§18.26 Wilson Class: Continued

Permalink:: http://dlmf.nist.gov/18.26

§18.26(i) Representations as Generalized Hypergeometric Functions
§18.26(ii) Limit Relations
§18.26(iii) Difference Relations
§18.26(iv) Generating Functions
§18.26(v) Asymptotic Approximations

§18.26(i) Representations as Generalized Hypergeometric 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 (2005, (6.2.19)).
Keywords:: Wilson class orthogonal polynomials, generalized hypergeometric functions
Referenced by:: §18.25(iv), §18.26(iii)
Permalink:: http://dlmf.nist.gov/18.26.i

For the definition of generalized hypergeometric functions see §16.2.

18.26.1 $\mathop{W_{{n}}\/}\nolimits\!\left(y^{2};a,b,c,d\right)=\left(a+b\right)_{{n}}\left(a+c\right)_{{n}}\left(a+d\right)_{{n}}\*\mathop{{{}_{{4}}F_{{3}}}\/}\nolimits\!\left({-n,n+a+b+c+d-1,a+iy,a-iy\atop a+b,a+c,a+d};1\right).$

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, $\mathop{W_{{n}}\/}\nolimits\!\left(x;a,b,c,d\right)$ : Wilson 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.E1
Encodings:: TeX, pMML, png

18.26.2 $\frac{\mathop{S_{{n}}\/}\nolimits\!\left(y^{2};a,b,c\right)}{\left(a+b\right)_{{n}}\left(a+c\right)_{{n}}}=\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({-n,a+iy,a-iy\atop a+b,a+c};1\right).$

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, $\mathop{S_{{n}}\/}\nolimits\!\left(x;a,b,c\right)$ : 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

18.26.3 $\mathop{R_{{n}}\/}\nolimits\!\left(y(y+\gamma+\delta+1);\alpha,\beta,\gamma,\delta\right)=\mathop{{{}_{{4}}F_{{3}}}\/}\nolimits\!\left({-n,n+\alpha+\beta+1,-y,y+\gamma+\delta+1\atop\alpha+1,\beta+\delta+1,\gamma+1};1\right),$ $\alpha+1$ or $\beta+\delta+1$ or $\gamma+1=-N$ ; $n=0,1,\dots,N$ .

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{R_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,\gamma,\delta\right)$ : Racah polynomial, $y$ : real variable, $n$ : nonnegative integer, $N$ : positive integer and $\delta$ : arbitary small positive constant
Referenced by:: §18.26(i), §18.26(ii), §18.26(iv)
Permalink:: http://dlmf.nist.gov/18.26.E3
Encodings:: TeX, pMML, png

18.26.4 $\mathop{R_{{n}}\/}\nolimits\!\left(y(y+\gamma+\delta+1);\gamma,\delta,N\right)=\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({-n,-y,y+\gamma+\delta+1\atop\gamma+1,-N};1\right),$ $n=0,1,\dots,N$ .

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{R_{{n}}\/}\nolimits\!\left(x;\gamma,\delta,N\right)$ : dual Hahn polynomial, $y$ : real variable, $n$ : nonnegative integer, $N$ : positive integer and $\delta$ : arbitary small positive constant
Referenced by:: §18.26(i), §18.26(ii)
Permalink:: http://dlmf.nist.gov/18.26.E4
Encodings:: TeX, pMML, png

§18.26(ii) Limit Relations

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).
Keywords:: Wilson class orthogonal polynomials
Referenced by:: ¶ ‣ §18.21(ii)
Permalink:: http://dlmf.nist.gov/18.26.ii

¶ Wilson $\to$ Continuous Dual Hahn

18.26.5 $\lim _{{d\to\infty}}\frac{\mathop{W_{{n}}\/}\nolimits\!\left(x;a,b,c,d\right)}{\left(a+d\right)_{{n}}}=\mathop{S_{{n}}\/}\nolimits\!\left(x;a,b,c\right).$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{W_{{n}}\/}\nolimits\!\left(x;a,b,c,d\right)$ : Wilson polynomial, $\mathop{S_{{n}}\/}\nolimits\!\left(x;a,b,c\right)$ : continuous dual Hahn polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.26(ii)
Permalink:: http://dlmf.nist.gov/18.26.E5
Encodings:: TeX, pMML, png

