18 Orthogonal Polynomials Askey Scheme 18.25 Wilson Class: Definitions 18.27

q

§18.26 Wilson Class: Continued

ⓘ

Referenced by:: Erratum (V1.2.0) §18.26
Permalink:: http://dlmf.nist.gov/18.26
See also:: Annotations for Ch.18

§18.26(i) Representations as Generalized Hypergeometric Functions and Dualities

ⓘ

Keywords:: Wilson class orthogonal polynomials, generalized hypergeometric functions, relations to other functions
Notes:: For (18.26.1) see Andrews et al. (1999, Definition 3.8.1). For (18.26.2) and (18.26.3) see Wilson (1980). For (18.26.4) see Ismail (2009, (6.2.19)).
Referenced by:: §18.25(iv), §18.26(iii), Erratum (V1.2.0) §18.26
Permalink:: http://dlmf.nist.gov/18.26.i
Modification (effective with 1.2.0):: The title of this subsection was updated to include “and Dualities”. A sentence was added just before (18.26.1) explaning some conventions.
Addition (effective with 1.2.0):: Equations (18.26.4_1), (18.26.4_2) were added.
See also:: Annotations for §18.26 and Ch.18

For the definition of generalized hypergeometric functions see §16.2. Here we use as convention for (16.2.1) with $b_{q}=-N$ , $a_{1}=-n$ , and $n=0,1,\ldots,N$ that the summation on the right-hand side ends at $k=n$ .

18.26.1		$W_{n}\left(y^{2};a,b,c,d\right)={\left(a+b\right)_{n}}{\left(a+c\right)_{n}}{% \left(a+d\right)_{n}}\*{{}_{4}F_{3}}\left({-n,n+a+b+c+d-1,a+iy,a-iy\atop a+b,a% +c,a+d};1\right).$
ⓘ 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), $W_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\right)$ : Wilson polynomial, $y$ : real variable and $n$ : nonnegative integer Referenced by: §18.26(i), §18.26(ii), §18.26(iv), (18.28.29) Permalink: http://dlmf.nist.gov/18.26.E1 Encodings: TeX, pMML, png See also: Annotations for §18.26(i), §18.26 and Ch.18

18.26.2		$S_{n}\left(y^{2};a,b,c\right)={\left(a+b\right)_{n}}{\left(a+c\right)_{n}}{{}_% {3}F_{2}}\left({-n,a+iy,a-iy\atop a+b,a+c};1\right).$
ⓘ 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), $S_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$ : continuous dual Hahn polynomial, $y$ : real variable and $n$ : nonnegative integer Referenced by: §18.26(i), §18.26(ii), §18.26(iv) Permalink: http://dlmf.nist.gov/18.26.E2 Encodings: TeX, pMML, png See also: Annotations for §18.26(i), §18.26 and Ch.18

18.26.3		$R_{n}\left(y(y+\gamma+\delta+1);\alpha,\beta,\gamma,\delta\right)={{}_{4}F_{3}% }\left({-n,n+\alpha+\beta+1,-y,y+\gamma+\delta+1\atop\alpha+1,\beta+\delta+1,% \gamma+1};1\right),$
$\alpha+1$ or $\beta+\delta+1$ or $\gamma+1=-N$ ; $n=0,1,\dots,N$ .
ⓘ 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, $R_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{% \delta}\right)$ : Racah polynomial, $y$ : real variable, $N$ : positive integer, $\delta$ : arbitrary small positive constant and $n$ : nonnegative integer Referenced by: §18.26(i), §18.26(ii), §18.26(iv), §18.38(iii) Permalink: http://dlmf.nist.gov/18.26.E3 Encodings: TeX, pMML, png See also: Annotations for §18.26(i), §18.26 and Ch.18

18.26.4		$R_{n}\left(y(y+\gamma+\delta+1);\gamma,\delta,N\right)={{}_{3}F_{2}}\left({-n,% -y,y+\gamma+\delta+1\atop\gamma+1,-N};1\right),$
$n=0,1,\dots,N$ .
ⓘ 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, $R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{\delta},\NVar{N}\right)$ : dual Hahn polynomial, $y$ : real variable, $N$ : positive integer, $\delta$ : arbitrary small positive constant and $n$ : nonnegative integer Referenced by: §18.26(i), §18.26(ii) Permalink: http://dlmf.nist.gov/18.26.E4 Encodings: TeX, pMML, png See also: Annotations for §18.26(i), §18.26 and Ch.18

