(For other notation see Notation for the Special Functions.)
|real variable such that , unless stated otherwise.|
|nonnegative integer, except in §18.30.|
|Dirac delta (§1.17).|
|arbitrary small positive constant.|
|polynomial in of degree .|
|weight function on an open interval .|
|weights at points of a finite or countably infinite subset of .|
Central differences in imaginary direction:
The main functions treated in this chapter are:
Ultraspherical (or Gegenbauer): .
Chebyshev of first, second, third, and fourth kinds: , , , .
Shifted Chebyshev of first and second kinds: , .
Shifted Legendre: .
Laguerre: and . ( with is also called Generalized Laguerre.)
Hermite: , .
Continuous Hahn: .
Continuous Dual Hahn: .
Dual Hahn: .
Big -Jacobi: .
Little -Jacobi: .
Discrete -Hermite I: .
Discrete -Hermite II: .
Continuous -Ultraspherical: .
Continuous -Hermite: .
In Szegö (1975, §4.7) the ultraspherical polynomials are denoted by . The ultraspherical polynomials will not be considered for . They are defined in the literature by and
Nor do we consider the shifted Jacobi polynomials:
or the dilated Chebyshev polynomials of the first and second kinds:
In Koekoek et al. (2010) denotes the operator .
In Mason and Handscomb (2003), the definitions of the Chebyshev polynomials of the third and fourth kinds and are the converse of the definitions in this chapter.