¶ Wilson $\to$ Continuous Hahn

18.26.6 $\lim _{{t\to\infty}}\frac{\mathop{W_{{n}}\/}\nolimits\!\left((x+t)^{2};a-it,b-it,\conj{a}+it,\conj{b}+it\right)}{(-2t)^{n}n!}=\mathop{p_{{n}}\/}\nolimits\!\left(x;a,b,\conj{a},\conj{b}\right).$

Symbols:: $\mathop{W_{{n}}\/}\nolimits\!\left(x;a,b,c,d\right)$ : Wilson polynomial, $\conj{z}$ : complex conjugate, $\mathop{p_{{n}}\/}\nolimits\!\left(x;a,b,\conj{a},\conj{b}\right)$ : continuous Hahn polynomial, $!$ : $n!$ : factorial, $n$ : 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

¶ Wilson $\to$ Jacobi

18.26.7 $\lim _{{t\to\infty}}\frac{\mathop{W_{{n}}\/}\nolimits\!\left(\tfrac{1}{2}(1-x)t^{2};\tfrac{1}{2}\alpha+\tfrac{1}{2},\tfrac{1}{2}\alpha+\tfrac{1}{2},\tfrac{1}{2}\beta+\tfrac{1}{2}+it,\tfrac{1}{2}\beta+\tfrac{1}{2}-it\right)}{t^{{2n}}n!}=\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\mathop{W_{{n}}\/}\nolimits\!\left(x;a,b,c,d\right)$ : Wilson polynomial, $!$ : $n!$ : factorial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.26(ii)
Permalink:: http://dlmf.nist.gov/18.26.E7
Encodings:: TeX, pMML, png

¶ Continuous Dual Hahn $\to$ Meixner–Pollaczek

18.26.8 $\lim _{{t\to\infty}}\ifrac{\mathop{S_{{n}}\/}\nolimits\!\left((x-t)^{2};\lambda+it,\lambda-it,t\mathop{\cot\/}\nolimits\phi\right)}{t^{n}}=n!(\mathop{\csc\/}\nolimits\phi)^{n}\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right).$

Symbols:: $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right)$ : Meixner–Pollaczek polynomial, $\mathop{S_{{n}}\/}\nolimits\!\left(x;a,b,c\right)$ : continuous dual Hahn polynomial, $\mathop{\csc\/}\nolimits z$ : cosecant function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $!$ : $n!$ : factorial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.26(ii)
Permalink:: http://dlmf.nist.gov/18.26.E8
Encodings:: TeX, pMML, png

¶ Racah $\to$ Dual Hahn

18.26.9 $\lim _{{\beta\to\infty}}\mathop{R_{{n}}\/}\nolimits\!\left(x;-N-1,\beta,\gamma,\delta\right)=\mathop{R_{{n}}\/}\nolimits\!\left(x;\gamma,\delta,N\right).$

Symbols:: $\mathop{R_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,\gamma,\delta\right)$ : Racah polynomial, $\mathop{R_{{n}}\/}\nolimits\!\left(x;\gamma,\delta,N\right)$ : dual Hahn polynomial, $n$ : nonnegative integer, $N$ : positive integer, $\delta$ : arbitary small positive constant and $x$ : real variable
Referenced by:: §18.26(ii)
Permalink:: http://dlmf.nist.gov/18.26.E9
Encodings:: TeX, pMML, png

¶ Racah $\to$ Hahn

18.26.10 $\lim _{{\delta\to\infty}}\mathop{R_{{n}}\/}\nolimits\!\left(x(x+\gamma+\delta+1);\alpha,\beta,-N-1,\delta\right)=\mathop{Q_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,N\right).$

Symbols:: $\mathop{Q_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,N\right)$ : Hahn polynomial, $\mathop{R_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,\gamma,\delta\right)$ : Racah polynomial, $n$ : nonnegative integer, $N$ : positive integer, $\delta$ : arbitary small positive constant and $x$ : real variable
Referenced by:: §18.26(ii)
Permalink:: http://dlmf.nist.gov/18.26.E10
Encodings:: TeX, pMML, png

¶ Dual Hahn $\to$ Krawtchouk

18.26.11 $\lim _{{t\to\infty}}\mathop{R_{{n}}\/}\nolimits\!\left(x(x+t+1);pt,(1-p)t,N\right)=\mathop{K_{{n}}\/}\nolimits\!\left(x;p,N\right).$

