Hahn

<< Continuous Hahn

Hahn

Dual Hahn >>

Hypergeometric representation

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

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

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\binomial{\alpha+x}{x}\binomial{\beta+N-x}{N-x}\Hahn{m}@{x}{\alpha}{\beta}{N}\Hahn{n}@{x}{\alpha}{\beta}{N}{}=\frac{(-1)^{n}\pochhammer{n+\alpha+\beta+1}{N+1}\pochhammer{\beta+1}{n}n!}{(2n+\alpha+\beta+1)\pochhammer{\alpha+1}{n}\pochhammer{-N}{n}N!}\,\Kronecker{m}{n}}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle-x\Hahn{n}@@{x}{\alpha}{\beta}{N}=A_{n}\Hahn{n+1}@@{x}{\alpha}{\beta}{N}-\left(A_{n}+C_{n}\right)\Hahn{n}@@{x}{\alpha}{\beta}{N}+C_{n}\Hahn{n-1}@@{x}{\alpha}{\beta}{N}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\Hahn{n}@@{x}{\alpha}{\beta}{N}:=\Hahn{n}@{x}{\alpha}{\beta}{N}}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x\monicHahn{n}@@{x}{\alpha}{\beta}{N}=\monicHahn{n+1}@@{x}{\alpha}{\beta}{N}+\left(A_{n}+C_{n}\right)\monicHahn{n}@@{x}{\alpha}{\beta}{N}+A_{n-1}C_{n}\monicHahn{n-1}@@{x}{\alpha}{\beta}{N}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\Hahn{n}@{x}{\alpha}{\beta}{N}=\frac{\pochhammer{n+\alpha+\beta+1}{n}}{\pochhammer{\alpha+1}{n}\pochhammer{-N}{n}}\monicHahn{n}@@{x}{\alpha}{\beta}{N}}}}$

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

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

${\displaystyle{\displaystyle{\displaystyle y(x)=\Hahn{n}@{x}{\alpha}{\beta}{N}}}}$

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle\Hahn{n}@{x+1}{\alpha}{\beta}{N}-\Hahn{n}@{x}{\alpha}{\beta}{N}{}=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}\Hahn{n-1}@{x}{\alpha+1}{\beta+1}{N-1}}}}$
${\displaystyle{\displaystyle{\displaystyle\Delta\Hahn{n}@{x}{\alpha}{\beta}{N}=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}\Hahn{n-1}@{x}{\alpha+1}{\beta+1}{N-1}}}}$

Backward shift operator

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

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

Rodrigues-type formula

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

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

Generating functions

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

Limit relations

Racah polynomial to Hahn polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\delta\rightarrow\infty}\Racah{n}@{\lambda(x)}{\alpha}{\beta}{-N-1}{\delta}=\Hahn{n}@{x}{\alpha}{\beta}{N}}}}$
${\displaystyle{\displaystyle{\displaystyle\lim_{\gamma\rightarrow\infty}\Racah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{-\beta-N-1}=\Hahn{n}@{x}{\alpha}{\beta}{N}}}}$
${\displaystyle{\displaystyle{\displaystyle\lim_{\delta\rightarrow\infty}\Racah{n}@{\lambda(x)}{-N-1}{\beta+\gamma+N+1}{\gamma}{\delta}=\Hahn{n}@{x}{\gamma}{\beta}{N}}}}$

Hahn polynomial to Jacobi polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}\Hahn{n}@{Nx}{\alpha}{\beta}{N}=\frac{\Jacobi{\alpha}{\beta}{n}@{1-2x}}{\Jacobi{\alpha}{\beta}{n}@{1}}}}}$

Hahn polynomial to Meixner polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}\Hahn{n}@{x}{b-1}{N(1-c)c^{-1}}{N}=\Meixner{n}@{x}{b}{c}}}}$

Hahn polynomial to Krawtchouk polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{t\rightarrow\infty}\Hahn{n}@{x}{pt}{(1-p)t}{N}=\Krawtchouk{n}@{x}{p}{N}}}}$

Remark

${\displaystyle{\displaystyle{\displaystyle\Hahn{n}@{x}{\alpha}{\beta}{N}=\dualHahn{x}@{\lambda(n)}{\alpha}{\beta}{N}}}}$
${\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{N}\frac{(2n+\alpha+\beta+1)\pochhammer{\alpha+1}{n}\pochhammer{-N}{n}N!}{(-1)^{n}\pochhammer{n+\alpha+\beta+1}{N+1}\pochhammer{\beta+1}{n}n!}\Hahn{n}@{x}{\alpha}{\beta}{N}\Hahn{n}@{y}{\alpha}{\beta}{N}{}=\frac{\Kronecker{x}{y}}{\dbinom{\alpha+x}{x}\dbinom{\beta+N-x}{N-x}}}}}$

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

Koornwinder Addendum: Hahn

Hahn: Special values

${\displaystyle{\displaystyle{\displaystyle\Hahn{n}@{0}{\alpha}{\beta}{N}=1\Hahn{n}@{N}{\alpha}{\beta}{N}}}}$
${\displaystyle{\displaystyle{\displaystyle\Hahn{n}@{0}{\alpha}{\beta}{N}=\frac{(-1)^{n}\pochhammer{\beta+1}{n}}{\pochhammer{\alpha+1}{n}}}}}$
${\displaystyle{\displaystyle{\displaystyle\Hahn{2n}@{N}{\alpha}{\alpha}{2N}=\frac{\pochhammer{\frac{1}{2}}{n}\pochhammer{N+\alpha+1}{n}}{\pochhammer{-N+\frac{1}{2}}{n}\pochhammer{\alpha+1}{n}}}}}$
${\displaystyle{\displaystyle{\displaystyle\Hahn{N}@{x}{\alpha}{\beta}{N}=\frac{\pochhammer{-N-\beta}{x}}{\pochhammer{\alpha+1}{x}}\qquad(x}}}$
${\displaystyle{\displaystyle{\displaystyle\Hahn{N}@{x}{\alpha}{\beta}{N}=0,1,\ldots,N)}}}$

Symmetries

${\displaystyle{\displaystyle{\displaystyle\frac{\Hahn{n}@{N-x}{\alpha}{\beta}{N}}{\Hahn{n}@{N}{\alpha}{\beta}{N}}=\Hahn{n}@{x}{\beta}{\alpha}{N}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{\Hahn{N-n}@{x}{\alpha}{\beta}{N}}{\Hahn{N}@{x}{\alpha}{\beta}{N}}=\Hahn{n}@{x}{-N-\beta-1}{-N-\alpha-1}{N}\qquad(x}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{\Hahn{N-n}@{x}{\alpha}{\beta}{N}}{\Hahn{N}@{x}{\alpha}{\beta}{N}}=0,1,\ldots,N)}}}$

Duality

${\displaystyle{\displaystyle{\displaystyle\Hahn{n}@{x}{\alpha}{\beta}{N}=\dualHahn{x}@{n(n+\alpha+\beta+1)}{\alpha}{\beta}{N}(n,x\in\{0,1,\ldots N\})}}}$

<< Continuous Hahn

Hahn

Dual Hahn >>