Classical Orthogonal Polynomials - Definition

Definition

In general, the orthogonal polynomials Pn with respect to a weight W:R→R+ on the real line are defined by

\begin{align} &\deg P_n = n~, \quad n = 0,1,2,\ldots\\ &\int P_m(x) \, P_n(x) \, W(x)\,dx = 0~, \quad m \neq n~. \end{align}

The relations above define Pn up to multiplication by a number. Various normalisations are used to fix the constant, e.g.

The classical orthogonal polynomials correspond to the three families of weights:

\begin{align} \text{(Jacobi)}\quad &W(x) = \begin{cases} (1 - x)^\alpha (1+x)^\beta~, & -1 \leq x \leq 1 \\ 0~, &\text{otherwise} \end{cases} \\ \text{(Hermite)}\quad &W(x) = \exp(- x^2) \\ \text{(Laguerre)}\quad &W(x) = \begin{cases} x^\alpha \exp(- x)~, &\quad x \geq 0 \\ 0~, &\text{otherwise} \end{cases} \end{align}

The standard normalisation (also called standardization) is detailed below.

Read more about this topic:  Classical Orthogonal Polynomials

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