§18.30 Associated OP’s

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).
Keywords:: associated orthogonal polynomials
Referenced by:: §18.1(i)
Permalink:: http://dlmf.nist.gov/18.30

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:: $C_{n}$ : real constant, $A_{n}$ : real constant and $n$ : continuous nonnegative real
Permalink:: http://dlmf.nist.gov/18.30.E1
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Then the associated orthogonal polynomials $p_{n}(x;c)$ are defined by

18.30.2

$p_{{-1}}(x;c)=0,$

$p_{0}(x;c)=1,$

Symbols:: $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable
Referenced by:: ¶ ‣ §18.30
Permalink:: http://dlmf.nist.gov/18.30.E2
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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 $n$ , $B_{n}$ : real constant, $C_{n}$ : real constant, $x$ : real variable, $A_{n}$ : real constant and $n$ : continuous nonnegative real
Referenced by:: ¶ ‣ §18.30, §18.30
Permalink:: http://dlmf.nist.gov/18.30.E3
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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

Keywords:: Jacobi polynomials, associated orthogonal polynomials, generalized hypergeometric functions

These are defined by

18.30.4 $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x;c\right)=p_{n}(x;c),$ $n=0,1,\dots$ ,

Defines:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x;c\right)$ : 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

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

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x;c\right)$ : associated Jacobi polynomial, $!$ : $n!$ : factorial, $\ell$ : nonnegative integer, $x$ : real variable and $n$ : continuous nonnegative real
Referenced by:: §18.30
Permalink:: http://dlmf.nist.gov/18.30.E5
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where the generalized hypergeometric function $\mathop{{{}_{{4}}F_{{3}}}\/}\nolimits$ is defined by (16.2.1).

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

¶ Associated Legendre Polynomials

Keywords:: Legendre polynomials, associated orthogonal polynomials

These are defined by

18.30.6 $\mathop{P_{{n}}\/}\nolimits\!\left(x;c\right)=\mathop{P^{{(0,0)}}_{{n}}\/}\nolimits\!\left(x;c\right),$ $n=0,1,\dots$ .

Defines:: $\mathop{P_{{n}}\/}\nolimits\!\left(x;c\right)$ : associated Legendre polynomial
Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x;c\right)$ : associated Jacobi polynomial, $x$ : real variable and $n$ : continuous nonnegative real
Permalink:: http://dlmf.nist.gov/18.30.E6
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Explicitly,

18.30.7 $\mathop{P_{{n}}\/}\nolimits\!\left(x;c\right)=\sum _{{\ell=0}}^{n}\frac{c}{\ell+c}\mathop{P_{{\ell}}\/}\nolimits\!\left(x\right)\mathop{P_{{n-\ell}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $\mathop{P_{{n}}\/}\nolimits\!\left(x;c\right)$ : 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

(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).