Dualities

ⓘ

Addition (effective with 1.2.0):: This paragraph was added.
See also:: Annotations for §18.26(i), §18.26 and Ch.18

18.26.4_1		$R_{n}\left(y(y+\gamma+\delta+1);\gamma,\delta,N\right)=Q_{y}\left(n;\gamma,% \delta,N\right),$
ⓘ Symbols: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{\delta},\NVar{N}\right)$ : dual Hahn polynomial, $y$ : real variable, $N$ : positive integer, $\delta$ : arbitrary small positive constant and $n$ : nonnegative integer Referenced by: §18.26(i), Erratum (V1.2.0) §18.26 Permalink: http://dlmf.nist.gov/18.26.E4_1 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.26(i), §18.26(i), §18.26 and Ch.18

compare (18.21.1).

18.26.4_2		$R_{n}\left(y(y+\gamma+\delta+1);\alpha,\beta,\gamma,\delta\right)=R_{y}\left(n% (n+\alpha+\beta+1);\gamma,\delta,\alpha,\beta\right).$
ⓘ Symbols: $R_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{% \delta}\right)$ : Racah polynomial, $y$ : real variable, $\delta$ : arbitrary small positive constant and $n$ : nonnegative integer Referenced by: §18.26(i), Erratum (V1.2.0) §18.26 Permalink: http://dlmf.nist.gov/18.26.E4_2 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.26(i), §18.26(i), §18.26 and Ch.18

§18.26(ii) Limit Relations

ⓘ

Keywords:: Wilson class orthogonal polynomials, interrelations with other orthogonal polynomials
Notes:: For (18.26.5) and (18.26.7) see Wilson (1980). (18.26.6) follows from (18.26.1) and (18.20.9). (18.26.8) follows from (18.26.2) and (18.20.10). (18.26.9) follows from (18.26.3) and (18.26.4). (18.26.10) follows from (18.26.3) and (18.20.5). For (18.26.11) see Karlin and McGregor (1961, (1.21)). (18.26.12), (18.26.13) follow from (18.26.4) and (18.20.7).
Referenced by:: §18.19, §18.21(ii)
Permalink:: http://dlmf.nist.gov/18.26.ii
See also:: Annotations for §18.26 and Ch.18

Wilson $\to$ Continuous Dual Hahn

ⓘ

See also:: Annotations for §18.26(ii), §18.26 and Ch.18

18.26.5		$\lim_{d\to\infty}\frac{W_{n}\left(x;a,b,c,d\right)}{{\left(a+d\right)_{n}}}=S_% {n}\left(x;a,b,c\right).$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $W_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\right)$ : Wilson polynomial, $S_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$ : continuous dual Hahn polynomial, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.26(ii) Permalink: http://dlmf.nist.gov/18.26.E5 Encodings: TeX, pMML, png See also: Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

Wilson $\to$ Continuous Hahn

ⓘ

See also:: Annotations for §18.26(ii), §18.26 and Ch.18

18.26.6		$\lim_{t\to\infty}\frac{W_{n}\left((x+t)^{2};a-it,b-it,\overline{a}+it,% \overline{b}+it\right)}{(-2t)^{n}n!}=p_{n}\left(x;a,b,\overline{a},\overline{b% }\right).$
ⓘ Symbols: $W_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\right)$ : Wilson polynomial, $\overline{\NVar{z}}$ : complex conjugate, $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{\overline{a}},\NVar{% \overline{b}}\right)$ : continuous Hahn polynomial, $!$ : factorial (as in $n!$ ), $t$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.22(i), §18.22(ii), §18.23, §18.26(ii) Permalink: http://dlmf.nist.gov/18.26.E6 Encodings: TeX, pMML, png See also: Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

