  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-Szeg\H{o} composition of
$n$ polynomials of the form $(x+1)^{n+k-1}(x+a_j)$. We study properties of the
affine mapping $\Phi _{n,k}$~:~$(c_1,\ldots ,c_n)$ $\mapsto$ $(\sigma _1,
\ldots ,\sigma _n)$, where $\sigma _i$ 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.
