On analytic continuation of multiple power series beyond the domain of convergence (Q898546)

Let \[ f(z)=\sum_{\alpha\in \mathbb{N}_0^d} f_\alpha z^\alpha \] denote a power series with \(\limsup_{|\alpha| \to \infty} (|f_\alpha|R^\alpha)^{1/|\alpha|}=1\) and \(R=R_1^{\alpha_1} \cdots R_d^{\alpha_d}\). The author characterizes the regularity condition on the coefficients \(f_\alpha\) under which the power series possesses an analytic continuation across a certain family of polyarcs on the boundary of the polydisk of convergence. The regularity condition is stated in terms of interpolation of the coefficients in the form \(\varphi(\alpha)=f_\alpha\), where \(\varphi\) denotes an entire function of \(d\) variables satisfying an appropriate growth condition.