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,

Read more about this topic:  Classical Orthogonal Polynomials

Famous quotes containing the words table and/or classical:

    “For who is greater, the one who is at the table or the one who serves? Is it not the one at the table? But I am among you as one who serves.”
    —Bible: New Testament, Luke 22:27.

    “Et in Arcadia ego.
    [I too am in Arcadia.]”
    —Anonymous, Anonymous.

    Tomb inscription, appearing in classical paintings by Guercino and Poussin, among others. The words probably mean that even the most ideal earthly lives are mortal. Arcadia, a mountainous region in the central Peloponnese, Greece, was the rustic abode of Pan, depicted in literature and art as a land of innocence and ease, and was the title of Sir Philip Sidney’s pastoral romance (1590)