18 Orthogonal Polynomials Askey Scheme 18.24 Hahn Class: Asymptotic Approximations 18.26 Wilson Class: Continued

§18.25 Wilson Class: Definitions

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Keywords:: Wilson class orthogonal polynomials, definitions
Referenced by:: §18.21(i), Erratum (V1.2.0) §18.25
Permalink:: http://dlmf.nist.gov/18.25
See also:: Annotations for Ch.18

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

§18.25(i) Preliminaries

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Notes:: For Table 18.25.1, Rows 2, 3, 4, see Wilson (1980); for Row 5 see Ismail (2009, (6.2.20)). (18.25.1) follows from (18.25.11).
Referenced by:: Erratum (V1.2.0) §18.25
Permalink:: http://dlmf.nist.gov/18.25.i
Addition (effective with 1.2.0):: A sentence was added, just before Table 18.25.1, at the end of the first paragraph of this subsection. Also, some explanatory text was added just below the table.
See also:: Annotations for §18.25 and Ch.18

For the Wilson class OP’s $p_{n}(x)$ with $x=\lambda(y)$ : if the -orthogonality set is $\{0,1,\dots,N\}$ , then the role of the differentiation operator $\ifrac{\mathrm{d}}{\mathrm{d}x}$ 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 -orthogonality interval is $(0,\infty)$ , then the role of $\ifrac{\mathrm{d}}{\mathrm{d}x}$ is played by the operator $\delta_{y}$ followed by division by $\delta_{y}(\lambda(y))$ . The Wilson class consists of two discrete families (Racah and dual Hahn) and two continuous families (Wilson and continuous dual Hahn).

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 $W_{n}\left(x;a,b,c,d\right)$ , continuous dual Hahn polynomials $S_{n}\left(x;a,b,c\right)$ , Racah polynomials $R_{n}\left(x;\alpha,\beta,\gamma,\delta\right)$ , and dual Hahn polynomials $R_{n}\left(x;\gamma,\delta,N\right)$ .

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

OP	$p_{n}(x)$	$x=\lambda(y)$	Orthogonality range for	Constraints
Wilson	$W_{n}\left(x;a,b,c,d\right)$	$y^{2}$	$(0,\infty)$	$\Re(a,b,c,d)>0$ ; nonreal parameters in conjugate pairs
continuous dual Hahn	$S_{n}\left(x;a,b,c\right)$	$y^{2}$	$(0,\infty)$	$\Re(a,b,c)>0$ ; nonreal parameters in conjugate pairs
Racah	$R_{n}\left(x;\alpha,\beta,\gamma,\delta\right)$	$y(y+\gamma+\delta+1)$	$\{0,1,\dots,N\}$	$\alpha+1$ or $\beta+\delta+1$ or $\gamma+1=-N;$ for further constraints see (18.25.1)
dual Hahn	$R_{n}\left(x;\gamma,\delta,N\right)$	$y(y+\gamma+\delta+1)$	$\{0,1,\dots,N\}$	$\gamma,\delta>-1$ or

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Symbols:: $R_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{% \delta}\right)$ : Racah polynomial, $W_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\right)$ : Wilson polynomial, $S_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$ : continuous dual Hahn polynomial, $R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{\delta},\NVar{N}\right)$ : dual Hahn polynomial, $(\NVar{a},\NVar{b})$ : open interval, $\Re$ : real part, : real variable, : positive integer, $\delta$ : arbitrary small positive constant, $p_{n}(x)$ : polynomial of degree , : nonnegative integer and : real variable
Keywords:: Wilson class orthogonal polynomials, orthogonality properties, transformations of variable
Referenced by:: §18.25(i), §18.25(i)
Permalink:: http://dlmf.nist.gov/18.25.T1
Modification (effective with 1.2.0):: A column was added on the left-hand side of the table which gives the names of the polynomials.
See also:: Annotations for §18.25(i), §18.25 and Ch.18

Under certain conditions on their parameters the orthogonality range for the Wilson polynomials and continuous dual Hahn polynomials is $(0,\infty)\cup S$ , where is a specific finite set, e.g., for the case and , , are positive or a pair of complex conjugates with positive real parts, see Wilson (1980, (3.3)) or Koekoek et al. (2010, (9.1.3)).

Further Constraints for Racah Polynomials

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See also:: Annotations for §18.25(i), §18.25 and Ch.18

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

18.25.1

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

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

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

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

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

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

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

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

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Symbols:: : positive integer and $\delta$ : arbitrary small positive constant
Referenced by:: Table 18.25.1, §18.25(i)
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
See also:: Annotations for §18.25(i), §18.25(i), §18.25 and Ch.18

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

§18.25(ii) Weights and Standardizations: Continuous Cases

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Keywords:: Wilson class orthogonal polynomials, standardizations, weight functions
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).
Referenced by:: Erratum (V1.2.0) §18.25
Permalink:: http://dlmf.nist.gov/18.25.ii
Modification (effective with 1.2.0):: The title of this subsection was updated such that “Normalizations” is now given by “Standardizations”.
See also:: Annotations for §18.25 and Ch.18

18.25.2 $\int_{0}^{\infty}p_{n}(x)p_{m}(x)w(x)\,\mathrm{d}x=h_{n}\delta_{n,m}.$

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $p_{n}(x)$ : polynomial of degree , : weight function, : nonnegative integer, : nonnegative integer, : real variable and $h_{n}$
Permalink:: http://dlmf.nist.gov/18.25.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §18.25(ii), §18.25 and Ch.18

Wilson

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See also:: Annotations for §18.25(ii), §18.25 and Ch.18

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

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Symbols:: $W_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\right)$ : Wilson polynomial, $p_{n}(x)$ : polynomial of degree , : nonnegative integer and : real variable
Referenced by:: §18.25(ii)
Permalink:: http://dlmf.nist.gov/18.25.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §18.25(ii), §18.25(ii), §18.25 and Ch.18

