Jacobi-Stirling数的二项式展开及Jacobi-Stirling下降多项式的实根性
Binomial expansions of Jacobi-Stirling numbers and real-rootedness of Jacobi-Stirling descent polynomials
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中文总结 AI 辅助
本文研究Jacobi-Stirling数的二项式展开，证明其下降多项式在特定删除或保留横杠字母数下仅有简单负零点，验证了Gessel-Lin-Zeng猜想的五个无限族，并揭示其顶部重及左半递增性质。
中文摘要 AI 辅助
本文首先在二项式基下展开Jacobi-Stirling数的固定对角线。对于第二类Jacobi-Stirling数，展开系数是$z+1$的多项式，且具有非负整数系数。我们给出了这些系数的递推关系及带符号划分解释。相应的第一类与第二类系数之差也具有同样的性质。随后我们证明，当删除的带横杠字母数为1、2或3时，Jacobi-Stirling排列上的下降多项式的每个非零非负线性组合仅有简单负零点。当恰好保留1或2个带横杠字母时，同样的结论也成立。由此我们验证了Gessel、Lin和Zeng猜想中的五个无限族。最后，利用插入算子，我们发现对于固定删除带横杠字母数的Jacobi-Stirling排列，其每个下降多项式都是顶部重的，且左半部分递增。
英文摘要
In this paper, we first expand fixed diagonals of the Jacobi-Stirling numbers in a binomial basis. For the second kind, the expansion coefficients are polynomials in $z+1$ with nonnegative integer coefficients. We give a recurrence and a signed-partition interpretation for these coefficients. The same holds for the differences between corresponding unsigned first-kind and second-kind coefficients. We then prove that every nonzero nonnegative linear combination of the descent polynomials over Jacobi-Stirling permutations with a fixed number of deleted barred letters has only simple negative zeros when that number is one, two, or three. The same holds when exactly one or two barred letters are retained. Thus we verify five infinite families in a conjecture of Gessel, Lin and Zeng. Finally, using insertion operators, we find that every descent polynomial over Jacobi-Stirling permutations with a fixed number of deleted barred letters is top heavy and has an increasing left half.
发表机构
- School of Mathematics and Statistics, Shandong University of Technology（山东理工大学数学与统计学院）
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