Biorthogonal polynomial


In mathematics, a biorthogonal polynomial is a polynomial that is orthogonal to several different measures. Biorthogonal polynomials are a generalization of orthogonal polynomials and share many of their properties. There are two different concepts of biorthogonal polynomials in the literature: introduced the concept of polynomials biorthogonal with respect to a sequence of measures, while Szegő introduced the concept of two sequences of polynomials that are biorthogonal with respect to each other.

Polynomials biorthogonal with respect to a sequence of measures

A polynomial p is called biorthogonal with respect to a sequence of measures μ1, μ2,... if

Biorthogonal pairs of sequences

Two sequences ψ0, ψ1,... and φ0, φ1,... of polynomials are called biorthogonal if
whenever m ≠ n.
The definition of biorthogonal pairs of sequences is in some sense a special case of the definition of biorthogonality with respect to a sequence of measures. More precisely two sequences ψ0, ψ1,... and φ0, φ1,... of polynomials are biorthogonal for the measure μ if and only if the sequence ψ0, ψ1,... is biorthogonal for the sequence of measures φ0μ, φ1μ,..., and the sequence φ0, φ1,... is biorthogonal for the sequence of measures ψ0μ, ψ1μ,....