On entire functions of $n$ variables being quasipolynomials in one the variables

L. I. Ronkin

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 3 (1996) no. 1, pp. 131-141

Résumé

A general form is found for entire functions $f(z_1,{}^{'}z)$, $z_1\in C$, ${}^{'}z\in C^{n-1}$, of a finite order $p$ that are $M$-quasipolynomials in $z_1$ for every ${}^{'}z$ from a non-pluripolar set $E\in C^{n-1}$, i.e. $f(z_1,{} ^{'}z)=\sum_{j=1}^m\alpha_j(z_1)e^{\lambda_j z_1}$, ${}^{'}z\in E$. Here $m$, $\lambda_j$ and $\alpha_j(z_1)$ depend on ${}^{'}z$ a priori arbitrarily and $\alpha_j(z_1)$ belong to the class $M$ of entire functions of the type $0$ with respect to the order $1$.