Symbols:: $\mathop{K_{{n}}\/}\nolimits\!\left(x;p,N\right)$ : Krawtchouk polynomial, $\mathop{R_{{n}}\/}\nolimits\!\left(x;\gamma,\delta,N\right)$ : dual Hahn polynomial, $n$ : nonnegative integer, $N$ : positive integer and $x$ : real variable
Referenced by:: §18.26(ii)
Permalink:: http://dlmf.nist.gov/18.26.E11
Encodings:: TeX, pMML, png

¶ Dual Hahn $\to$ Meixner

Keywords:: Wilson class orthogonal polynomials

With

18.26.12 $r(x;\beta,c,N)=x(x+\beta+c^{{-1}}(1-c)N),$

Symbols:: $N$ : positive integer, $x$ : real variable and $r(x;\beta,c,N)$
Referenced by:: §18.26(ii)
Permalink:: http://dlmf.nist.gov/18.26.E12
Encodings:: TeX, pMML, png

18.26.13 $\lim _{{N\to\infty}}\mathop{R_{{n}}\/}\nolimits\!\left(r(x;\beta,c,N);\beta-1,c^{{-1}}(1-c)N,N\right)=\mathop{M_{{n}}\/}\nolimits\!\left(x;\beta,c\right).$

Symbols:: $\mathop{M_{{n}}\/}\nolimits\!\left(x;\beta,c\right)$ : Meixner polynomial, $\mathop{R_{{n}}\/}\nolimits\!\left(x;\gamma,\delta,N\right)$ : dual Hahn polynomial, $n$ : nonnegative integer, $N$ : positive 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

§18.26(iii) Difference Relations

Notes:: (18.26.14)–(18.26.17) follow from §18.26(i).
Keywords:: Wilson class orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.26.iii

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 1) for further formulas.

18.26.14 $\ifrac{\delta _{{y}}\left(\mathop{W_{{n}}\/}\nolimits\!\left(y^{2};a,b,c,d\right)\right)}{\delta _{{y}}(y^{2})}=-n(n+a+b+c+d-1)\*\mathop{W_{{n-1}}\/}\nolimits\!\left(y^{2};a+\tfrac{1}{2},b+\tfrac{1}{2},c+\tfrac{1}{2},d+\tfrac{1}{2}\right).$

Symbols:: $\mathop{W_{{n}}\/}\nolimits\!\left(x;a,b,c,d\right)$ : Wilson polynomial, $\delta _{{x}}$ : 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

18.26.15 $\ifrac{\delta _{{y}}\left(\mathop{S_{{n}}\/}\nolimits\!\left(y^{2};a,b,c\right)\right)}{\delta _{{y}}(y^{2})}=-n\mathop{S_{{n-1}}\/}\nolimits\!\left(y^{2};a+\tfrac{1}{2},b+\tfrac{1}{2},c+\tfrac{1}{2}\right).$

Symbols:: $\delta _{{x}}$ : central difference, $\mathop{S_{{n}}\/}\nolimits\!\left(x;a,b,c\right)$ : continuous dual Hahn polynomial, $y$ : real variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.26.E15
Encodings:: TeX, pMML, png

18.26.16 $\frac{\Delta _{{y}}\left(\mathop{R_{{n}}\/}\nolimits\!\left(y(y+\gamma+\delta+1);\alpha,\beta,\gamma,\delta\right)\right)}{\Delta _{{y}}\left(y(y+\gamma+\delta+1)\right)}=\frac{n(n+\alpha+\beta+1)}{(\alpha+1)(\beta+\delta+1)(\gamma+1)}\*\mathop{R_{{n-1}}\/}\nolimits\!\left(y(y+\gamma+\delta+2);\alpha+1,\beta+1,\gamma+1,\delta\right).$

Symbols:: $\mathop{R_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,\gamma,\delta\right)$ : Racah polynomial, $\Delta$ : forward difference operator, $y$ : real variable, $n$ : nonnegative integer and $\delta$ : arbitary small positive constant
Referenced by:: §16.4(iii)
Permalink:: http://dlmf.nist.gov/18.26.E16
Encodings:: TeX, pMML, png

