On some classes of noncontinuable analytic functions

Abstract

Here, f(n) is a real valued function, I I an| } is a bounded sequence, such that infk E Ar I an| ?->0 if N-> oo. Using a lemma due to Riesz and certain results on uniform distributions (mod 1), simple conditions are obtained sufficient in order that F(z) have I z I = 1 as its natural boundary. Some of the more applicable such conditions are (cf. ?4): (i) From some positive integer r, Arf(n)>oo in a monotone fashion, Ar+lf(n)O_>o; (ii) I an I= 1 and f(n) is a finite sum of terms Cna(log n)#(log log n)Y, at least one of these terms being of higher order than n and not of the form Cna with C rational and a integral; (iii) Ian I = 1 and f(n) =An sin Bna, where A 0, B O0, 0 1. As is well-known, such classes of noncontinuable functions can be enlarged considerably by using Hadamard's multiplication theorem. For, let bn $O (n=O,1, * ... ) be such that L(z) = J bnzn has a radius of convergence 1 with only one singularity z1 on its circle of convergence. Then, assuming that F,(z) = b I ane2Tif(n)zn has also a radius of convergence 1, each singularity zo of F(z) with I zo I = 1 is equal to Zz2, Z2 denoting a singularity of F,(z) with I z21 = 1. Consequently, if F(z) has I zI =1 as its natural boundary then so has F1(z). In ?5, cf. Theorem 6, this principle is applied in showing that certain generalized hypergeometric series represent a function analytic in a circle I zI <R, and having I zI =R as a natural boundary. 2. Principal results. In this paper, {ao, alp a, ... } denotes a bounded sequence of complex numbers satisfying