New theory and results from an algebraic application of householder root finding schemes

Abstract

In this paper we develop an analysis of the generalised Householder root finding scheme, and generate auto-identities for so called (Large) Schröder and Motzkin polynomials associated with their namesake sequences - they are but two examples of polynomial families whose origin sequences each have an ordinary generating function satisfying a quadratic equation with functional coefficients. Properties of these polynomials are also given within the context of some new general theory. Connections with previous work involving Catalan polynomials are made throughout where appropriate.