§15.9 Relations to Other Functions
Contents
- §15.9(i) Orthogonal Polynomials
- §15.9(ii) Jacobi Function
- §15.9(iii) Gegenbauer Function
- §15.9(iv) Associated Legendre Functions; Ferrers Functions
§15.9(i) Orthogonal Polynomials
¶ Jacobi
¶ Gegenbauer (or Ultraspherical)
¶ Chebyshev
¶ Legendre
¶ Krawtchouk
compare also §15.2(ii).
¶ Meixner
¶ Meixner–Pollaczek
§15.9(ii) Jacobi Function
This is a generalization of Jacobi polynomials (§18.3) and has the representation
The Jacobi transform is defined as
with inverse
where the contour of integration is located to the right of the poles of the gamma functions in the integrand, and
For this result, together with restrictions on the functions and , see Koornwinder (1984a).
§15.9(iii) Gegenbauer Function
This is a generalization of Gegenbauer (or ultraspherical) polynomials (§18.3). It is defined by:
§15.9(iv) Associated Legendre Functions; Ferrers Functions
Any hypergeometric function for which a quadratic transformation exists can be expressed in terms of associated Legendre functions or Ferrers functions. For examples see §§14.3(i)–14.3(iii) and 14.21(iii).
The following formulas apply with principal branches of the hypergeometric functions, associated Legendre functions, and fractional powers.
where the sign in the exponential is according as .
where the sign in the exponential is according as .