Functions with Dominant Polar Singularities on the Circle of Convergence

Extract

1. Introduction. In an earlier paper¹ the function

was considered, having on its circle of convergence, taken to be |z|=1 only isolated essential points of finite exponential order, situated at the points e_ia^v (v = 1, 2, .., k). It was there proved,² in the cases k = 2, k = 3, that the upper density of small coefficients is positive only if the points e_ia^v are situated at some of the vertices of a regular polygon inscribed in the circle of convergence and if the singularities are virtually identical or linearly related. In this case the sequence of small coefficients possesses a density. The coefficients c_n can be interpolated in the form

where the G_v(z) are integral functions of order less than 1, so that the result stated may be regarded as expressing an arithmetical property of a certain type of integral function at positive integer points.