Wilson $\to$ Jacobi

ⓘ

See also:: Annotations for §18.26(ii), §18.26 and Ch.18

18.26.7		$\lim_{t\to\infty}\frac{W_{n}\left(\tfrac{1}{2}(1-x)t^{2};\tfrac{1}{2}\alpha+% \tfrac{1}{2},\tfrac{1}{2}\alpha+\tfrac{1}{2},\tfrac{1}{2}\beta+\tfrac{1}{2}+it% ,\tfrac{1}{2}\beta+\tfrac{1}{2}-it\right)}{t^{2n}n!}=P^{(\alpha,\beta)}_{n}% \left(x\right).$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $W_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\right)$ : Wilson polynomial, $!$ : factorial (as in $n!$ ), $t$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.26(ii) Permalink: http://dlmf.nist.gov/18.26.E7 Encodings: TeX, pMML, png See also: Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

Continuous Dual Hahn $\to$ Meixner–Pollaczek

ⓘ

See also:: Annotations for §18.26(ii), §18.26 and Ch.18

18.26.8		$\lim_{t\to\infty}\ifrac{S_{n}\left((x-t)^{2};\lambda+it,\lambda-it,t\cot\phi% \right)}{t^{n}}=n!(\csc\phi)^{n}P^{(\lambda)}_{n}\left(x;\phi\right).$
ⓘ Symbols: $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{\phi}\right)$ : Meixner–Pollaczek polynomial, $S_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$ : continuous dual Hahn polynomial, $\csc\NVar{z}$ : cosecant function, $\cot\NVar{z}$ : cotangent function, $!$ : factorial (as in $n!$ ), $t$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.26(ii) Permalink: http://dlmf.nist.gov/18.26.E8 Encodings: TeX, pMML, png See also: Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

Racah $\to$ Dual Hahn

ⓘ

See also:: Annotations for §18.26(ii), §18.26 and Ch.18

18.26.9		$\lim_{\beta\to\infty}R_{n}\left(x;-N-1,\beta,\gamma,\delta\right)=R_{n}\left(x% ;\gamma,\delta,N\right).$
ⓘ Symbols: $R_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{% \delta}\right)$ : Racah polynomial, $R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{\delta},\NVar{N}\right)$ : dual Hahn polynomial, $N$ : positive integer, $\delta$ : arbitrary small positive constant, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.26(ii) Permalink: http://dlmf.nist.gov/18.26.E9 Encodings: TeX, pMML, png See also: Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

Racah $\to$ Hahn

ⓘ

See also:: Annotations for §18.26(ii), §18.26 and Ch.18

18.26.10		$\lim_{\delta\to\infty}R_{n}\left(x(x+\gamma+\delta+1);\alpha,\beta,-N-1,\delta% \right)=Q_{n}\left(x;\alpha,\beta,N\right).$
ⓘ Symbols: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $R_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{% \delta}\right)$ : Racah polynomial, $N$ : positive integer, $\delta$ : arbitrary small positive constant, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.26(ii) Permalink: http://dlmf.nist.gov/18.26.E10 Encodings: TeX, pMML, png See also: Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

Dual Hahn $\to$ Krawtchouk

ⓘ

See also:: Annotations for §18.26(ii), §18.26 and Ch.18

18.26.11		$\lim_{t\to\infty}R_{n}\left(x(x+t+1);pt,(1-p)t,N\right)=K_{n}\left(x;p,N\right).$
ⓘ Symbols: $K_{\NVar{n}}\left(\NVar{x};\NVar{p},\NVar{N}\right)$ : Krawtchouk polynomial, $R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{\delta},\NVar{N}\right)$ : dual Hahn polynomial, $N$ : positive integer, $t$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.26(ii) Permalink: http://dlmf.nist.gov/18.26.E11 Encodings: TeX, pMML, png See also: Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

Dual Hahn $\to$ Meixner

ⓘ

Keywords:: Wilson class orthogonal polynomials, interrelations with other orthogonal polynomials
See also:: Annotations for §18.26(ii), §18.26 and Ch.18

With

18.26.12		$r(x;\beta,c,N)=x(x+\beta+c^{-1}(1-c)N),$
ⓘ Defines: $r(x;\beta,c,N)$ (locally) Symbols: $N$ : positive integer and $x$ : real variable Referenced by: §18.26(ii) Permalink: http://dlmf.nist.gov/18.26.E12 Encodings: TeX, pMML, png See also: Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

