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
18.30.1 | |||
. | |||
Then the associated orthogonal polynomials are defined by
18.30.2 | ||||
and
18.30.3 | |||
. | |||
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.
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,
18.30.5 | |||
where the generalized hypergeometric function is defined by (16.2.1).
For corresponding corecursive associated Jacobi polynomials see Letessier (1995).
These are defined by
18.30.6 | |||
. | |||
Explicitly,
18.30.7 | |||
(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).