# §15.9 Relations to Other Functions

## §15.9(i) Orthogonal Polynomials

For the notation see §§18.3 and 18.19.

## §15.9(ii) Jacobi Function

This is a generalization of Jacobi polynomials (§18.3) and has the representation

15.9.11

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.

15.9.22, ,

where the sign in the exponential is according as .

15.9.23, ,

where the sign in the exponential is according as .