Dual q-Hahn

<< q-Hahn

Dual q-Hahn

Al-Salam-Chihara >>

Basic hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}=\qHyperrphis{3}{2}@@{q^{-n},q^{-x},\gamma\delta q^{x+1}}{\gamma q,q^{-N}}{q}{q}}}}$

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\frac{\left(\gamma q,\gamma\delta q,q^{-N};q\right)_{x}}{\left(q,\gamma\delta q^{N+2},\delta q;q\right)_{x}}\frac{(1-\gamma\delta q^{2x+1})}{(1-\gamma\delta q)(-\gamma q)^{x}}q^{Nx-\binomial{x}{2}}{}R_{m}\!\left(\mu(x);\gamma,\delta,N\right){q}R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}{}=\frac{\left(\gamma\delta q^{2};q\right)_{N}}{\left(\delta q;q\right)_{N}}(\gamma q)^{-N}\frac{\left(q,\delta^{-1}q^{-N};q\right)_{n}}{\left(\gamma q,q^{-N};q\right)_{n}}(\gamma\delta q)^{n}\,\delta_{m,n}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle-\left(1-q^{-x}\right)\left(1-\gamma\delta q^{x+1}\right)R_{n}\!\left(\mu(x)\right){q}{}=A_{n}R_{n+1}\!\left(\mu(x)\right){q}-\left(A_{n}+C_{n}\right)R_{n}\!\left(\mu(x)\right){q}+C_{n}R_{n-1}\!\left(\mu(x)\right){q}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle C_{n}=\gamma q\left(1-q^{n}\right)\left(\delta-q^{n-N-1}\right)}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=\left(1-q^{n-N}\right)\left(1-\gamma q^{n+1}\right)}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x)\right){q}:=R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{R}}_{n}\!\left(x\right)={\widehat{R}}_{n+1}\!\left(x\right)+\left[1+\gamma\delta q-(A_{n}+C_{n})\right]{\widehat{R}}_{n}\!\left(x\right){}+\gamma q(1-q^{n})(1-\gamma q^{n}){}(1-q^{n-N-1})(\delta-q^{n-N-1}){\widehat{R}}_{n-1}\!\left(x\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle C_{n}=\gamma q\left(1-q^{n}\right)\left(\delta-q^{n-N-1}\right)}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=\left(1-q^{n-N}\right)\left(1-\gamma q^{n+1}\right)}}}$

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}=\frac{1}{\left(\gamma q,q^{-N};q\right)_{n}}{\widehat{R}}_{n}\!\left(\mu(x)\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

q-Difference equation

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle D(x)=-\frac{\gamma q^{x-N}(1-q^{x})(1-\gamma\delta q^{x+N+1})(1-\delta q^{x})}{(1-\gamma\delta q^{2x})(1-\gamma\delta q^{2x+1})}}}}$ &

${\displaystyle{\displaystyle{\displaystyle B(x)=\frac{(1-q^{x-N})(1-\gamma q^{x+1})(1-\gamma\delta q^{x+1})}{(1-\gamma\delta q^{2x+1})(1-\gamma\delta q^{2x+2})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle y(x)=R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x+1);\gamma,\delta,N\right){q}-R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}{}=\frac{q^{-n-x}(1-q^{n})(1-\gamma\delta q^{2x+2})}{(1-\gamma q)(1-q^{-N})}R_{n-1}\!\left(\mu(x);\gamma q,\delta,N-1\right){q}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

${\displaystyle{\displaystyle{\displaystyle\frac{\Delta R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}}{\Delta\mu(x)}{}=\frac{q^{-n+1}(1-q^{n})}{(1-q)(1-\gamma q)(1-q^{-N})}R_{n-1}\!\left(\mu(x);\gamma q,\delta,N-1\right){q}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle(1-\gamma q^{x})(1-\gamma\delta q^{x})(1-q^{x-N-1})R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}{}+\gamma q^{x-N-1}(1-q^{x})(1-\gamma\delta q^{x+N+1})(1-\delta q^{x})R_{n}\!\left(\mu(x-1);\gamma,\delta,N\right){q}{}=q^{x}(1-\gamma)(1-q^{-N-1})(1-\gamma\delta q^{2x})R_{n+1}\!\left(\mu(x);\gamma q^{-1},\delta,N+1\right){q}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

