§18.25 Wilson Class: Definitions

Keywords:: Wilson class orthogonal polynomials
Referenced by:: ¶ ‣ §18.21(i)
Permalink:: http://dlmf.nist.gov/18.25

§18.25(i) Preliminaries
§18.25(ii) Weights and Normalizations: Continuous Cases
§18.25(iii) Weights and Normalizations: Discrete Cases
§18.25(iv) Leading Coefficients

§18.25(i) Preliminaries

Notes:: For Table 18.25.1, Rows 2, 3, 4, see Wilson (1980); for Row 5 see Ismail (2005, (6.2.20)). (18.25.1) follows from (18.25.11).
Permalink:: http://dlmf.nist.gov/18.25.i

For the Wilson class OP’s $p_{n}(x)$ with $x=\lambda(y)$ : if the $y$ -orthogonality set is $\{ 0,1,\dots,N\}$ , then the role of the differentiation operator $\ifrac{d}{dx}$ in the Jacobi, Laguerre, and Hermite cases is played by the operator $\Delta _{{y}}$ followed by division by $\Delta _{{y}}(\lambda(y))$ , or by the operator $\nabla _{{y}}$ followed by division by $\nabla _{{y}}(\lambda(y))$ . Alternatively if the $y$ -orthogonality interval is $(0,\infty)$ , then the role of $\ifrac{d}{dx}$ is played by the operator $\delta _{{y}}$ followed by division by $\delta _{{y}}(\lambda(y))$ .

Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials $\mathop{W_{{n}}\/}\nolimits\!\left(x;a,b,c,d\right)$ , continuous dual Hahn polynomials $\mathop{S_{{n}}\/}\nolimits\!\left(x;a,b,c\right)$ , Racah polynomials $\mathop{R_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,\gamma,\delta\right)$ , and dual Hahn polynomials $\mathop{R_{{n}}\/}\nolimits\!\left(x;\gamma,\delta,N\right)$ .

Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.

$p_{n}(x)$	$x=\lambda(y)$	Orthogonality range for $y$	Constraints
$\mathop{W_{{n}}\/}\nolimits\!\left(x;a,b,c,d\right)$	$y^{2}$	$(0,\infty)$	$\realpart{(a,b,c,d)}>0$ ; nonreal parameters in conjugate pairs
$\mathop{S_{{n}}\/}\nolimits\!\left(x;a,b,c\right)$	$y^{2}$	$(0,\infty)$	$\realpart{(a,b,c)}>0$ ; nonreal parameters in conjugate pairs
$\mathop{R_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,\gamma,\delta\right)$	$y(y+\gamma+\delta+1)$	$\{ 0,1,\dots,N\}$	$\alpha+1$ or $\beta+\delta+1$ or $\gamma+1=-N;$ for further constraints see (18.25.1)
$\mathop{R_{{n}}\/}\nolimits\!\left(x;\gamma,\delta,N\right)$	$y(y+\gamma+\delta+1)$	$\{ 0,1,\dots,N\}$	$\gamma,\delta>-1$ or $<-N$

Defines:: $\mathop{R_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,\gamma,\delta\right)$ : Racah polynomial, $\mathop{W_{{n}}\/}\nolimits\!\left(x;a,b,c,d\right)$ : Wilson polynomial, $\mathop{S_{{n}}\/}\nolimits\!\left(x;a,b,c\right)$ : continuous dual Hahn polynomial and $\mathop{R_{{n}}\/}\nolimits\!\left(x;\gamma,\delta,N\right)$ : dual Hahn polynomial
Symbols:: $(a,b)$ : open interval, $\realpart{}$ : real part, $\{\ldots\}$ : sequence, asymptotic sequence (or scale), or enumerable set, $y$ : real variable, $n$ : nonnegative integer, $N$ : positive integer, $\delta$ : arbitary small positive constant, $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable
Keywords:: Wilson class orthogonal polynomials
Referenced by:: §18.25(i), §18.25(i)
Permalink:: http://dlmf.nist.gov/18.25.T1

¶ Further Constraints for Racah Polynomials

If $\alpha+1=-N$ , then the weights will be positive iff one of the following eight sets of inequalities holds:

18.25.1

$-\delta-1<\beta<\gamma+1<-N+1.$

$N-1<-\delta-1<\beta<\gamma+1.$

$\gamma,\delta>-1,\quad\beta>N+\gamma.$

$\gamma,\delta>-1,\quad\beta<-N-\delta.$

$N-1<N+\gamma<\beta<-N-\delta.$

$N+\gamma<\beta<-N-\delta<-N-1.$

$\gamma,\delta<-N,\quad\beta>-1-\delta.$

$\gamma,\delta<-N,\quad\beta<\gamma+1.$

Symbols:: $N$ : positive integer and $\delta$ : arbitary small positive constant
Referenced by:: §18.25(i), Table 18.25.1
Permalink:: http://dlmf.nist.gov/18.25.E1
Encodings:: TeX, TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png, png

The first four sets imply $\gamma+\delta>-2$ , and the last four imply $\gamma+\delta<-2N$ .

