On the structure of an entire function vanishing on a given sequence of points (Q2565298)

We make use of the denotations of the author. Let \(F\) be an entire function of exponential type, \(\gamma_F\) be the Borel transformation of \(F\) and \((S(\gamma_F))\) be a set of singularities of \(\gamma_F\). The main result is the following. Let \(F(z)\) be an entire function of exponential type, \[ F(\pm\lambda_n)=0,\quad \lambda_n>0,\quad\lim_{n\to\infty} {n\over\lambda_n}=\sigma, \] and let \[ S(\gamma_F)\subset\{z:|\text{Im }z|<\pi\sigma\}\cup \{K+\alpha+i\pi\}\cup \{K+\alpha-i\pi\}, \] where \(\alpha\in(-\infty,\infty)\), \(K\) be a compact set satisfying certain restrictions. Then \(F(z)=A(z)e^{\alpha z}\Pi(1-{z^2\over\lambda^2_n})\), where \(A(z)\) be an entire function of exponential type such that \(S(\gamma_A)\subset K\). The case \(\lambda_n=n\) was investigated by Yu. Kazmin (1976). The analytic extension of the Borel transformation may be a multifunction. For the validity of the statement of the author it is necessary to give the precise definition of the set \(S(\gamma_F)\).