Al-Salam-Carlitz II

<< Al-Salam-Carlitz I

Al-Salam-Carlitz II

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Basic hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \AlSalamCarlitzII{a}{n}@{x}{q}= (-a)^nq^{-\binomial{n}{2}}\,\qHyperrphis{2}{0}@@{q^{-n},x}{-}{q}{\frac{q^n}{a}} }}

Orthogonality relation(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{k=0}^{\infty}\frac{q^{k^2}a^k}{\qPochhammer{q}{q}{k}\qPochhammer{aq}{q}{k}} \AlSalamCarlitzII{a}{m}@{q^{-k}}{q}\AlSalamCarlitzII{a}{n}@{q^{-k}}{q} {}=\frac{\qPochhammer{q}{q}{n}a^n}{\qPochhammer{aq}{q}{\infty}q^{n^2}}\,\Kronecker{m}{n} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle 0<aq<1}}

Recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\AlSalamCarlitzII{a}{n}@{x}{q}=\AlSalamCarlitzII{a}{n+1}@{x}{q}+(a+1)q^{-n}\AlSalamCarlitzII{a}{n}@{x}{q} {}+aq^{-2n+1}(1-q^n)\AlSalamCarlitzII{a}{n-1}@{x}{q} }}

Monic recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicAlSalamCarlitzII{a}{n}@@{x}{q}=\monicAlSalamCarlitzII{a}{n+1}@@{x}{q}+(a+1)q^{-n}\monicAlSalamCarlitzII{a}{n}@@{x}{q}+aq^{-2n+1}(1-q^n)\monicAlSalamCarlitzII{a}{n-1}@@{x}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \AlSalamCarlitzII{a}{n}@{x}{q}=\monicAlSalamCarlitzII{a}{n}@@{x}{q} }}

q-Difference equation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle -(1-q^n)x^2y(x) {}=(1-x)(a-x)y(qx)-\left[(1-x)(a-x)+aq\right]y(x)+aqy(q^{-1}x) }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\AlSalamCarlitzII{a}{n}@{x}{q}}}

Forward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \AlSalamCarlitzII{a}{n}@{x}{q}-\AlSalamCarlitzII{a}{n}@{qx}{q}=q^{-n+1}(1-q^n)x\AlSalamCarlitzII{a}{n-1}@{qx}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qderiv{q}\AlSalamCarlitzII{a}{n}@{x}{q}=\frac{q^{-n+1}(1-q^n)}{1-q} \AlSalamCarlitzII{a}{n-1}@{qx}{q} }}

Backward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle a\AlSalamCarlitzII{a}{n}@{x}{q}-(1-x)(a-x)\AlSalamCarlitzII{a}{n}@{qx}{q}=-q^nx\AlSalamCarlitzII{a}{n+1}@{x}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qderiv{q}\left[w(x;a;q)\AlSalamCarlitzII{a}{n}@{x}{q}\right] =-\frac{q^n}{a(1-q)}w(x;a;q)\AlSalamCarlitzII{a}{n+1}@{x}{q} }}

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

Rodrigues-type formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;a;q)\AlSalamCarlitzII{a}{n}@{x}{q}=a^n(q-1)^nq^{-\binomial{n}{2}} \left(\qderiv{q}\right)^n\left[w(x;a;q)\right] }}

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

Generating functions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\qPochhammer{xt}{q}{\infty}}{\qPochhammer{t,at}{q}{\infty}}=\sum_{n=0}^{\infty} \frac{(-1)^nq^{\binomial{n}{2}}}{\qPochhammer{q}{q}{n}}\AlSalamCarlitzII{a}{n}@{x}{q}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qPochhammer{at}{q}{\infty}\cdot\qHyperrphis{1}{1}@@{x}{at}{q}{t}=\sum_{n=0}^{\infty} \frac{q^{n(n-1)}}{\qPochhammer{q}{q}{n}}\AlSalamCarlitzII{a}{n}@{x}{q}t^n }}

Limit relations

q-Meixner polynomial to Al-Salam-Carlitz II polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{c\rightarrow 0}\qMeixner{n}@{x}{-ac^{-1}}{c}{q}= \left(-\frac{1}{a}\right)^nq^{\binomial{n}{2}}\AlSalamCarlitzII{a}{n}@{x}{q} }}

Quantum q-Krawtchouk polynomial to Al-Salam-Carlitz II polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{N\rightarrow\infty}\qtmqKrawtchouk{n}@{x}{a^{-1}q^{-N-1}}{N}{q}= \left(-\frac{1}{a}\right)^nq^{\binomial{n}{2}}\AlSalamCarlitzII{a}{n}@{x}{q} }}

Al-Salam-Carlitz II polynomial to Discrete q-Hermite II polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \iunit^{-n}\AlSalamCarlitzII{-1}{n}@{\iunit x}{q}=\discrqHermiteII{n}@{x}{q} }}

Al-Salam-Carlitz II polynomial to Charlier / Hermite polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}\frac{\AlSalamCarlitzII{a(1-q)}{n}@{q^{-x}}{q}}{(q-1)^n}= a^n\Charlier{n}@{x}{a} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}\frac{\AlSalamCarlitzII{\iunit a\sqrt{1-q^2}-1}{n}@{\iunit x\sqrt{1-q^2}}{q}} {\iunit^n(1-q^2)^{\frac{n}{2}}}=\frac{\Hermite{n}@{x-a}}{2^n} }}

Remark

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \AlSalamCarlitzII{a}{n}@{x}{q^{-1}}=\AlSalamCarlitzI{a}{n}@{x}{q} }}

<< Al-Salam-Carlitz I

Al-Salam-Carlitz II

Continuous q-Hermite >>