${\displaystyle{\displaystyle{\displaystyle\frac{\nabla\left[w(x;\gamma,\delta,N|q)R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}\right]}{\nabla\mu(x)}{}=\frac{1}{(1-q)(1-\gamma\delta)}w(x;\gamma q^{-1},\delta,N+1|q){}R_{n+1}\!\left(\mu(x);\gamma q^{-1},\delta,N+1\right){q}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

${\displaystyle{\displaystyle{\displaystyle w(x;\gamma,\delta,N|q)=\frac{\left(\gamma q,\gamma\delta q,q^{-N};q\right)_{x}}{\left(q,\gamma\delta q^{N+2},\delta q;q\right)_{x}}(-\gamma^{-1})^{x}q^{Nx-\binomial{x}{2}}}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle w(x;\gamma,\delta,N|q)R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}{}=(1-q)^{n}\left(\gamma\delta q;q\right)_{n}\left(\nabla_{\mu}\right)^{n}\left[w(x;\gamma q^{n},\delta,N-n|q)\right]}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

${\displaystyle{\displaystyle{\displaystyle\nabla_{\mu}:=\frac{\nabla}{\nabla\mu(x)}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

Generating functions

${\displaystyle{\displaystyle{\displaystyle\left(q^{-N}t;q\right)_{N-x}\cdot\qHyperrphis{2}{1}@@{q^{-x},\delta^{-1}q^{-x}}{\gamma q}{q}{\gamma\delta q^{x+1}t}{}=\sum_{n=0}^{N}\frac{\left(q^{-N};q\right)_{n}}{\left(q;q\right)_{n}}R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}t^{n}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

${\displaystyle{\displaystyle{\displaystyle\left(\gamma\delta qt;q\right)_{x}\cdot\qHyperrphis{2}{1}@@{q^{x-N},\gamma q^{x+1}}{\delta^{-1}q^{-N}}{q}{q^{-x}t}{}=\sum_{n=0}^{N}\frac{\left(q^{-N},\gamma q;q\right)_{n}}{\left(\delta^{-1}q^{-N},q;q\right)_{n}}R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}t^{n}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

Limit relations

q-Racah polynomial to Dual q-Hahn polynomial

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);q^{-N-1},0,\gamma,\delta\,|\,q\right)=R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);0,\delta^{-1}q^{-N-1},\gamma,\delta\,|\,q\right)=R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);\alpha,0,q^{-N-1},\alpha\delta q^{N+1}\,|\,q\right)=R_{n}\!\left({\tilde{\mu}}(x);\alpha,\delta,N\right){q}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle{\tilde{\mu}}(x)=q^{-x}+\alpha\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

Dual q-Hahn polynomial to Affine q-Krawtchouk polynomial

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);p,0,N\right){q}=K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

Dual q-Hahn polynomial to Dual q-Krawtchouk polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\gamma\rightarrow 0}R_{n}\!\left(\mu(x);\gamma,c\gamma^{-1}q^{-N-1},N\right){q}=K_{n}\!\left(\lambda(x);c,N|q\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

Dual q-Hahn polynomial to Dual Hahn polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}R_{n}\!\left(\mu(x);q^{\gamma},q^{\delta},N\right){q}=R_{n}\!\left(\lambda(x);\gamma,\delta,N\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

Remark

${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(q^{-x};\alpha,\beta,N;q\right)=R_{x}\!\left(\mu(n);\alpha,\beta,N\right){q}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}=Q_{x}\!\left(q^{-n};\gamma,\delta,N;q\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=q^{-x}+q^{x+\gamma+\delta+1}=q^{-x}+\gamma\delta q^{x+1}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\mu(n)=q^{-n}+\alpha\beta q^{n+1}}}}$

Koornwinder Addendum: Dual q-Hahn

Symmetry

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(x;\gamma,\delta,N\right){q}=\frac{\left(\delta^{-1}q^{-N};q\right)_{n}}{\left(\gamma q;q\right)_{n}}\big{(}\gamma\delta q^{N+1}\big{)}^{n}R_{n}\!\left(\gamma^{-1}\delta^{-1}q^{-1-N}x;\delta^{-1}q^{-N-1},\gamma^{-1}q^{-N-1},N\right){q}}}}$

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