18 Orthogonal Polynomials Other Orthogonal Polynomials 18.29 Asymptotic Approximations for

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-Hahn and Askey–Wilson Classes 18.31 Bernstein–Szegő Polynomials

§18.30 Associated OP’s

ⓘ

Keywords:: associated orthogonal polynomials, corecursive
Notes:: For (18.30.5) see Wimp (1987, Theorem 1). (18.30.7) is mentioned in Chihara (1978, Chapter VI, (12.6)), and proved in Barrucand and Dickinson (1968).
Referenced by:: §18.1(i)
Permalink:: http://dlmf.nist.gov/18.30
See also:: Annotations for Ch.18

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 Ch.18

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

18.30.2	$\displaystyle p_{-1}(x;c)$	$\displaystyle=0,$
18.30.2	$\displaystyle p_{0}(x;c)$	$\displaystyle=1,$
ⓘ 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 Ch.18

and

18.30.3		$p_{n+1}(x;c)=(A_{n+c}x+B_{n+c})p_{n}(x;c)-C_{n+c}p_{n-1}(x;c),$
$n=0,1,\dots$ .
ⓘ Symbols: $p_{n}(x)$ : polynomial of degree $n$ , $x$ : real variable, $n$ : continuous nonnegative real, $A_{n}$ : coefficient, $B_{n}$ : coefficient and $C_{n}$ : coefficient Referenced by: §18.30, §18.30 Permalink: http://dlmf.nist.gov/18.30.E3 Encodings: TeX, pMML, png See also: Annotations for §18.30 and Ch.18

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

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Keywords:: Jacobi, Jacobi polynomials, associated, associated Jacobi polynomials, associated orthogonal polynomials, generalized hypergeometric functions, relations to other functions
See also:: Annotations for §18.30 and Ch.18

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 Ch.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		$\frac{(-1)^{n}{\left(\alpha+\beta+c+1\right)_{n}}n!\,P^{(\alpha,\beta)}_{n}% \left(x;c\right)}{{\left(\alpha+\beta+2c+1\right)_{n}}{\left(\beta+c+1\right)_% {n}}}=\sum_{\ell=0}^{n}\frac{{\left(-n\right)_{\ell}}{\left(n+\alpha+\beta+2c+% 1\right)_{\ell}}}{{\left(c+1\right)_{\ell}}{\left(\beta+c+1\right)_{\ell}}}% \left(\tfrac{1}{2}x+\tfrac{1}{2}\right)^{\ell}\*{{}_{4}F_{3}}\left({\ell-n,n+% \ell+\alpha+\beta+2c+1,\beta+c,c\atop\beta+\ell+c+1,\ell+c+1,\alpha+\beta+2c};% 1\right),$
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$ generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)$ : associated Jacobi polynomial, $!$ : factorial (as in $n!$ ), $\ell$ : nonnegative integer, $x$ : real variable and $n$ : continuous nonnegative real Referenced by: §18.30 Permalink: http://dlmf.nist.gov/18.30.E5 Encodings: TeX, pMML, png See also: Annotations for §18.30, §18.30 and Ch.18

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

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Keywords:: Legendre, Legendre polynomials, associated, associated orthogonal polynomials
See also:: Annotations for §18.30 and Ch.18

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 Ch.18

Explicitly,

18.30.7		$P_{n}\left(x;c\right)=\sum_{\ell=0}^{n}\frac{c}{\ell+c}P_{\ell}\left(x\right)P% _{n-\ell}\left(x\right).$
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $P_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)$ : associated Legendre polynomial, $\ell$ : nonnegative integer, $x$ : real variable and $n$ : continuous nonnegative real Referenced by: §18.30 Permalink: http://dlmf.nist.gov/18.30.E7 Encodings: TeX, pMML, png See also: Annotations for §18.30, §18.30 and Ch.18

(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).

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-Hahn and Askey–Wilson Classes 18.31 Bernstein–Szegő Polynomials