§18.25(ii) Weights and Normalizations: Continuous Cases

Notes:: For (18.25.3)–(18.25.5) see Wilson (1980) and Andrews et al. (1999, (3.8.3)). For (18.25.6)–(18.25.8) see Wilson (1980).
Keywords:: Wilson class orthogonal polynomials, weight functions
Permalink:: http://dlmf.nist.gov/18.25.ii

18.25.2 $\int _{0}^{\infty}p_{n}(x)p_{m}(x)w(x)dx=h_{n}\delta _{{n,m}}.$

Symbols:: $\delta _{{j,k}}$ : Kronecker delta, $dx$ : differential of $x$ , $\int$ : integral, $w(x)$ : weight function, $m$ : nonnegative integer, $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ , $x$ : real variable and $h_{n}$
Permalink:: http://dlmf.nist.gov/18.25.E2
Encodings:: TeX, pMML, png

¶ Wilson

18.25.3 $p_{n}(x)=\mathop{W_{{n}}\/}\nolimits\!\left(x;a_{1},a_{2},a_{3},a_{4}\right),$

Symbols:: $\mathop{W_{{n}}\/}\nolimits\!\left(x;a,b,c,d\right)$ : Wilson polynomial, $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable
Referenced by:: §18.25(ii)
Permalink:: http://dlmf.nist.gov/18.25.E3
Encodings:: TeX, pMML, png

18.25.4 $w(y^{2})=\frac{1}{2y}\left|\frac{\prod _{j}\mathop{\Gamma\/}\nolimits\!\left(a_{j}+iy\right)}{\mathop{\Gamma\/}\nolimits\!\left(2iy\right)}\right|^{2},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $y$ : real variable and $w(x)$ : weight function
Permalink:: http://dlmf.nist.gov/18.25.E4
Encodings:: TeX, pMML, png

18.25.5 $h_{n}=\frac{n!\, 2\pi\prod _{{j<\ell}}\mathop{\Gamma\/}\nolimits\!\left(n+a_{j}+a_{\ell}\right)}{(2n-1+\sum _{j}a_{j})\mathop{\Gamma\/}\nolimits\!\left(n-1+\sum _{j}a_{j}\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $!$ : $n!$ : factorial, $\ell$ : nonnegative integer, $n$ : nonnegative integer and $h_{n}$
Referenced by:: §18.25(ii)
Permalink:: http://dlmf.nist.gov/18.25.E5
Encodings:: TeX, pMML, png

¶ Continuous Dual Hahn

18.25.6 $p_{n}(x)=\mathop{S_{{n}}\/}\nolimits\!\left(x;a_{1},a_{2},a_{3}\right),$

Symbols:: $\mathop{S_{{n}}\/}\nolimits\!\left(x;a,b,c\right)$ : continuous dual Hahn polynomial, $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable
Referenced by:: §18.25(ii)
Permalink:: http://dlmf.nist.gov/18.25.E6
Encodings:: TeX, pMML, png

18.25.7 $w(y^{2})=\frac{1}{2y}\left|\frac{\prod _{j}\mathop{\Gamma\/}\nolimits\!\left(a_{j}+iy\right)}{\mathop{\Gamma\/}\nolimits\!\left(2iy\right)}\right|^{2},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $y$ : real variable and $w(x)$ : weight function
Permalink:: http://dlmf.nist.gov/18.25.E7
Encodings:: TeX, pMML, png

18.25.8 $h_{n}=n!\, 2\pi\prod _{{j<\ell}}\mathop{\Gamma\/}\nolimits\!\left(n+a_{j}+a_{\ell}\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $!$ : $n!$ : factorial, $\ell$ : nonnegative integer, $n$ : nonnegative integer and $h_{n}$
Referenced by:: §18.25(ii)
Permalink:: http://dlmf.nist.gov/18.25.E8
Encodings:: TeX, pMML, png

§18.25(iii) Weights and Normalizations: Discrete Cases

Notes:: For (18.25.10)–(18.25.12) see Wilson (1980). For (18.25.13)–(18.25.15) see Ismail (2005, (6.2.20)).
Permalink:: http://dlmf.nist.gov/18.25.iii

18.25.9 $\sum _{{y=0}}^{N}p_{n}(y(y+\gamma+\delta+1))p_{m}(y(y+\gamma+\delta+1))\*\frac{\gamma+\delta+1+2y}{\gamma+\delta+1+y}\omega _{y}=h_{n}\delta _{{n,m}}.$

Symbols:: $\delta _{{j,k}}$ : Kronecker delta, $y$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer, $N$ : positive integer, $\delta$ : arbitary small positive constant, $p_{n}(x)$ : polynomial of degree $n$ and $h_{n}$
Permalink:: http://dlmf.nist.gov/18.25.E9
Encodings:: TeX, pMML, png

¶ Racah

18.25.10 $p_{n}(x)=\mathop{R_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,\gamma,\delta\right),$ $\alpha+1=-N$ ,

