Classical Orthogonal Polynomials - Table of Classical Orthogonal Polynomials

Table of Classical Orthogonal Polynomials

The following table summarises the properties of the classical orthogonal polynomials.

Name, and conventional symbol Chebyshev, Chebyshev
(second kind),
Legendre, Hermite,
Limits of orthogonality
Weight,
Standardization Lead term =
Square of norm \left\{
\begin{matrix}
\pi &:~n=0 \\
\pi/2 &:~n\ne 0
\end{matrix}\right.
Leading term
Second term,
Constant in diff. equation,
Constant in Rodrigues' formula,
Recurrence relation,
Recurrence relation,
Recurrence relation,
Name, and conventional symbol Associated Laguerre, Laguerre,
Limits of orthogonality
Weight,
Standardization Lead term = Lead term =
Square of norm,
Leading term,
Second term,
Constant in diff. equation,
Constant in Rodrigues' formula,
Recurrence relation,
Recurrence relation,
Recurrence relation,
Name, and conventional symbol Gegenbauer, Jacobi,
Limits of orthogonality
Weight,
Standardization if
Square of norm, \frac{2^{\alpha+\beta+1}\,\Gamma(n\!+\!\alpha\!+\!1)\,\Gamma(n\!+\!\beta\!+\!1)}
{n!(2n\!+\!\alpha\!+\!\beta\!+\!1)\Gamma(n\!+\!\alpha\!+\!\beta\!+\!1)}
Leading term,
Second term,
Constant in diff. equation,
Constant in Rodrigues' formula, \frac{(-2)^n\,n!\,\Gamma(2\alpha)\,\Gamma(n\!+\!1/2\!+\!\alpha)}
{\Gamma(n\!+\!2\alpha)\Gamma(\alpha\!+\!1/2)}
Recurrence relation,
Recurrence relation,
Recurrence relation,

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