New proofs of two $q$-analogues of Koshy's formula
New proofs of two $q$-analogues of Koshy's formula
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DOI：
10.1090/proc/12627
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发表时间：
2013-09
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通讯作者：
E. Y. Jin;M. Nebel
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文献类型：
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作者：
E. Y. Jin;M. Nebel
在本文中，我们证明了一个$q$-模拟的Koshy的公式的Narayana多项式由于Lassalle和一个$q$-模拟的Koshy的公式的$q$-超几何级数由于安德鲁斯通过应用的包含-排除原则的Dyck道路和分区。我们推广这两个$q$-类似物的Koshy的公式为$q$-Catalan数，为$q$-选票数。这项工作还回答了Lassalle提出的一个开放性问题和Andrews在2010年提出的两个问题。我们猜想如果$n$是奇数，那么对于$m\geN 1$，多项式$（1+q^n）{m\brackn-1}_q$是单峰的。如果$n$是偶数，对于任何偶数$j\ne 0$和$m\ge n\ge 1$，多项式$（1+q^n）[j]_q{m\brack n-1}_q$是单峰的。这意味着答案的第二个问题所提出的安德鲁斯。
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.