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 of, table and/or classical:

    “Remember thee?
    Ay, thou poor ghost, whiles memory holds a seat
    In this distracted globe. Remember thee?
    Yea, from the table of my memory
    I’ll wipe away all trivial fond records,
    All saws of books, all forms, all pressures past
    That youth and observation copied there,
    And thy commandment all alone shall live
    Within the book and volume of my brain,”
    —William Shakespeare (1564–1616)

    “But hospitality must be for service, and not for show, or it pulls down the host. The brave soul rates itself too high to value itself by the splendor of its table and draperies. It gives what it hath, and all it hath, but its own majesty can lend a better grace to bannocks and fair water than belong to city feasts.”
    —Ralph Waldo Emerson (1803–1882)

    “Several classical sayings that one likes to repeat had quite a different meaning from the ones later times attributed to them.”
    —Johann Wolfgang Von Goethe (1749–1832)