迭代序列与零点几何
Iterated sequences and the geometry of zeros
摘要：
本文研究了某些非线性变换对序列生成函数零点的影响。Pólya-Schur型的特征给出的变换，保持只有实和非正零点的性质。特别地，如果多项式$a_0+a_1z +\cdots+a_nz^n$仅具有实零和非正零，则多项式$a_0^2+ (a_1^2-a_0a_2)z+...+ (a_{n-1}^2-a_{n-2}a_n)z^{n-1}+a_n^2z^n$也是如此。这证实了菲斯克、麦克纳马拉-萨根和斯坦利的猜想。一个推论是，如果一个多项式只有实零点和非正零点，那么它的泰勒系数形成一个无限对数凹序列。我们将结果推广到Laguerre-Pólya类中的超越整函数，并讨论了Craven和Csordas研究的迭代Turán不等式问题的结果.最后，我们提出了一个新的方法来解决Boros和Moll的猜想。
英文摘要：
We study the effect on the zeros of generating functions of sequences under certain non-linear transformations. Characterizations of Pólya--Schur type are given of the transformations that preserve the property of having only real and non-positive zeros. In particular, if a polynomial $a_0+a_1z +\cdots+a_nz^n$ has only real and non-positive zeros, then so does the polynomial $a_0^2+ (a_1^2-a_0a_2)z+...+ (a_{n-1}^2-a_{n-2}a_n)z^{n-1}+a_n^2z^n$. This confirms a conjecture of Fisk, McNamara-Sagan and Stanley, respectively. A consequence is that if a polynomial has only real and non-positive zeros, then its Taylor coefficients form an infinitely log-concave sequence. We extend the results to transcendental entire functions in the Laguerre-Pólya class, and discuss the consequences to problems on iterated Turán inequalities, studied by Craven and Csordas. Finally, we propose a new approach to a conjecture of Boros and Moll.
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