Hahn

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Hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;\alpha,\beta,N\right)=\HyperpFq{3}{2}@@{-n,n+\alpha+\beta+1,-x}{\alpha+1,-N}{1}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots,N}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\binomial{\alpha+x}{x}\binomial{\beta+N-x}{N-x}Q_{m}\!\left(x;\alpha,\beta,N\right)Q_{n}\!\left(x;\alpha,\beta,N\right){}=\frac{(-1)^{n}{\left(n+\alpha+\beta+1\right)_{N+1}}{\left(\beta+1\right)_{n}}n!}{(2n+\alpha+\beta+1){\left(\alpha+1\right)_{n}}{\left(-N\right)_{n}}N!}\,\delta_{m,n}}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle-xQ_{n}\!\left(x\right)=A_{n}Q_{n+1}\!\left(x\right)-\left(A_{n}+C_{n}\right)Q_{n}\!\left(x\right)+C_{n}Q_{n-1}\!\left(x\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{n(n+\alpha+\beta+N+1)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(n+\alpha+\beta+1)(n+\alpha+1)(N-n)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}}}}$

${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x\right):=Q_{n}\!\left(x;\alpha,\beta,N\right)}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{Q}}_{n}\!\left(x\right)={\widehat{Q}}_{n+1}\!\left(x\right)+\left(A_{n}+C_{n}\right){\widehat{Q}}_{n}\!\left(x\right)+A_{n-1}C_{n}{\widehat{Q}}_{n-1}\!\left(x\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{n(n+\alpha+\beta+N+1)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(n+\alpha+\beta+1)(n+\alpha+1)(N-n)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}}}}$

${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;\alpha,\beta,N\right)=\frac{{\left(n+\alpha+\beta+1\right)_{n}}}{{\left(\alpha+1\right)_{n}}{\left(-N\right)_{n}}}{\widehat{Q}}_{n}\!\left(x\right)}}}$

Difference equation

${\displaystyle{\displaystyle{\displaystyle n(n+\alpha+\beta+1)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle D(x)=x(x-\beta-N-1)}}}$ &

${\displaystyle{\displaystyle{\displaystyle B(x)=(x+\alpha+1)(x-N)}}}$ &

${\displaystyle{\displaystyle{\displaystyle y(x)=Q_{n}\!\left(x;\alpha,\beta,N\right)}}}$

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x+1;\alpha,\beta,N\right)-Q_{n}\!\left(x;\alpha,\beta,N\right){}=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}\!\left(x;\alpha+1,\beta+1,N-1\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\Delta Q_{n}\!\left(x;\alpha,\beta,N\right)=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}\!\left(x;\alpha+1,\beta+1,N-1\right)}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle(x+\alpha)(N+1-x)Q_{n}\!\left(x;\alpha,\beta,N\right)-x(\beta+N+1-x)Q_{n}\!\left(x-1;\alpha,\beta,N\right){}=\alpha(N+1)Q_{n+1}\!\left(x;\alpha-1,\beta-1,N+1\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\nabla\left[\omega(x;\alpha,\beta,N)Q_{n}\!\left(x;\alpha,\beta,N\right)\right]{}=\frac{N+1}{\beta}\omega(x;\alpha-1,\beta-1,N+1)Q_{n+1}\!\left(x;\alpha-1,\beta-1,N+1\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\omega(x;\alpha,\beta,N)=\binomial{\alpha+x}{x}\binomial{\beta+N-x}{N-x}}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle\omega(x;\alpha,\beta,N)Q_{n}\!\left(x;\alpha,\beta,N\right){}=\frac{(-1)^{n}{\left(\beta+1\right)_{n}}}{{\left(-N\right)_{n}}}\nabla^{n}\left[\omega(x;\alpha+n,\beta+n,N-n)\right]}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\omega(x;\alpha,\beta,N)=\binomial{\alpha+x}{x}\binomial{\beta+N-x}{N-x}}}}$

Generating functions

${\displaystyle{\displaystyle{\displaystyle\HyperpFq{1}{1}@@{-x}{\alpha+1}{-t}\,\HyperpFq{1}{1}@@{x-N}{\beta+1}{t}=\sum_{n=0}^{N}\frac{{\left(-N\right)_{n}}}{{\left(\beta+1\right)_{n}}n!}Q_{n}\!\left(x;\alpha,\beta,N\right)t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\HyperpFq{2}{0}@@{-x,-x+\beta+N+1}{-}{-t}\,\HyperpFq{2}{0}@@{x-N,x+\alpha+1}{-}{t}{}=\sum_{n=0}^{N}\frac{{\left(-N\right)_{n}}{\left(\alpha+1\right)_{n}}}{n!}Q_{n}\!\left(x;\alpha,\beta,N\right)t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\left[(1-t)^{-\alpha-\beta-1}\,\HyperpFq{3}{2}@@{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2),-x}{\alpha+1,-N}{-\frac{4t}{(1-t)^{2}}}\right]_{N}{}=\sum_{n=0}^{N}\frac{{\left(\alpha+\beta+1\right)_{n}}}{n!}Q_{n}\!\left(x;\alpha,\beta,N\right)t^{n}}}}$

