ACTA issues

The zero distribution of Nörlund-transforms of power series

Wolfgang Luh, Karin Stadtmüller

Acta Sci. Math. (Szeged) 58:1-4(1993), 243-251

Abstract. A sequence $c = (c_\nu )$ of nonnegative numbers with $$C_n := \sum ^n_{\nu =0} c_\nu >0 \hbox{ for all } n\in{\msbm N}_0 \hbox{ and } \lim_{n\to\infty }{c_n\over C_n}=0$$ generates a regular Nörlund-method. The Nörlund-transforms of a power series $\sum ^\infty_{\nu =0} a_\nu z^\nu $ with radius of convergence $1$ are given by $$\sigma_n(z):={1\over C_n}\sum ^n_{\nu =0}c_{n-\nu }\sum ^\nu_{\mu =0} a_\mu z^\mu.$$ We investigate the distribution of zeros of the polynomials $\sigma_n$. If $A_n (R)$ denotes the number of zeros of $\sigma_n$ in $| z| \le R$ we have $\limsup_{n\to\infty }{A_n(R)\over n}=1$ for all $R>1$. It is the object of the present paper to characterize those power series which satisfy $$\liminf_{n\to\infty }{A_n (R)\over n}< 1\quad\hbox{for an }\ R>1. \quad (\ast)$$ One of our main results is, that $(\ast )$ holds if and only if the power series has Ostrowski-gaps.

AMS Subject Classification (1991): 40G05, 30C15, 30B30

Received November 30, 1992. (Registered under 5546/2009.)