Schur-Szegő composition of entire functions

Abstract

For any pair of algebraic polynomials $A(x)=\sum_{k=0}^{n} {n \choose k} a_{k} x^{k}$ and $B(x)=\sum_{k=0}^{n} {n \choose k} b_{k} x^{k}$, their Schur-Szegő composition is defined by $(A~^{\ast}_{n}~B)(x)= \sum_{k=0}^{n}{n \choose k} a_{k} b_{k} x^{k}$. Motivated by some recent results which show that every polynomial P(x) of degree n with P(−1)=0 can be represented as $K_{a_{1}}~^{\ast}_{n}~ \cdots ~^{\ast}_{n}~ K_{a_{n-1}}$ with K _a :=(x+1)ⁿ⁻¹(x+a), we introduce the notion of Schur-Szegő composition of formal power series and study its properties in the case when the series represents an entire function. We also concentrate on the special case of composition of functions of the form e ^x P(x), where P(x) is an algebraic polynomial and investigate its properties in detail.