Bi-orthogonal polynomials

Abstract

Given a monotone measure α(x), a positive function ω(x,μ), μεΩ and a sequence μ₁,μ₂ ,... εΩ, we consider monic polynomials that satisfy the bi-orthogonality conditions

$$\int {p_m \left( x \right)\omega \left( {x,\mu _k } \right)d\alpha \left( x \right) = 0,} 1 \leqslant k \leqslant m, p_m \in \pi _m \left[ x \right].$$

Questions of existence, uniqueness, location of zeros and existence of Rodrigues-type formulae are investigated.

Polynomials of this type arise in numerical analysis of two-step multistage methods for ordinary differential equations.