On Schur-Szego composition of polynomials

Narayana numbers and Schur-Szego composition

In the present paper we find a new interpretation of Narayana polynomials N_n(x) which are the generating polynomials for the Narayana numbers N_{n,k} counting Dyck paths of length n and with exactly k peaks. Strangely enough Narayana polynomials also occur as limits as n->oo of the sequences of eigenpolynomials of the Schur-Szego composition map sending (n-1)-tuples of polynomials of the form (x+1)^{n-1}(x+a) to their Schur-Szego product, see below. As a corollary we obtain that every N_n(x) has all roots real and non-positive. Additionally, we present an explicit formula for the density and …

A mapping defined by the Schur-Szegő composition

Each degree $n+k$ polynomial of the form $(x+1)^k(x^n+c_1x^{n-1}+\cdots +c_n)$, $k\in \mathbb{N}$, is representable as Schur-Szegő composition of $n$ polynomials of the form $(x+1)^{n+k-1}(x+a_j)$. We study properties of the affine mapping $Φ_{n,k}$~:~$(c_1,\ldots ,c_n)$ $\mapsto$ $(σ_1, \ldots ,σ_n)$, where $σ_i$ are the elementary symmetric polynomials of the numbers $a_j$. We study also properties of a similar mapping for functions of the form $e^xP$, where $P$ is a polynomial, $P(0)=1$, and we extend the Descartes rule to them.

A refined realization theorem in the context of the Schur-Szegő composition

Every polynomial of the form $P=(x+1)(x^{n-1}+c_1x^{n-2}+\cdots +c_{n-1})$ is representable as Schur-Szegő composition of $n-1$ polynomials of the form $(x+1)^{n-1}(x+a_i)$, where the numbers $a_i$ are unique up to permutation. We give necessary and sufficient conditions upon the possible values of the $8$-vector whose components are the number of positive, zero, negative and complex roots of a real polynomial $P$ and the number of positive, zero, negative and complex among the quantities $a_i$ corresponding to $P$. A similar result is proved about entire functions of the form $e^xR$, where $R…

Interlacing properties and the Schur-Szegő composition

Each degree $n$ polynomial in one variable of the form $(x+1)(x^{n-1}+c_1x^{n-2}+\cdots +c_{n-1})$ is representable in a unique way as a Schur-Szegő composition of $n-1$ polynomials of the form $(x+1)^{n-1}(x+a_i)$, see \cite{Ko1}, \cite{AlKo} and \cite{Ko2}. Set $σ_j:=\sum _{1\leq i_1<\cdots <i_j\leq n-1}a_{i_1}\cdots a_{i_j}$. The eigenvalues of the affine mapping $(c_1,\ldots ,c_{n-1})\mapsto (σ_1,\ldots ,σ_{n-1})$ are positive rational numbers and its eigenvectors are defined by hyperbolic polynomials (i.e. with real roots only). In the present paper we prove interlacing properties of the …

正交多项式，Szego-Verblunsky定理与Baxter定理在四元数球面上的应用

Orthogonal Polynomials, a Szegő--Verblunsky Theorem and Baxter's Theorem on the Quaternionic Sphere

本文研究四元数单位球上的正交多项式理论，扩展了Szego递推关系、正交多项式零点定理、Szego-Verblunsky定理和Baxter定理，引入了Alpay等人提出的q-正测量概念及矩阵值Schur递推关系和Verblunsky公式。

Multivariate Polya-Schur classification problems in the Weyl algebra

A multivariate polynomial is {\em stable} if it is nonvanishing whenever all variables have positive imaginary parts. We classify all linear partial differential operators in the Weyl algebra $\A_n$ that preserve stability. An important tool that we develop in the process is the higher dimensional generalization of Pólya-Schur's notion of multiplier sequence. We characterize all multivariate multiplier sequences as well as those of finite order. Next, we establish a multivariate extension of the Cauchy-Poincaré interlacing theorem and prove a natural analog of the Lax conjecture for real stabl…