# 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: Iserles & Nørsett (1988) 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 *μ*_{1}, *μ*_{2}, ... if

- whenever
*i*≤ deg(*p*).

## Biorthogonal pairs of sequences

Two sequences *ψ*_{0}, *ψ*_{1}, ... and *φ*_{0}, *φ*_{1}, ... of polynomials are called biorthogonal (for some measure *μ*) 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 ψ_{0}, ψ_{1}, ... and φ_{0}, φ_{1}, ... of polynomials are biorthogonal for the measure μ if and only if the sequence ψ_{0}, ψ_{1}, ... is biorthogonal for the sequence of measures φ_{0}μ, φ_{1}μ, ..., and the sequence φ_{0}, φ_{1}, ... is biorthogonal for the sequence of measures ψ_{0}μ, ψ_{1}μ,....

## References

- Iserles, Arieh; Nørsett, Syvert Paul (1988), "On the theory of biorthogonal polynomials",
*Transactions of the American Mathematical Society*,**306**(2): 455–474, doi:10.2307/2000806, ISSN 0002-9947, JSTOR 2000806, MR 0933301