18.26.13		$\lim_{N\to\infty}R_{n}\left(r(x;\beta,c,N);\beta-1,c^{-1}(1-c)N,N\right)=M_{n}% \left(x;\beta,c\right).$
ⓘ Symbols: $M_{\NVar{n}}\left(\NVar{x};\NVar{\beta},\NVar{c}\right)$ : Meixner polynomial, $R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{\delta},\NVar{N}\right)$ : dual Hahn polynomial, $N$ : positive integer, $n$ : nonnegative integer, $x$ : real variable and $r(x;\beta,c,N)$ Referenced by: §18.26(ii) Permalink: http://dlmf.nist.gov/18.26.E13 Encodings: TeX, pMML, png See also: Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

§18.26(iii) Difference Relations

ⓘ

Keywords:: Wilson class orthogonal polynomials, differences
Notes:: (18.26.14)–(18.26.17) follow from §18.26(i).
Permalink:: http://dlmf.nist.gov/18.26.iii
See also:: Annotations for §18.26 and Ch.18

For comments on the use of the forward-difference operator $\Delta_{x}$ , the backward-difference operator $\nabla_{x}$ , and the central-difference operator $\delta_{x}$ , see §18.2(ii).

For each family only the $y$ -difference that lowers $n$ is given. See Koekoek et al. (2010, Chapter 9) for further formulas.

18.26.14		$\ifrac{\delta_{y}\left(W_{n}\left(y^{2};a,b,c,d\right)\right)}{\delta_{y}(y^{2% })}=-n(n+a+b+c+d-1)\*W_{n-1}\left(y^{2};a+\tfrac{1}{2},b+\tfrac{1}{2},c+\tfrac% {1}{2},d+\tfrac{1}{2}\right).$
ⓘ Symbols: $W_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\right)$ : Wilson polynomial, $\delta_{\NVar{x}}$ : central difference, $y$ : real variable and $n$ : nonnegative integer Referenced by: §16.4(iii), §18.26(iii) Permalink: http://dlmf.nist.gov/18.26.E14 Encodings: TeX, pMML, png See also: Annotations for §18.26(iii), §18.26 and Ch.18

18.26.15		$\ifrac{\delta_{y}\left(S_{n}\left(y^{2};a,b,c\right)\right)}{\delta_{y}(y^{2})% }=-nS_{n-1}\left(y^{2};a+\tfrac{1}{2},b+\tfrac{1}{2},c+\tfrac{1}{2}\right).$
ⓘ Symbols: $\delta_{\NVar{x}}$ : central difference, $S_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$ : continuous dual Hahn polynomial, $y$ : real variable and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/18.26.E15 Encodings: TeX, pMML, png See also: Annotations for §18.26(iii), §18.26 and Ch.18

18.26.16		$\frac{\Delta_{y}\left(R_{n}\left(y(y+\gamma+\delta+1);\alpha,\beta,\gamma,% \delta\right)\right)}{\Delta_{y}\left(y(y+\gamma+\delta+1)\right)}=\frac{n(n+% \alpha+\beta+1)}{(\alpha+1)(\beta+\delta+1)(\gamma+1)}\*R_{n-1}\left(y(y+% \gamma+\delta+2);\alpha+1,\beta+1,\gamma+1,\delta\right).$
ⓘ Symbols: $R_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{% \delta}\right)$ : Racah polynomial, $\Delta$ : forward difference operator, $y$ : real variable, $\delta$ : arbitrary small positive constant and $n$ : nonnegative integer Referenced by: §16.4(iii) Permalink: http://dlmf.nist.gov/18.26.E16 Encodings: TeX, pMML, png See also: Annotations for §18.26(iii), §18.26 and Ch.18

