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§18.30 Associated OP’s

In the recurrence relation (18.2.8) assume that the coefficients An, Bn, and Cn+1 are defined when n is a continuous nonnegative real variable, and let c be an arbitrary positive constant. Assume also

18.30.1 AnAn+1Cn+1>0,
n0.

Then the associated orthogonal polynomials pn(x;c) are defined by

18.30.2 p-1(x;c) =0,
p0(x;c) =1,

and

18.30.3 pn+1(x;c)=(An+cx+Bn+c)pn(x;c)-Cn+cpn-1(x;c),
n=0,1,.

Assume also that Eq. (18.30.3) continues to hold, except that when n=0, Bc is replaced by an arbitrary real constant. Then the polynomials pn(x,c) generated in this manner are called corecursive associated OP’s.

Associated Jacobi Polynomials

These are defined by

18.30.4 Pn(α,β)(x;c)=pn(x;c),
n=0,1,,

where pn(x;c) is given by (18.30.2) and (18.30.3), with An, Bn, and Cn as in (18.9.2). Explicitly,

18.30.5 (-1)n(α+β+c+1)nn!Pn(α,β)(x;c)(α+β+2c+1)n(β+c+1)n==0n(-n)(n+α+β+2c+1)(c+1)(β+c+1)(12x+12)F34(-n,n++α+β+2c+1,β+c,cβ++c+1,+c+1,α+β+2c;1),

where the generalized hypergeometric function F34 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 Pn(x;c)=Pn(0,0)(x;c),
n=0,1,.

Explicitly,

18.30.7 Pn(x;c)==0nc+cP(x)Pn-(x).

(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).