q-Hahn

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q-Hahn

Basic hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}=\qHyperrphis{3}{2}@@{q^{-n},\alpha\beta q^{n+1},q^{-x}}{\alpha q,q^{-N}}{q}{q} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle n=0,1,2,\ldots,N}}


Orthogonality relation(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{x=0}^N\frac{\qPochhammer{\alpha q,q^{-N}}{q}{x}}{\qPochhammer{q,\beta^{-1}q^{-N}}{q}{x}}(\alpha\beta q)^{-x} \qHahn{m}@{q^{-x}}{\alpha}{\beta}{N}{q}\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q} {}=\frac{\qPochhammer{\alpha\beta q^2}{q}{N}}{\qPochhammer{\beta q}{q}{N}(\alpha q)^N} \frac{\qPochhammer{q,\alpha\beta q^{N+2},\beta q}{q}{n}}{\qPochhammer{\alpha q,\alpha\beta q,q^{-N}}{q}{n}}\ \frac{(1-\alpha\beta q)(-\alpha q)^n}{(1-\alpha\beta q^{2n+1})} q^{\binomial{n}{2}-Nn}\,\Kronecker{m}{n} }}

Recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle -\left(1-q^{-x}\right)\qHahn{n}@@{q^{-x}}{\alpha}{\beta}{N}{q}=A_n\qHahn{n+1}@@{q^{-x}}{\alpha}{\beta}{N}{q}-\left(A_n+C_n\right)\qHahn{n}@@{q^{-x}}{\alpha}{\beta}{N}{q} {}+C_n\qHahn{n-1}@@{q^{-x}}{\alpha}{\beta}{N}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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})}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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})}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHahn{n}@@{q^{-x}}{\alpha}{\beta}{N}{q}:=\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q} }}

Monic recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicqHahn{n}@@{x}{\alpha}{\beta}{N}{q}=\monicqHahn{n+1}@@{x}{\alpha}{\beta}{N}{q}+\left[1-(A_n+C_n)\right]\monicqHahn{n}@@{x}{\alpha}{\beta}{N}{q}+A_{n-1}C_n\monicqHahn{n-1}@@{x}{\alpha}{\beta}{N}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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})}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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})}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}= \frac{\qPochhammer{\alpha\beta q^{n+1}}{q}{n}}{\qPochhammer{\alpha q,q^{-N}}{q}{n}}\monicqHahn{n}@@{q^{-x}}{\alpha}{\beta}{N}{q} }}

q-Difference equation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle D(x)=\alpha q(1-q^x)(\beta-q^{x-N-1})}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle B(x)=(1-q^{x-N})(1-\alpha q^{x+1})}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}}}


Forward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHahn{n}@{q^{-x-1}}{\alpha}{\beta}{N}{q}-\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q} {}=\frac{q^{-n-x}(1-q^n)(1-\alpha\beta q^{n+1})}{(1-\alpha q)(1-q^{-N})} \qHahn{n-1}@{q^{-x}}{\alpha q}{\beta q}{N-1}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\Delta \qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}}{\Delta q^{-x}} =\frac{q^{-n+1}(1-q^n)(1-\alpha\beta q^{n+1})}{(1-q)(1-\alpha q)(1-q^{-N})} {} \qHahn{n-1}@{q^{-x}}{\alpha q}{\beta q}{N-1}{q} }}

Backward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-\alpha q^x)(1-q^{x-N-1})\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q} {}-\alpha(1-q^x)(\beta-q^{x-N-1})\qHahn{n}@{q^{-x+1}}{\alpha}{\beta}{N}{q} {}=q^x(1-\alpha)(1-q^{-N-1})Q_{n+1}(q^{-x};\alpha q^{-1},\beta q^{-1},N+1|q) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\nabla\left[w(x;\alpha,\beta,N|q)\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;\alpha,\beta,N|q)=\frac{\qPochhammer{\alpha q,q^{-N}}{q}{x}}{\qPochhammer{q,\beta^{-1}q^{-N}}{q}{x}}(\alpha\beta)^{-x}}}


Rodrigues-type formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;\alpha,\beta,N|q)\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q} {}=(1-q)^n\left(\nabla_q\right)^n\left[w(x;\alpha q^n,\beta q^n,N-n|q)\right] }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;\alpha,\beta,N|q)=\frac{\qPochhammer{\alpha q,q^{-N}}{q}{x}}{\qPochhammer{q,\beta^{-1}q^{-N}}{q}{x}}(\alpha\beta)^{-x}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \nabla_q:=\frac{\nabla}{\nabla q^{-x}} }}

Generating functions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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{\qPochhammer{q^{-N}}{q}{n}}{\qPochhammer{\beta q,q}{q}{n}}\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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{\qPochhammer{\alpha q,q^{-N}}{q}{n}}{\qPochhammer{q}{q}{n}}q^{-\binomial{n}{2}}\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}t^n }}

Limit relations

q-Racah polynomial to q-Hahn polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qRacah{n}@{\mu(x)}{\alpha}{\beta}{q^{-N-1}}{0}{q}=\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(n)=q^{-n}+\alpha\beta q^{n+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qRacah{n}@{\mu(x)}{\alpha}{\beta}{0}{\beta^{-1}q^{-N-1}}{q}=\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(n)=q^{-n}+\alpha\beta q^{n+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qRacah{n}@{\mu(x)}{q^{-N-1}}{\beta\gamma q^{N+1}}{\gamma}{0}{q}=\qHahn{n}@{q^{-x}}{\gamma}{\beta}{N}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(n)=q^{-n}+\alpha\beta q^{n+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1}}}


q-Hahn polynomial to Little q-Jacobi polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim _{N\rightarrow\infty} \qHahn{n}@{q^{x-N}}{\alpha}{\beta}{N}{q}=\littleqJacobi{n}@{q^x}{\alpha}{\beta}{q} }}

q-Hahn polynomial to q-Meixner polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{N\rightarrow\infty} Q_n(q^{-x};b,-b^{-1}c^{-1}q^{-N-1},N|q)=\qMeixner{n}@{q^{-x}}{b}{c}{q} }}

q-Hahn polynomial to Quantum q-Krawtchouk polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{\alpha\rightarrow\infty}\qHahn{n}@{q^{-x}}{\alpha}{p}{N}{q}=\qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q} }}

q-Hahn polynomial to q-Krawtchouk polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{\alpha\rightarrow 0} Q_n(q^{-x};\alpha,-\alpha^{-1}q^{-1}p,N|q)=\qKrawtchouk{n}@{q^{-x}}{p}{N}{q} }}

q-Hahn polynomial to Affine q-Krawtchouk polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHahn{n}@{q^{-x}}{p}{0}{N}{q}=\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q} }}

q-Hahn polynomial to Hahn polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}Q_n(q^{-x};q^{\alpha},q^{\beta},N|q)=\Hahn{n}@{x}{\alpha}{\beta}{N} }}

Remark

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q}=\dualqHahn{x}@{\mu(n)}{\alpha}{\beta}{N}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(n)=q^{-n}+\alpha\beta q^{n+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \dualqHahn{n}@{\mu(x)}{\gamma}{\delta}{N}{q}=\qHahn{x}@{q^{-n}}{\gamma}{\delta}{N}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(n)=q^{-n}+\alpha\beta q^{n+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1}}}


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