Limit relations

Racah polynomial to Hahn polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\delta\rightarrow\infty}R_{n}\!\left(\lambda(x);\alpha,\beta,-N-1,\delta\right)=Q_{n}\!\left(x;\alpha,\beta,N\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\lim_{\gamma\rightarrow\infty}R_{n}\!\left(\lambda(x);\alpha,\beta,\gamma,-\beta-N-1\right)=Q_{n}\!\left(x;\alpha,\beta,N\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\lim_{\delta\rightarrow\infty}R_{n}\!\left(\lambda(x);-N-1,\beta+\gamma+N+1,\gamma,\delta\right)=Q_{n}\!\left(x;\gamma,\beta,N\right)}}}$

Hahn polynomial to Jacobi polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}Q_{n}\!\left(Nx;\alpha,\beta,N\right)=\frac{P^{(\alpha,\beta)}_{n}\left(1-2x\right)}{P^{(\alpha,\beta)}_{n}\left(1\right)}}}}$

Hahn polynomial to Meixner polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}Q_{n}\!\left(x;b-1,N(1-c)c^{-1},N\right)=M_{n}\!\left(x;b,c\right)}}}$

Hahn polynomial to Krawtchouk polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{t\rightarrow\infty}Q_{n}\!\left(x;pt,(1-p)t,N\right)=K_{n}\!\left(x;p,N\right)}}}$

Remark

${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;\alpha,\beta,N\right)=R_{x}\!\left(\lambda(n);\alpha,\beta,N\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{N}\frac{(2n+\alpha+\beta+1){\left(\alpha+1\right)_{n}}{\left(-N\right)_{n}}N!}{(-1)^{n}{\left(n+\alpha+\beta+1\right)_{N+1}}{\left(\beta+1\right)_{n}}n!}Q_{n}\!\left(x;\alpha,\beta,N\right)Q_{n}\!\left(y;\alpha,\beta,N\right){}=\frac{\delta_{x,y}}{\dbinom{\alpha+x}{x}\dbinom{\beta+N-x}{N-x}}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle x,y\in\{0,1,2,\ldots,N\}}}}$

Koornwinder Addendum: Hahn

Hahn: Special values

${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(0;\alpha,\beta,N\right)=1Q_{n}\!\left(N;\alpha,\beta,N\right)}}}$
${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(0;\alpha,\beta,N\right)=\frac{(-1)^{n}{\left(\beta+1\right)_{n}}}{{\left(\alpha+1\right)_{n}}}}}}$
${\displaystyle{\displaystyle{\displaystyle Q_{2n}\!\left(N;\alpha,\alpha,2N\right)=\frac{{\left(\frac{1}{2}\right)_{n}}{\left(N+\alpha+1\right)_{n}}}{{\left(-N+\frac{1}{2}\right)_{n}}{\left(\alpha+1\right)_{n}}}}}}$
${\displaystyle{\displaystyle{\displaystyle Q_{N}\!\left(x;\alpha,\beta,N\right)=\frac{{\left(-N-\beta\right)_{x}}}{{\left(\alpha+1\right)_{x}}}\qquad(x}}}$
${\displaystyle{\displaystyle{\displaystyle Q_{N}\!\left(x;\alpha,\beta,N\right)=0,1,\ldots,N)}}}$

Symmetries

${\displaystyle{\displaystyle{\displaystyle\frac{Q_{n}\!\left(N-x;\alpha,\beta,N\right)}{Q_{n}\!\left(N;\alpha,\beta,N\right)}=Q_{n}\!\left(x;\beta,\alpha,N\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{Q_{N-n}\!\left(x;\alpha,\beta,N\right)}{Q_{N}\!\left(x;\alpha,\beta,N\right)}=Q_{n}\!\left(x;-N-\beta-1,-N-\alpha-1,N\right)\qquad(x}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{Q_{N-n}\!\left(x;\alpha,\beta,N\right)}{Q_{N}\!\left(x;\alpha,\beta,N\right)}=0,1,\ldots,N)}}}$

Duality

${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;\alpha,\beta,N\right)=R_{x}\!\left(n(n+\alpha+\beta+1);\alpha,\beta,N\right)(n,x\in\{0,1,\ldots N\})}}}$

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