Special Functions – Orthogonal polynomials – Classical orthogonal polynomials<

The classical orthogonal polynomials are named after Hermite, Laguerre and Jacobi.

The Hermite polynomials are orthogonal on the interval \((-\infty,\infty)=\mathbb{R}\) with respect to the normal distribution \(w(x)=e^{-x^2}\), the Laguerre polynomials are orthogonal on the interval \((0,\infty)\) with respect to the gamma distribution \(w(x)=e^{-x}x^{\alpha}\) and the Jacobi polynomials are orthogonal on the interval \((-1,1)\) with respect to the beta distribution \(w(x)=(1-x)^{\alpha}(1+x)^{\beta}\).

The Legendre polynomials form a special case (\(\alpha=\beta=0\)) of the Jacobi polynomials.

These classical orthogonal polynomials satisfy an orthogonality relation, a three-term recurrence relation, a second-order linear differential equation and a so-called Rodrigues formula. Moreover, for each family of classical orthogonal polynomials we have a generating function.

In the sequel we will often use the Kronecker delta which is defined by

\[\delta_{mn}:=\left\{\begin{array}{ll}0, & m\neq n\\[2.5mm]1, & m=n\end{array}\right.\]

for \(m,n\in\{0,1,2,\ldots\}\) and the notation

\[D=\displaystyle\frac{d}{dx}\]

for the differentiation operator. Then we have Leibniz' rule

\[D^n\left[f(x)g(x)\right]=\sum_{k=0}^n\binom{n}{k}D^kf(x)D^{n-k}g(x),\quad n=0,1,2,\ldots\]

which is a generalization of the product rule. The proof is by mathematical induction and by use of Pascal's triangle identity

\[\binom{n}{k}+\binom{n}{k-1}=\binom{n+1}{k},\quad k=1,2,\ldots,n.\]

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Last modified on May 22, 2021