Zeros of polynomials generated by 4-term recurrence relations

Abstract

In Padé approximation the study of the location of the poles of the approximants, i.e. the zeros of the denominators, is of some importance. Loosely speaking, one might say that a (compact) set in the complex plane which is zero-free for a sequence of Padé approximants is the set where this sequence converges to the function it is derived from (under some suitable restrictions of course). As the most widely used sequences (e.g. steplines, diagonals) lead to certain "well-balanced" recurrence relations containing three terms, the study of the behaviour of the zeros of polynomials satisfying 3-term recurrence relations without thinking about a connection with any Padé table whatsoever has received some attention (cf. the famous Parabola theorem due to Saff & Varga [9] and Henrici [5], Leopold [6], Runckel [8]). The aim of this paper is to study the location of the zeros of polynomials generated by certain recurrence relations arising from sequences of approximants in a simultaneous Padé table.