Absolutely continuous case

Sometimes we have

$${\displaystyle d\alpha (x)=W(x)\,dx}$$

where

$${\displaystyle W:[x_{1},x_{2}]\to \mathbb {R} }$$

is a non-negative function with support on some interval [x₁, x₂] in the real line (where x₁ = −∞ and x₂ = ∞ are allowed). Such a W is called a weight function.^[1] Then the inner product is given by

$${\displaystyle \langle f,g\rangle =\int _{x_{1}}^{x_{2}}f(x)g(x)W(x)\,dx.}$$

However, there are many examples of orthogonal polynomials where the measure dα(x) has points with non-zero measure where the function α is discontinuous, so cannot be given by a weight function W as above.

Relation to moments

The orthogonal polynomials P_n can be expressed in terms of the moments

${\displaystyle m_{n}=\int x^{n}\,d\alpha (x)}$

as follows:

${\displaystyle P_{n}(x)=c_{n}\,\det {\begin{bmatrix}m_{0}&m_{1}&m_{2}&\cdots &m_{n}\\m_{1}&m_{2}&m_{3}&\cdots &m_{n+1}\\\vdots &\vdots &\vdots &\ddots &\vdots \\m_{n-1}&m_{n}&m_{n+1}&\cdots &m_{2n-1}\\1&x&x^{2}&\cdots &x^{n}\end{bmatrix}}~,}$

where the constants c_n are arbitrary (depend on the normalization of P_n).

This comes directly from applying the Gram–Schmidt process to the monomials, imposing each polynomial to be orthogonal with respect to the previous ones. For example, orthogonality with ${\displaystyle P_{0}}$ prescribes that ${\displaystyle P_{1}}$ must have the form

$${\displaystyle P_{1}(x)=c_{1}\left(x-{\frac {\langle P_{0},x\rangle P_{0}}{\langle P_{0},P_{0}\rangle }}\right)=c_{1}(x-m_{1}),}$$

which can be seen to be consistent with the previously given expression with the determinant.

Recurrence relation

The polynomials P_n satisfy a recurrence relation of the form

${\displaystyle P_{n}(x)=(A_{n}x+B_{n})P_{n-1}(x)+C_{n}P_{n-2}(x)}$

where A_n is not 0. The converse is also true; see Favard's theorem.

Christoffel–Darboux formula

Zeros

If the measure dα is supported on an interval [a, b], all the zeros of P_n lie in [a, b]. Moreover, the zeros have the following interlacing property: if m < n, there is a zero of P_n between any two zeros of P_m. Electrostatic interpretations of the zeros can be given.

Combinatorial interpretation

From the 1980s, with the work of X. G. Viennot, J. Labelle, Y.-N. Yeh, D. Foata, and others, combinatorial interpretations were found for all the classical orthogonal polynomials. ^[3]

Multivariate orthogonal polynomials

The Macdonald polynomials are orthogonal polynomials in several variables, depending on the choice of an affine root system. They include many other families of multivariable orthogonal polynomials as special cases, including the Jack polynomials, the Hall–Littlewood polynomials, the Heckman–Opdam polynomials, and the Koornwinder polynomials. The Askey–Wilson polynomials are the special case of Macdonald polynomials for a certain non-reduced root system of rank 1.

Multiple orthogonal polynomials

Multiple orthogonal polynomials are polynomials in one variable that are orthogonal with respect to a finite family of measures.

Sobolev orthogonal polynomials

These are orthogonal polynomials with respect to a Sobolev inner product, i.e. an inner product with derivatives. Including derivatives has big consequences for the polynomials, in general they no longer share some of the nice features of the classical orthogonal polynomials.

Orthogonal polynomials with matrices

Orthogonal polynomials with matrices have either coefficients that are matrices or the indeterminate is a matrix.