Continuous big q-Hermite
Continuous big q-Hermite
Basic hypergeometric representation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsbigqHermite{n}@{x}{a}{q}=a^{-n}\,\qHyperrphis{3}{2}@@{q^{-n},a\expe^{\iunit\theta},a\expe^{-\iunit\theta}}{0,0}{q}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsbigqHermite{n}@{x}{a}{q}=\expe^{\iunit n\theta}\,\qHyperrphis{2}{0}@@{q^{-n},a\expe^{\iunit\theta}}{-}{q}{q^n\expe^{-2\iunit\theta}} }}
Orthogonality relation(s)
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{1}{2\cpi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}\ctsbigqHermite{m}@{x}{a}{q}\ctsbigqHermite{n}@{x}{a}{q}\,dx =\frac{\,\Kronecker{m}{n}}{\qPochhammer{q^{n+1}}{q}{\infty}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit\theta},\alpha\expe^{-\iunit\theta}}{q}{\infty}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{1}{2\cpi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}\ctsbigqHermite{m}@{x}{a}{q}\ctsbigqHermite{n}@{x}{a}{q}\,dx {}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 1<aq^k\leq a\end{array}} w_k\ctsbigqHermite{m}@{x_k}{a}{q}\ctsbigqHermite{n}@{x_k}{a}{q}=\frac{\,\Kronecker{m}{n}}{\qPochhammer{q^{n+1}}{q}{\infty}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x_k=\frac{aq^k+\left(aq^k\right)^{-1}}{2}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;a|q)=\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit\theta}}{q}{\infty}}\right|^2= \frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit\theta},\alpha\expe^{-\iunit\theta}}{q}{\infty}}}
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Recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle 2x\ctsbigqHermite{n}@{x}{a}{q}=\ctsbigqHermite{n+1}@{x}{a}{q}+aq^n\ctsbigqHermite{n}@{x}{a}{q}+(1-q^n)\ctsbigqHermite{n-1}@{x}{a}{q} }}
Monic recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicctsbigqHermite{n}@@{x}{a}{q}=\monicctsbigqHermite{n+1}@@{x}{a}{q}+\frac{1}{2}aq^n\monicctsbigqHermite{n}@@{x}{a}{q}+\frac{1}{4}(1-q^n)\monicctsbigqHermite{n-1}@@{x}{a}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsbigqHermite{n}@{x}{a}{q}=2^n\monicctsbigqHermite{n}@@{x}{a}{q} }}
q-Difference equations
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-q)^2D_q\left[{\tilde w}(x;aq^{\frac{1}{2}}|q)D_qy(x)\right] +4q^{-n+1}(1-q^n){\tilde w}(x;a|q)y(x)=0 }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\ctsbigqHermite{n}@{x}{a}{q}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;a|q)=\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit\theta}}{q}{\infty}}\right|^2= \frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit\theta},\alpha\expe^{-\iunit\theta}}{q}{\infty}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \LegendrePoly{n}@{z}:=a^{-n}\,\qHyperrphis{3}{2}@@{q^{-n},az,az^{-1}}{0,0}{q}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle q^{-n}(1-q^n)\LegendrePoly{n}@{z}=A(z)\LegendrePoly{n}@{qz}-\left[A(z)+A(z^{-1})\right]\LegendrePoly{n}@{z} {}+A(z^{-1})\LegendrePoly{n}@{q^{-1}z} }}
Forward shift operator
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \delta_q\ctsbigqHermite{n}@{x}{a}{q}=-q^{-\frac{1}{2}n}(1-q^n)(\expe^{\iunit\theta}-\expe^{-\iunit\theta}) \ctsbigqHermite{n-1}@{x}{aq^{\frac{1}{2}}}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle D_q\ctsbigqHermite{n}@{x}{a}{q}=\frac{2q^{-\frac{1}{2}(n-1)}(1-q^n)}{1-q} \ctsbigqHermite{n-1}@{x}{aq^{\frac{1}{2}}}{q} }}
Backward shift operator
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \delta_q\left[{\tilde w}(x;a|q)\ctsbigqHermite{n}@{x}{a}{q}\right] {}=q^{-\frac{1}{2}(n+1)}(\expe^{\iunit\theta}-\expe^{-\iunit\theta}) {} {\tilde w}(x;aq^{-\frac{1}{2}}|q)\ctsbigqHermite{n+1}@{x}{aq^{-\frac{1}{2}}}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;a|q)=\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit\theta}}{q}{\infty}}\right|^2= \frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit\theta},\alpha\expe^{-\iunit\theta}}{q}{\infty}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle D_q\left[{\tilde w}(x;a|q)\ctsbigqHermite{n}@{x}{a}{q}\right]= -\frac{2q^{-\frac{1}{2}n}}{1-q}{\tilde w}(x;aq^{-\frac{1}{2}}|q)\ctsbigqHermite{n+1}@{x}{aq^{-\frac{1}{2}}}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;a|q)=\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit\theta}}{q}{\infty}}\right|^2= \frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit\theta},\alpha\expe^{-\iunit\theta}}{q}{\infty}}}
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Rodrigues-type formula
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;a|q)\ctsbigqHermite{n}@{x}{a}{q}=\left(\frac{q-1}{2}\right)^nq^{\frac{1}{4}n(n-1)} \left(D_q\right)^n\left[w(x;aq^{\frac{1}{2}n}|q)\right] }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit\theta},\alpha\expe^{-\iunit\theta}}{q}{\infty}}}
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Generating functions
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\qPochhammer{at}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t,\expe^{-\iunit\theta}t}{q}{\infty}} =\sum_{n=0}^{\infty}\frac{\ctsbigqHermite{n}@{x}{a}{q}}{\qPochhammer{q}{q}{n}}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}\cdot\qHyperrphis{1}{1}@@{a\expe^{\iunit\theta}}{\expe^{\iunit\theta}t}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binomial{n}{2}}}{\qPochhammer{q}{q}{n}}\ctsbigqHermite{n}@{x}{a}{q}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\qPochhammer{\gamma\expe^{\iunit\theta}t}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\ \qHyperrphis{2}{1}@@{\gamma,a\expe^{\iunit\theta}}{\gamma\expe^{\iunit\theta}t}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{\gamma}{q}{n}}{\qPochhammer{q}{q}{n}}\ctsbigqHermite{n}@{x}{a}{q}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \gamma}} arbitrary
Limit relations
Al-Salam-Chihara polynomial to Continuous big q-Hermite polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \AlSalamChihara{n}@{x}{a}{0}{q}=\ctsbigqHermite{n}@{x}{a}{q} }}
Continuous big q-Hermite polynomial to Continuous q-Hermite polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsbigqHermite{n}@{x}{0}{q}=\ctsqHermite{n}@{x}{q} }}
Continuous big q-Hermite polynomial to Hermite polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1} \frac{\ctsbigqHermite{n}@{x\sqrt{\frac{1}{2}(1-q)}}{0}{q}} {\left(\frac{1-q}{2}\right)^{\frac{n}{2}}}=\Hermite{n}@{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1} \frac{\ctsbigqHermite{n}@{x\sqrt{\frac{1}{2}(1-q)}}{a\sqrt{2(1-q)}}{q}} {\left(\frac{1-q}{2}\right)^{\frac{n}{2}}}=\Hermite{n}@{x-a} }}