18 Orthogonal PolynomialsOther Orthogonal Polynomials18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes18.31 Bernstein–Szegö Polynomials

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 | $${A}_{n}{A}_{n+1}{C}_{n+1}>0,$$ | ||

$n\ge 0$. | |||

Then the *associated orthogonal polynomials* ${p}_{n}\left(x;c\right)$ are defined by

18.30.2 | ${p}_{-1}\left(x;c\right)$ | $=0,$ | ||

${p}_{0}\left(x;c\right)$ | $=1,$ | |||

and

18.30.3 | $${p}_{n+1}\left(x;c\right)=\left({A}_{n+c}x+{B}_{n+c}\right){p}_{n}\left(x;c\right)-{C}_{n+c}{p}_{n-1}\left(x;c\right),$$ | ||

$n=0,1,\mathrm{\dots}$. | |||

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
${p}_{n}\left(x,c\right)$ generated in this manner are called *corecursive associated
OP’s*.

These are defined by

18.30.4 | $${P}_{n}^{\left(\alpha ,\beta \right)}\left(x;c\right)={p}_{n}\left(x;c\right),$$ | ||

$n=0,1,\mathrm{\dots}$, | |||

where ${p}_{n}\left(x;c\right)$ 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 | $$\begin{array}{l}\frac{{\left(-1\right)}^{n}{\left(\alpha +\beta +c+1\right)}_{n}n\mathrm{!}{P}_{n}^{\left(\alpha ,\beta \right)}\left(x;c\right)}{{\left(\alpha +\beta +2c+1\right)}_{n}{\left(\beta +c+1\right)}_{n}}\\ \phantom{\rule{2em}{0ex}}=\sum _{\mathrm{\ell}=0}^{n}\frac{{\left(-n\right)}_{\mathrm{\ell}}{\left(n+\alpha +\beta +2c+1\right)}_{\mathrm{\ell}}}{{\left(c+1\right)}_{\mathrm{\ell}}{\left(\beta +c+1\right)}_{\mathrm{\ell}}}{\left(\frac{1}{2}x+\frac{1}{2}\right)}^{\mathrm{\ell}}{}_{4}F_{3}\left(\begin{array}{c}\mathrm{\ell}-n,n+\mathrm{\ell}+\alpha +\beta +2c+1,\beta +c,c\\ \beta +\mathrm{\ell}+c+1,\mathrm{\ell}+c+1,\alpha +\beta +2c\end{array};1\right),\end{array}$$ | ||

where the generalized hypergeometric function ${}_{4}F_{3}$ is defined by (16.2.1).

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

These are defined by

18.30.6 | $${P}_{n}\left(x;c\right)={P}_{n}^{\left(0,0\right)}\left(x;c\right),$$ | ||

$n=0,1,\mathrm{\dots}$. | |||

Explicitly,

18.30.7 | $${P}_{n}\left(x;c\right)=\sum _{\mathrm{\ell}=0}^{n}\frac{c}{\mathrm{\ell}+c}{P}_{\mathrm{\ell}}\left(x\right){P}_{n-\mathrm{\ell}}\left(x\right).$$ | ||

(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).