n\ge 2}$是递增的和正的。受他的猜想和安德鲁斯对科希身份的开放性问题的启发，我们主要关注$q$——加泰罗尼亚数字交替符号的两种递归的类似物。这里，$q$ -类似于一个数列。恒等式是对参数$q$的扩展，在极限$q\rightarrow 1$中自然地还原为原来的恒等式。我们的主要目标是将Zeilberger关于序列$\{A_n\}_{n\ge 1}$的猜想推广到$\{A_n\}_{n\ge 1}$的$q$ -类似物的组合和概率研究中。通过进一步分析fuss - catalalan数的类似递归的$q$ -类似物，我们旨在证明$q$ -类似物的系数的普遍正性。还有很多超出了Zeilberger的猜想。事实上，我们的研究将有助于接近两个看似不相关的$q$ -系列具有正系数的内在联系。一个是$\{A_n\}_{n\ge 1}$的$q$ -类比，另一个是由于Andrews对Koshy的身份的$q$ -类比。在我们的过程中，我们也期望获得一些相关的中间结果，包括在$q$ -超几何级数的背景下完全解决Andrew关于$q$ -类似Koshy恒等式的开放问题，用Narayana多项式对$q$ -类似Koshy恒等式的组合证明，利用解析组合工具给出Zeilberger猜想的新证明，并回答了拉萨尔关于由加泰罗尼亚数的指数生成函数推广的完备函数所关联的Schur函数正性的开放性问题。

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.