Schur-Szeg\H{o} composition of entire functions
Résumé
For any pair of algebraic polynomials $A(x) =\sum _{k=0}^ n{n\choose k} a_kx^k$ and $B(x) =\sum _{k=0}^ n{n\choose k} b_kx^k$, their Schur-Szeg\H{o} composition is defined by $(A^*_nB)(x) =\sum _{k=0}^ n{n\choose k} a_kb_kx^k$. Motivated by some recent results which show that every polynomial $P(x)$ of degree $n$ with $P(−1) = 0$ can be represented as $K_{a_1}^∗_n\cdots ^∗_n K_{a_{n−1}}$ with $K_a := (x + 1)^{n−1}(x + a)$, we introduce the notion of Schur-Szeg\H{o} 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 $e^xP(x)$, where $P(x)$ is an algebraic polynomial and investigate its properties in detail.