§ 2 Orthogonal polynomials

1.        Legendre polynomial

[ Generating function of Legendre polynomial ] is expanded by function press :

to define the sequence of Legendre polynomials

The function is called a generating or generating function .

[ Expression of Legendre polynomial ]

･･････････････････････････････････

(Merfeit expression)

[ Legendre differential equations ]

[ Orthogonality of Legendre Polynomials ]

[ Inequalities and Special Values ]

[ Recursion formula and derivative formula ]

(recursion relationship)

2.        Chebyshev polynomials of the first kind

[ Generating function of Chebyshev polynomials of the first kind ] is expanded   by the generating function :

to define a sequence of Chebyshev polynomials of the first kind .

[ Expressions for Chebyshev polynomials of the first kind ]

･･･････････････････････････････････

[ Chebyshev differential equations of the first kind ]

[ Orthogonality of Chebyshev Polynomials of the First Kind ]

[ Inequalities and Special Values ]

[ Recursion formula and derivative formula ]

(recursive formula)

3.        Chebyshev polynomials of the second kind

[ Generating function of Chebyshev polynomials of the second kind ] is expanded by the generating function :

to define a sequence of Chebyshev polynomials of the second kind .

[ Expressions for Chebyshev polynomials of the second kind ]

……………………

[ Chebyshev differential equations of the second kind ]

[ Orthogonality of Chebyshev Polynomials of the Second Kind ]

[ Inequalities and Special Values ]

[ Recursive formula and related formulas ]

(recursive formula)

4.        Laguerre polynomials

1.        General Laguerre polynomials

[ Generic function of a general Laguerre polynomial ] is expanded by the generating function :

to define a general sequence of Laguerre polynomials .

[ Expression of general Laguerre polynomial ]

where is the Kummer function, which is a first-order Bessel function. very

[ General Laguerre Differential Equations ]

[ Orthogonality of General Laguerre Polynomials ]

[ Inequalities and Special Values ]

[ Recursive formula and related formulas ]

(recursive formula)

where is the Hermitian polynomial.

2.          Laguerre polynomials

In general Laguerre polynomials, then , define

is the Laguerre polynomial . Its corresponding formula is

(generating function expansion)

････････････････････････････････

(Laguerre differential equations)

(orthogonality)

(recursive formula)

5.        Hermitian polynomials

[ Generating function of Hermitian polynomial ] is expanded by the generating function :

to define a sequence of Hermitian polynomials .

[ Expression of Hermitian polynomial ]

･･････････

where is the Kummer function .

[ Asymptotic expressions for Hermitian polynomials ]

[ Hermitian differential equations ]

[ Orthogonality of Hermitian Polynomials ]

[ Inequalities and Special Values ]

[ Recursive formula and related formulas ]

(recursive formula)

[ weighted Hermitian polynomial ]   is the Hermitian polynomial of the weight function,

Its expression is

relationship with

Six,        Jacobi polynomial

[ Generating function of Jacobian polynomial ] is expanded by the generating function (where ):

to define a sequence of Jacobi polynomials .

[ Expression for Jacobian polynomial ]

where F is the hypergeometric function.

[ Jacobi Differential Equations ]

[ Orthogonality of Jacobian Polynomials ]

[ Inequalities and Special Values ]

where is one of the two maxima points closest to the point .

[ Recursive formula and related formulas ]

(recursive formula)

7.        Geigenberger polynomial

[ Generating function of Geigenberger polynomial ]   Expansion by the generating function

to define the sequence of Geigenberger polynomials, also known as special spherical polynomials .

[ Expression of Geigenberger polynomial ]

where is the hypergeometric function .

······························································································ ･･･

[ Gegenberg differential equations ]

[ Orthogonality of Geigenberg Polynomials ]

[ Inequalities and Special Values ]

and not an integer)

( not an integer, and

[ Recursive formula and related formulas ]

(recursive formula)