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Schur-Szeg\H{o} composition of entire functions

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-00866143
Contributor : Vladimir Kostov <>
Submitted on : Thursday, September 26, 2013 - 9:51:06 AM
Last modification on : Monday, October 12, 2020 - 2:28:05 PM

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  • HAL Id : hal-00866143, version 1

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Vladimir Kostov, Vladimir Kostov, Dimitar Dimitrov. Schur-Szeg\H{o} composition of entire functions. Revista Matematica Complutense, Universidad Complutense, 2012, 25 (2), pp.475-491. ⟨hal-00866143⟩

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