q-Hahn

<< Big q-Jacobi: Special case

q-Hahn

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Basic hypergeometric representation

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

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

Orthogonality relation(s)

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

Recurrence relation

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

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

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

Monic recurrence relation

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

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

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

q-Difference equation

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

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

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

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

Forward shift operator

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

Backward shift operator

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;\alpha,\beta,N|q)=\frac{\left(\alpha q,q^{-N};q\right)_{x}}{\left(q,\beta^{-1}q^{-N};q\right)_{x}}(\alpha\beta)^{-x}}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle w(x;\alpha,\beta,N|q)Q_{n}\!\left(q^{-x};\alpha,\beta,N;q\right){}=(1-q)^{n}\left(\nabla_{q}\right)^{n}\left[w(x;\alpha q^{n},\beta q^{n},N-n|q)\right]}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;\alpha,\beta,N|q)=\frac{\left(\alpha q,q^{-N};q\right)_{x}}{\left(q,\beta^{-1}q^{-N};q\right)_{x}}(\alpha\beta)^{-x}}}}$

${\displaystyle{\displaystyle{\displaystyle\nabla_{q}:=\frac{\nabla}{\nabla q^{-x}}}}}$

Generating functions

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

Limit relations

q-Racah polynomial to q-Hahn polynomial

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

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

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

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

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

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

q-Hahn polynomial to Little q-Jacobi polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}Q_{n}\!\left(q^{x-N};\alpha,\beta,N;q\right)=p_{n}\!\left(q^{x};\alpha,\beta;q\right)}}}$

q-Hahn polynomial to q-Meixner polynomial

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

q-Hahn polynomial to Quantum q-Krawtchouk polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow\infty}Q_{n}\!\left(q^{-x};\alpha,p,N;q\right)=K^{\mathrm{qtm}}_{n}\!\left(q^{-x};p,N;q\right)}}}$

q-Hahn polynomial to q-Krawtchouk polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow 0}Q_{n}(q^{-x};\alpha,-\alpha^{-1}q^{-1}p,N|q)=K_{n}\!\left(q^{-x};p,N;q\right)}}}$

q-Hahn polynomial to Affine q-Krawtchouk polynomial

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

q-Hahn polynomial to Hahn polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}Q_{n}(q^{-x};q^{\alpha},q^{\beta},N|q)=Q_{n}\!\left(x;\alpha,\beta,N\right)}}}$

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

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

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

<< Big q-Jacobi: Special case

q-Hahn

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