# §18.30 Associated OP’s

In the recurrence relation (18.2.8) assume that the coefficients $A_{n}$, $B_{n}$, and $C_{n+1}$ are defined when $n$ is a continuous nonnegative real variable, and let $c$ be an arbitrary positive constant. Assume also

 18.30.1 $A_{n}A_{n+1}C_{n+1}>0,$ $n\geq 0$. ⓘ Symbols: $n$: continuous nonnegative real, $A_{n}$: coefficient and $C_{n}$: coefficient Permalink: http://dlmf.nist.gov/18.30.E1 Encodings: TeX, pMML, png See also: Annotations for 18.30 and 18

Then the associated orthogonal polynomials $p_{n}(x;c)$ are defined by

 18.30.2 $\displaystyle p_{-1}(x;c)$ $\displaystyle=0,$ $\displaystyle p_{0}(x;c)$ $\displaystyle=1,$ ⓘ Symbols: $p_{n}(x)$: polynomial of degree $n$ and $x$: real variable Referenced by: §18.30 Permalink: http://dlmf.nist.gov/18.30.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 18.30 and 18

and

 18.30.3 $p_{n+1}(x;c)=(A_{n+c}x+B_{n+c})p_{n}(x;c)-C_{n+c}p_{n-1}(x;c),$ $n=0,1,\dots$.

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

## Associated Jacobi Polynomials

These are defined by

 18.30.4 $P^{(\alpha,\beta)}_{n}\left(x;c\right)=p_{n}(x;c),$ $n=0,1,\dots$, ⓘ Defines: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)$: associated Jacobi polynomial Symbols: $p_{n}(x)$: polynomial of degree $n$, $x$: real variable and $n$: continuous nonnegative real Permalink: http://dlmf.nist.gov/18.30.E4 Encodings: TeX, pMML, png See also: Annotations for 18.30, 18.30 and 18

where $p_{n}(x;c)$ is given by (18.30.2) and (18.30.3), with $A_{n}$, $B_{n}$, and $C_{n}$ as in (18.9.2). Explicitly,

 18.30.5 $\frac{(-1)^{n}{\left(\alpha+\beta+c+1\right)_{n}}n!\,P^{(\alpha,\beta)}_{n}% \left(x;c\right)}{{\left(\alpha+\beta+2c+1\right)_{n}}{\left(\beta+c+1\right)_% {n}}}=\sum_{\ell=0}^{n}\frac{{\left(-n\right)_{\ell}}{\left(n+\alpha+\beta+2c+% 1\right)_{\ell}}}{{\left(c+1\right)_{\ell}}{\left(\beta+c+1\right)_{\ell}}}% \left(\tfrac{1}{2}x+\tfrac{1}{2}\right)^{\ell}\*{{}_{4}F_{3}}\left({\ell-n,n+% \ell+\alpha+\beta+2c+1,\beta+c,c\atop\beta+\ell+c+1,\ell+c+1,\alpha+\beta+2c};% 1\right),$

where the generalized hypergeometric function ${{}_{4}F_{3}}$ 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 $P_{n}\left(x;c\right)=P^{(0,0)}_{n}\left(x;c\right),$ $n=0,1,\dots$. ⓘ Defines: $P_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)$: associated Legendre polynomial Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)$: associated Jacobi polynomial, $x$: real variable and $n$: continuous nonnegative real Permalink: http://dlmf.nist.gov/18.30.E6 Encodings: TeX, pMML, png See also: Annotations for 18.30, 18.30 and 18

Explicitly,

 18.30.7 $P_{n}\left(x;c\right)=\sum_{\ell=0}^{n}\frac{c}{\ell+c}P_{\ell}\left(x\right)P% _{n-\ell}\left(x\right).$

(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).