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

L. I. Ronkin

B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, 47, Lenin Ave., 310164, Kharkov, Ukraine

Abstract: 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 $1$.

Received: 17.04.1995

Language: English

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