18.26.17 $\frac{\Delta _{{y}}\left(\mathop{R_{{n}}\/}\nolimits\!\left(y(y+\gamma+\delta+1);\gamma,\delta,N\right)\right)}{\Delta _{{y}}\left(y(y+\gamma+\delta+1)\right)}=-\frac{n}{(\gamma+1)N}\*\mathop{R_{{n-1}}\/}\nolimits\!\left(y(y+\gamma+\delta+2);\gamma+1,\delta,N-1\right).$

Symbols:: $\mathop{R_{{n}}\/}\nolimits\!\left(x;\gamma,\delta,N\right)$ : dual Hahn polynomial, $\Delta$ : forward difference operator, $y$ : real variable, $n$ : nonnegative integer, $N$ : positive integer and $\delta$ : arbitary small positive constant
Referenced by:: §18.26(iii)
Permalink:: http://dlmf.nist.gov/18.26.E17
Encodings:: TeX, pMML, png

§18.26(iv) Generating 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 (2005, (6.2.31)).
Keywords:: Wilson class orthogonal polynomials, hypergeometric function
Permalink:: http://dlmf.nist.gov/18.26.iv

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

¶ Wilson

18.26.18 $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({a+iy,d+iy\atop a+d};z\right)\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({b-iy,c-iy\atop b+c};z\right)=\sum _{{n=0}}^{\infty}\frac{\mathop{W_{{n}}\/}\nolimits\!\left(y^{2};a,b,c,d\right)}{\left(a+d\right)_{{n}}\left(b+c\right)_{{n}}n!}z^{n},$ $|z|<1$ .

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, $\mathop{W_{{n}}\/}\nolimits\!\left(x;a,b,c,d\right)$ : Wilson polynomial, $!$ : $n!$ : factorial, $y$ : 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

¶ Continuous Dual Hahn

18.26.19 $(1-z)^{{-c+iy}}\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({a+iy,b+iy\atop a+b};z\right)=\sum _{{n=0}}^{\infty}\frac{\mathop{S_{{n}}\/}\nolimits\!\left(y^{2};a,b,c\right)}{\left(a+b\right)_{{n}}n!}z^{n},$ $|z|<1$ .

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, $\mathop{S_{{n}}\/}\nolimits\!\left(x;a,b,c\right)$ : continuous dual Hahn polynomial, $!$ : $n!$ : factorial, $y$ : 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

¶ Racah

18.26.20 $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({-y,-y+\beta-\gamma\atop\beta+\delta+1};z\right)\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({y-N,y+\gamma+1\atop-\delta-N};z\right)=\sum _{{n=0}}^{N}\frac{\left(-N\right)_{{n}}\left(\gamma+1\right)_{{n}}}{\left(-\delta-N\right)_{{n}}n!}\mathop{R_{{n}}\/}\nolimits\!\left(y(y+\gamma+\delta+1);-N-1,\beta,\gamma,\delta\right)z^{n}.$

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, $\mathop{R_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,\gamma,\delta\right)$ : Racah polynomial, $!$ : $n!$ : factorial, $y$ : real variable, $z$ : complex variable, $n$ : nonnegative integer, $N$ : positive integer and $\delta$ : arbitary small positive constant
Referenced by:: §18.26(iv)
Permalink:: http://dlmf.nist.gov/18.26.E20
Encodings:: TeX, pMML, png

¶ Dual Hahn

18.26.21 $(1-z)^{y}\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({y-N,y+\gamma+1\atop-\delta-N};z\right)=\sum _{{n=0}}^{N}\frac{\left(\gamma+1\right)_{{n}}\left(-N\right)_{{n}}}{\left(-\delta-N\right)_{{n}}n!}\*\mathop{R_{{n}}\/}\nolimits\!\left(y(y+\gamma+\delta+1);\gamma,\delta,N\right)z^{n}.$

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, $\mathop{R_{{n}}\/}\nolimits\!\left(x;\gamma,\delta,N\right)$ : dual Hahn polynomial, $!$ : $n!$ : factorial, $y$ : real variable, $z$ : complex variable, $n$ : nonnegative integer, $N$ : positive integer and $\delta$ : arbitary small positive constant
Referenced by:: §18.26(iv)
Permalink:: http://dlmf.nist.gov/18.26.E21
Encodings:: TeX, pMML, png

§18.26(v) Asymptotic Approximations

Keywords:: Wilson class orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.26.v

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).

§18.26 Wilson Class: Continued

Contents

§18.26(i) Representations as Generalized Hypergeometric Functions

§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