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 Szegà  introduced the concept of two sequences of polynomials that are biorthogonal with respect to each other.
A polynomial p is called biorthogonal with respect to a sequence of measures ü<sub>1</sub>, ü<sub>2</sub>, ... if
Two sequences ÃÂ<sub>0</sub>, ÃÂ<sub>1</sub>, ... and ÃÂ<sub>0</sub>, ÃÂ<sub>1</sub>, ... of polynomials are called biorthogonal (for some measure ü) if
whenever m â 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 ÃÂ<sub>0</sub>, ÃÂ<sub>1</sub>, ... and ÃÂ<sub>0</sub>, ÃÂ<sub>1</sub>, ... of polynomials are biorthogonal for the measure ü if and only if the sequence ÃÂ<sub>0</sub>, ÃÂ<sub>1</sub>, ... is biorthogonal for the sequence of measures ÃÂ<sub>0</sub>ü, ÃÂ<sub>1</sub>ü, ..., and the sequence ÃÂ<sub>0</sub>, ÃÂ<sub>1</sub>, ... is biorthogonal for the sequence of measures ÃÂ<sub>0</sub>ü, ÃÂ<sub>1</sub>ü,....