Zero-Free Parabolic Regions for Sequences of Polynomials

In this paper, we show that certain sequences of polynomials $\{ p_k (z)\} _{k = 0}^n $, generated from three-term recurrence relations, have no zeros in parabolic regions in the complex plane of the form $y^2 \leqq 4\alpha (x + \alpha )$, $x > - \alpha $. As a special case of this, no partial $s_n (z) = {{\sum _{k = 0}^n z^k } / {k!}}$ of $e^z $ has a zero in $ y^2 \leqq 4(x + 1)$, $x > - 1$, for any $n \geqq 1$. Such zero-free parabolic regions are obtained for Pade approximants of certain meromorphic functions, as well as for the partial sums of certain hypergeometric functions.