Al-Salam-Carlitz I

<< q-Charlier

Al-Salam-Carlitz I

Al-Salam-Carlitz II >>

Basic hypergeometric representation

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

Orthogonality relation(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_a^1\qPochhammer{qx,a^{-1}qx}{q}{\infty}\AlSalamCarlitzI{a}{m}@{x}{q}\AlSalamCarlitzI{a}{n}@{x}{q}\,d_qx {}=(-a)^n(1-q)\qPochhammer{q}{q}{n}\qPochhammer{q,a,a^{-1}q}{q}{\infty}q^{\binomial{n}{2}}\,\Kronecker{m}{n} }}

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

Recurrence relation

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

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

Forward shift operator

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

Backward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle a\AlSalamCarlitzI{a}{n}@{x}{q}-(1-x)(a-x)\AlSalamCarlitzI{a}{n}@{q^{-1}x}{q}=-q^{-n}x\AlSalamCarlitzI{a}{n+1}@{x}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mathcal{D}_{q^{-1}}\left[w(x;a;q)\AlSalamCarlitzI{a}{n}@{x}{q}\right]= \frac{q^{-n+1}}{a(1-q)}w(x;a;q)\AlSalamCarlitzI{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)=\qPochhammer{qx,a^{-1}qx}{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)\AlSalamCarlitzI{a}{n}@{x}{q}=a^nq^{\frac{1}{2}n(n-3)}(1-q)^n \left(\mathcal{D}_{q^{-1}}\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)=\qPochhammer{qx,a^{-1}qx}{q}{\infty}}}

Generating function

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

Limit relations

Big q-Laguerre polynomial to Al-Salam-Carlitz I polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{a\rightarrow 0}\frac{\bigqLaguerre{n}@{aqx}{a}{ab}{q}}{a^n}=q^n\AlSalamCarlitzI{b}{n}@{x}{q} }}

Dual q-Krawtchouk polynomial to Al-Salam-Carlitz I polynomial

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

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

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

Al-Salam-Carlitz I 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{\AlSalamCarlitzI{a(q-1)}{n}@{q^x}{q}}{(1-q)^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{\AlSalamCarlitzI{a\sqrt{1-q^2}-1}{n}@{x\sqrt{1-q^2}}{q}} {(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 \AlSalamCarlitzI{a}{n}@{x}{q^{-1}}=\AlSalamCarlitzII{a}{n}@{x}{q} }}

<< q-Charlier

Al-Salam-Carlitz I

Al-Salam-Carlitz II >>