New closed forms for Householder root finding functions and associated non-linear polynomial identities

Abstract

Constructed previously from the long established suite of Householder root finding schemes by algebraic computation, we analyse further those non-linear identities which exist for families of polynomials associated with sequences whose ordinary generating functions satisfy a quadratic equation with functional coefficients. We formulate theory which yields new closed forms describing both these identities and the iterated Householder functions from which they are derived.