Orthogonal polynomials

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.

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.

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.

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.

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.