# §18.30 Associated OP’s

In the recurrence relation (18.2.8) assume that the coefficients , , and are defined when is a continuous nonnegative real variable, and let be an arbitrary positive constant. Assume also

Then the associated orthogonal polynomials are defined by

18.30.2

and

Assume also that Eq. (18.30.3) continues to hold, except that when , is replaced by an arbitrary real constant. Then the polynomials generated in this manner are called corecursive associated OP’s.

## ¶ Associated Jacobi Polynomials

These are defined by

18.30.4,

where is given by (18.30.2) and (18.30.3), with , , and as in (18.9.2). Explicitly,

where the generalized hypergeometric function is defined by (16.2.1).

For corresponding corecursive associated Jacobi polynomials see Letessier (1995).

## ¶ Associated Legendre Polynomials

These are defined by

18.30.6.

Explicitly,

(These polynomials are not to be confused with associated Legendre functions §14.3(ii).)

For further results on associated Legendre polynomials see Chihara (1978, Chapter VI, §12); on associated Jacobi polynomials, see Wimp (1987) and Ismail and Masson (1991). For associated Pollaczek polynomials (compare §18.35) see Erdélyi et al. (1953b, §10.21). For associated Askey–Wilson polynomials see Rahman (2001).