ARITHMETICAL RESULTS ON CERTAIN MULTIVARIATE POWER SERIES

Abstract

Power series $\sum_{n=0}^\infty f(n)x^n$ with non-zero convergence radius $R(f)$ are considered, and the arithmetical nature (that is, irrationality, or even transcendence) of the corresponding multivariate series $\smash{\sum_{i_1,\ldots,i_m\!=0}^\infty f(i_1+\ldots +i_m)x_1^{i_1}\cdot\ldots \cdot x_m^{i_m}}$ is studied if $x_1,\ldots,x_m$ and the sequence $(f(n))$ satisfy appropriate arithmetical conditions. It follows that such arithmetical results can be written down easily if linear independence results on the function $F(x)$, defined in $|x|\,{<}\,R(f)$ by the original one-dimensional power series, and possibly its derivatives at the points $x_\mu$ are known. Some typical applications are explicitly stated.