18.25.4 $w(y^{2})=\frac{1}{2y}\left|\frac{\prod_{j}\Gamma\left(a_{j}+iy\right)}{\Gamma% \left(2iy\right)}\right|^{2},$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, : real variable and : weight function
Permalink:: http://dlmf.nist.gov/18.25.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §18.25(ii), §18.25(ii), §18.25 and Ch.18

18.25.5 $h_{n}=\frac{n!\,2\pi\prod_{j<\ell}\Gamma\left(n+a_{j}+a_{\ell}\right)}{(2n-1+% \sum_{j}a_{j})\Gamma\left(n-1+\sum_{j}a_{j}\right)}.$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, : factorial (as in ), $\ell$ : nonnegative integer, : nonnegative integer and $h_{n}$
Referenced by:: §18.25(ii)
Permalink:: http://dlmf.nist.gov/18.25.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §18.25(ii), §18.25(ii), §18.25 and Ch.18

Continuous Dual Hahn

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See also:: Annotations for §18.25(ii), §18.25 and Ch.18

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

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Symbols:: $S_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$ : continuous dual Hahn polynomial, $p_{n}(x)$ : polynomial of degree , : nonnegative integer and : real variable
Referenced by:: §18.25(ii)
Permalink:: http://dlmf.nist.gov/18.25.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §18.25(ii), §18.25(ii), §18.25 and Ch.18

18.25.7 $w(y^{2})=\frac{1}{2y}\left|\frac{\prod_{j}\Gamma\left(a_{j}+iy\right)}{\Gamma% \left(2iy\right)}\right|^{2},$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, : real variable and : weight function
Permalink:: http://dlmf.nist.gov/18.25.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §18.25(ii), §18.25(ii), §18.25 and Ch.18

18.25.8 $h_{n}=n!\,2\pi\prod_{j<\ell}\Gamma\left(n+a_{j}+a_{\ell}\right).$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, : factorial (as in ), $\ell$ : nonnegative integer, : nonnegative integer and $h_{n}$
Referenced by:: §18.25(ii)
Permalink:: http://dlmf.nist.gov/18.25.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §18.25(ii), §18.25(ii), §18.25 and Ch.18

§18.25(iii) Weights and Normalizations: Discrete Cases

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Notes:: For (18.25.10)–(18.25.12) see Wilson (1980). For (18.25.13)–(18.25.15) see Ismail (2009, (6.2.20)).
Referenced by:: §18.38(iii)
Permalink:: http://dlmf.nist.gov/18.25.iii
See also:: Annotations for §18.25 and Ch.18

18.25.9 $\sum_{y=0}^{N}p_{n}(y(y+\gamma+\delta+1))p_{m}(y(y+\gamma+\delta+1))\*\frac{% \gamma+\delta+1+2y}{\gamma+\delta+1+y}\omega_{y}=h_{n}\delta_{n,m}.$

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, : real variable, : positive integer, $\delta$ : arbitrary small positive constant, $p_{n}(x)$ : polynomial of degree , : nonnegative integer, : nonnegative integer and $h_{n}$
Referenced by:: §18.38(iii)
Permalink:: http://dlmf.nist.gov/18.25.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §18.25(iii), §18.25 and Ch.18

Racah

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See also:: Annotations for §18.25(iii), §18.25 and Ch.18

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

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Symbols:: $R_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{% \delta}\right)$ : Racah polynomial, : positive integer, $\delta$ : arbitrary small positive constant, $p_{n}(x)$ : polynomial of degree , : nonnegative integer and : real variable
Referenced by:: §18.25(iii)
Permalink:: http://dlmf.nist.gov/18.25.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §18.25(iii), §18.25(iii), §18.25 and Ch.18

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

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : real variable and $\delta$ : arbitrary small positive constant
Referenced by:: §18.25(i)
Permalink:: http://dlmf.nist.gov/18.25.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §18.25(iii), §18.25(iii), §18.25 and Ch.18

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

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : positive integer, $\delta$ : arbitrary small positive constant, : nonnegative integer and $h_{n}$
Referenced by:: §18.25(iii), §18.38(iii)
Permalink:: http://dlmf.nist.gov/18.25.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §18.25(iii), §18.25(iii), §18.25 and Ch.18

Dual Hahn

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Keywords:: Wilson class orthogonal polynomials, standardizations, weight functions
See also:: Annotations for §18.25(iii), §18.25 and Ch.18

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

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Symbols:: $R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{\delta},\NVar{N}\right)$ : dual Hahn polynomial, : positive integer, $\delta$ : arbitrary small positive constant, $p_{n}(x)$ : polynomial of degree , : nonnegative integer and : real variable
Referenced by:: §18.25(iii)
Permalink:: http://dlmf.nist.gov/18.25.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §18.25(iii), §18.25(iii), §18.25 and Ch.18

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

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : real variable, : positive integer and $\delta$ : arbitrary small positive constant
Permalink:: http://dlmf.nist.gov/18.25.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §18.25(iii), §18.25(iii), §18.25 and Ch.18

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

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : positive integer, $\delta$ : arbitrary small positive constant, : nonnegative integer and $h_{n}$
Referenced by:: §18.25(iii)
Permalink:: http://dlmf.nist.gov/18.25.E15
Encodings:: TeX, pMML, png
See also:: Annotations for §18.25(iii), §18.25(iii), §18.25 and Ch.18

§18.25(iv) Leading Coefficients

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Notes:: Table 18.25.2 follows from §18.26(i).
Permalink:: http://dlmf.nist.gov/18.25.iv
See also:: Annotations for §18.25 and Ch.18

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.

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