18.26.17		$\frac{\Delta_{y}\left(R_{n}\left(y(y+\gamma+\delta+1);\gamma,\delta,N\right)% \right)}{\Delta_{y}\left(y(y+\gamma+\delta+1)\right)}=-\frac{n}{(\gamma+1)N}\*% R_{n-1}\left(y(y+\gamma+\delta+2);\gamma+1,\delta,N-1\right).$
ⓘ Symbols: $R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{\delta},\NVar{N}\right)$ : dual Hahn polynomial, $\Delta$ : forward difference operator, $y$ : real variable, $N$ : positive integer, $\delta$ : arbitrary small positive constant and $n$ : nonnegative integer Referenced by: §18.26(iii) Permalink: http://dlmf.nist.gov/18.26.E17 Encodings: TeX, pMML, png See also: Annotations for §18.26(iii), §18.26 and Ch.18

§18.26(iv) Generating Functions

ⓘ

Keywords:: Wilson class orthogonal polynomials, generalized hypergeometric functions, generating functions, hypergeometric function, relations to other functions
Notes:: For (18.26.18) see Ismail et al. (1990, (6.1)). (18.26.19) follows by expanding both factors on the left as power series in $z$ , and substituting (18.26.2) on the right. (18.26.20) follows from (18.26.18), (18.26.1), (18.26.3). For (18.26.21) see Ismail (2009, (6.2.31)).
Permalink:: http://dlmf.nist.gov/18.26.iv
See also:: Annotations for §18.26 and Ch.18

For the hypergeometric function ${{}_{2}F_{1}}$ see §§15.1 and 15.2(i).

Wilson

ⓘ

See also:: Annotations for §18.26(iv), §18.26 and Ch.18

18.26.18		${{}_{2}F_{1}}\left({a+\mathrm{i}y,d+\mathrm{i}y\atop a+d};z\right){{}_{2}F_{1}% }\left({b-\mathrm{i}y,c-\mathrm{i}y\atop b+c};z\right)=\sum_{n=0}^{\infty}% \frac{W_{n}\left(y^{2};a,b,c,d\right)}{{\left(a+d\right)_{n}}{\left(b+c\right)% _{n}}n!}z^{n},$
$\|z\|<1$ .
ⓘ Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $W_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\right)$ : Wilson polynomial, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $y$ : real variable, $z$ : complex variable and $n$ : nonnegative integer Referenced by: §18.23, §18.26(iv) Permalink: http://dlmf.nist.gov/18.26.E18 Encodings: TeX, pMML, png See also: Annotations for §18.26(iv), §18.26(iv), §18.26 and Ch.18

Continuous Dual Hahn

ⓘ

See also:: Annotations for §18.26(iv), §18.26 and Ch.18

18.26.19		$(1-z)^{-c+\mathrm{i}y}{{}_{2}F_{1}}\left({a+\mathrm{i}y,b+\mathrm{i}y\atop a+b% };z\right)=\sum_{n=0}^{\infty}\frac{S_{n}\left(y^{2};a,b,c\right)}{{\left(a+b% \right)_{n}}n!}z^{n},$
$\|z\|<1$ .
ⓘ Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $S_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$ : continuous dual Hahn polynomial, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $y$ : real variable, $z$ : complex variable and $n$ : nonnegative integer Referenced by: §18.26(iv) Permalink: http://dlmf.nist.gov/18.26.E19 Encodings: TeX, pMML, png See also: Annotations for §18.26(iv), §18.26(iv), §18.26 and Ch.18

Racah

ⓘ

See also:: Annotations for §18.26(iv), §18.26 and Ch.18

18.26.20		${{}_{2}F_{1}}\left({-y,-y+\beta-\gamma\atop\beta+\delta+1};z\right){{}_{2}F_{1% }}\left({y-N,y+\gamma+1\atop-\delta-N};z\right)=\sum_{n=0}^{N}\frac{{\left(-N% \right)_{n}}{\left(\gamma+1\right)_{n}}}{{\left(-\delta-N\right)_{n}}n!}R_{n}% \left(y(y+\gamma+\delta+1);-N-1,\beta,\gamma,\delta\right)z^{n}.$
ⓘ Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $R_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{% \delta}\right)$ : Racah polynomial, $!$ : factorial (as in $n!$ ), $y$ : real variable, $N$ : positive integer, $\delta$ : arbitrary small positive constant, $z$ : complex variable and $n$ : nonnegative integer Referenced by: §18.26(iv) Permalink: http://dlmf.nist.gov/18.26.E20 Encodings: TeX, pMML, png See also: Annotations for §18.26(iv), §18.26(iv), §18.26 and Ch.18