Symbols:: $\mathop{R_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,\gamma,\delta\right)$ : Racah polynomial, $n$ : nonnegative integer, $N$ : positive integer, $\delta$ : arbitary small positive constant, $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable
Referenced by:: §18.25(iii)
Permalink:: http://dlmf.nist.gov/18.25.E10
Encodings:: TeX, pMML, png

18.25.11 $\omega _{y}=\frac{\left(\alpha+1\right)_{{y}}\left(\beta+\delta+1\right)_{{y}}\left(\gamma+1\right)_{{y}}\left(\gamma+\delta+2\right)_{{y}}}{\left(-\alpha+\gamma+\delta+1\right)_{{y}}\left(-\beta+\gamma+1\right)_{{y}}\left(\delta+1\right)_{{y}}y!},$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $y$ : real variable and $\delta$ : arbitary small positive constant
Referenced by:: §18.25(i)
Permalink:: http://dlmf.nist.gov/18.25.E11
Encodings:: TeX, pMML, png

18.25.12 $h_{n}=\frac{\left(-\beta\right)_{{N}}\left(\gamma+\delta+2\right)_{{N}}}{\left(-\beta+\gamma+1\right)_{{N}}\left(\delta+1\right)_{{N}}}\frac{\left(n+\alpha+\beta+1\right)_{{n}}n!}{\left(\alpha+\beta+2\right)_{{2n}}}\*\frac{\left(\alpha+\beta-\gamma+1\right)_{{n}}\left(\alpha-\delta+1\right)_{{n}}\left(\beta+1\right)_{{n}}}{\left(\alpha+1\right)_{{n}}\left(\beta+\delta+1\right)_{{n}}\left(\gamma+1\right)_{{n}}}.$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $n$ : nonnegative integer, $N$ : positive integer, $\delta$ : arbitary small positive constant and $h_{n}$
Referenced by:: §18.25(iii)
Permalink:: http://dlmf.nist.gov/18.25.E12
Encodings:: TeX, pMML, png

¶ Dual Hahn

Keywords:: Wilson class orthogonal polynomials, weight functions

18.25.13 $p_{n}(x)=\mathop{R_{{n}}\/}\nolimits\!\left(x;\gamma,\delta,N\right),$

Symbols:: $\mathop{R_{{n}}\/}\nolimits\!\left(x;\gamma,\delta,N\right)$ : dual Hahn polynomial, $n$ : nonnegative integer, $N$ : positive integer, $\delta$ : arbitary small positive constant, $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable
Referenced by:: §18.25(iii)
Permalink:: http://dlmf.nist.gov/18.25.E13
Encodings:: TeX, pMML, png

18.25.14 $\omega _{y}=\frac{(-1)^{y}\left(-N\right)_{{y}}\left(\gamma+1\right)_{{y}}\left(\gamma+\delta+1\right)_{{2}}}{\left(N+\gamma+\delta+2\right)_{{y}}\left(\delta+1\right)_{{y}}y!},$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $y$ : real variable, $N$ : positive integer and $\delta$ : arbitary small positive constant
Permalink:: http://dlmf.nist.gov/18.25.E14
Encodings:: TeX, pMML, png

18.25.15 $h_{n}=\frac{n!\,(N-n)!\,\left(\gamma+\delta+2\right)_{{N}}}{N!\,\left(\gamma+1\right)_{{n}}\left(\delta+1\right)_{{N-n}}}.$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $n$ : nonnegative integer, $N$ : positive integer, $\delta$ : arbitary small positive constant and $h_{n}$
Referenced by:: §18.25(iii)
Permalink:: http://dlmf.nist.gov/18.25.E15
Encodings:: TeX, pMML, png

§18.25(iv) Leading Coefficients

Notes:: Table 18.25.2 follows from §18.26(i).
Permalink:: http://dlmf.nist.gov/18.25.iv

Table 18.25.2 provides the leading coefficients $k_{n}$ (§18.2(iii)) for the Wilson, continuous dual Hahn, Racah, and dual Hahn polynomials.

Table 18.25.2: Wilson class OP’s: leading coefficients.

$p_{n}(x)$	$k_{n}$
$\mathop{W_{{n}}\/}\nolimits\!\left(x;a,b,c,d\right)$	$(-1)^{n}\left(n+a+b+c+d-1\right)_{{n}}$
$\mathop{S_{{n}}\/}\nolimits\!\left(x;a,b,c\right)$	$(-1)^{n}$
$\mathop{R_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,\gamma,\delta\right)$	$\dfrac{\left(n+\alpha+\beta+1\right)_{{n}}}{\left(\alpha+1\right)_{{n}}\left(\beta+\delta+1\right)_{{n}}\left(\gamma+1\right)_{{n}}}$
$\mathop{R_{{n}}\/}\nolimits\!\left(x;\gamma,\delta,N\right)$	$\dfrac{1}{\left(\gamma+1\right)_{{n}}\left(-N\right)_{{n}}}$