Dual Hahn

ⓘ

See also:: Annotations for §18.26(iv), §18.26 and Ch.18

18.26.21		$(1-z)^{y}{{}_{2}F_{1}}\left({y-N,y+\gamma+1\atop-\delta-N};z\right)=\sum_{n=0}% ^{N}\frac{{\left(\gamma+1\right)_{n}}{\left(-N\right)_{n}}}{{\left(-\delta-N% \right)_{n}}n!}\*R_{n}\left(y(y+\gamma+\delta+1);\gamma,\delta,N\right)z^{n}.$
ⓘ Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{\delta},\NVar{N}\right)$ : dual Hahn polynomial, $!$ : factorial (as in $n!$ ), $y$ : real variable, $N$ : positive integer, $\delta$ : arbitrary small positive constant, $z$ : complex variable and $n$ : nonnegative integer Referenced by: §18.26(iv) Permalink: http://dlmf.nist.gov/18.26.E21 Encodings: TeX, pMML, png See also: Annotations for §18.26(iv), §18.26(iv), §18.26 and Ch.18

§18.26(v) Asymptotic Approximations

ⓘ

Keywords:: Wilson class orthogonal polynomials, asymptotic approximations
Referenced by:: Erratum (V1.0.5) for References
Permalink:: http://dlmf.nist.gov/18.26.v
Addition (effective with 1.0.5):: The reference to Koornwinder (2009), with associated paragraph, has been added at the end of this subsection.
See also:: Annotations for §18.26 and Ch.18

For asymptotic expansions of Wilson polynomials of large degree see Wilson (1991), and for asymptotic approximations to their largest zeros see Chen and Ismail (1998).

Koornwinder (2009) rescales and reparametrizes Racah polynomials and Wilson polynomials in such a way that they are continuous in their four parameters, provided that these parameters are nonnegative. Moreover, if one or more of the new parameters becomes zero, then the polynomial descends to a lower family in the Askey scheme.

18.25 Wilson Class: Definitions 18.27

q

-Hahn Class

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

§18.26 Wilson Class: Continued

Contents

§18.26(i) Representations as Generalized Hypergeometric Functions and Dualities

Dualities

§18.26(ii) Limit Relations

Wilson $\to$ Continuous Dual Hahn

Wilson $\to$ Continuous Hahn

Wilson $\to$ Jacobi

Continuous Dual Hahn $\to$ Meixner–Pollaczek

Racah $\to$ Dual Hahn

Racah $\to$ Hahn

Dual Hahn $\to$ Krawtchouk

Dual Hahn $\to$ Meixner

§18.26(iii) Difference Relations

§18.26(iv) Generating Functions

Wilson

Continuous Dual Hahn

Racah

Dual Hahn

§18.26(v) Asymptotic Approximations

pFad - (p)hone/(F)rame/(a)nonymizer/(d)eclutterfier! Saves Data!

§18.26 Wilson Class: Continued

Contents

§18.26(i) Representations as Generalized Hypergeometric Functions and Dualities

Dualities

§18.26(ii) Limit Relations

Wilson → Continuous Dual Hahn

Wilson → Continuous Hahn

Wilson → Jacobi

Continuous Dual Hahn → Meixner–Pollaczek

Racah → Dual Hahn

Racah → Hahn

Dual Hahn → Krawtchouk

Dual Hahn → Meixner

§18.26(iii) Difference Relations

§18.26(iv) Generating Functions

Wilson

Continuous Dual Hahn

Racah

Dual Hahn

§18.26(v) Asymptotic Approximations

pFad - (p)hone/(F)rame/(a)nonymizer/(d)eclutterfier! Saves Data!

Wilson $\to$ Continuous Dual Hahn

Wilson $\to$ Continuous Hahn

Wilson $\to$ Jacobi

Continuous Dual Hahn $\to$ Meixner–Pollaczek

Racah $\to$ Dual Hahn

Racah $\to$ Hahn

Dual Hahn $\to$ Krawtchouk

Dual Hahn $\to$ Meixner