Definition:bigqLaguerre and Q-Charlier: Difference between pages

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{{DISPLAYTITLE:''q''-Charlier}}
<div id="drmf_head">
<div id="drmf_head">
<div id="alignleft"> << [[Definition:bigqJacobiIVparam|Definition:bigqJacobiIVparam]] </div>
<div id="alignleft"> << [[q-Bessel|q-Bessel]] </div>
<div id="aligncenter"> [[Main_Page|Main Page]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#q-Charlier|q-Charlier]] </div>
<div id="alignright"> [[Definition:bigqLegendre|Definition:bigqLegendre]] >> </div>
<div id="alignright"> [[Al-Salam-Carlitz I|Al-Salam-Carlitz I]] >> </div>
</div>
</div>


The LaTeX DLMF and DRMF macro '''\bigqLaguerre''' represents big <math>q</math>-Laguerre polynomial.
== ''q''-Charlier ==


This macro is in the category of polynomials.
== Basic hypergeometric representation ==


In math mode, this macro can be called in the following ways:
<math id="KLS:14.23:01">{\displaystyle
\qCharlier{n}@{q^{-x}}{a}{q}=\qHyperrphis{2}{1}@@{q^{-n},q^{-x}}{0}{q}{-\frac{q^{n+1}}{a}}
}</math><br />
<math id="KLS:14.23:02">{\displaystyle
\qCharlier{n}@{q^{-x}}{a}{q}=\qPochhammer{-a^{-1}q}{q}{n}\cdot\qHyperrphis{1}{1}@@{q^{-n}}{-a^{-1}q}{q}{-\frac{q^{n+1-x}}{a}}
}</math>
 
== Orthogonality relation(s) ==
 
<math id="KLS:14.23:03">{\displaystyle
\sum_{x=0}^{\infty}\frac{a^x}{\qPochhammer{q}{q}{x}}q^{\binomial{x}{2}}\qCharlier{m}@{q^{-x}}{a}{q}\qCharlier{n}@{q^{-x}}{a}{q}
{}=q^{-n}\qPochhammer{-a}{q}{\infty}\qPochhammer{-a^{-1}q,q}{q}{n}\,\Kronecker{m}{n}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle a>0}</math></div><br />
 
== Recurrence relation ==
 
<math id="KLS:14.23:04">{\displaystyle
q^{2n+1}(1-q^{-x})\qCharlier{n}@@{q^{-x}}{a}{q}
{}=a\qCharlier{n+1}@@{q^{-x}}{a}{q}-\left[a+q(1-q^n)(a+q^n)\right]\qCharlier{n}@@{q^{-x}}{a}{q}
{}+q(1-q^n)(a+q^n)\qCharlier{n-1}@@{q^{-x}}{a}{q}
}</math><br />
<math id="KLS:14.23:05">{\displaystyle
\qCharlier{n}@@{q^{-x}}{a}{q}:=\qCharlier{n}@{q^{-x}}{a}{q}
}</math>
 
== Monic recurrence relation ==
 
<math id="KLS:14.23:06">{\displaystyle
x\monicqCharlier{n}@@{x}{a}{q}=\monicqCharlier{n+1}@@{x}{a}{q}+\left[1+q^{-2n-1}\left\{a+q(1-q^n)(a+q^n)\right\}\right]\monicqCharlier{n}@@{x}{a}{q}
{}+aq^{-4n+1}(1-q^n)(a+q^n)\monicqCharlier{n-1}@@{x}{a}{q}
}</math><br />
<math id="KLS:14.23:07">{\displaystyle
\qCharlier{n}@{q^{-x}}{a}{q}=\frac{(-1)^nq^{n^2}}{a^n}\monicqCharlier{n}@@{q^{-x}}{a}{q}
}</math>
 
== ''q''-Difference equation ==
 
<math id="KLS:14.23:08">{\displaystyle
q^ny(x)=aq^xy(x+1)-q^x(a-1)y(x)+(1-q^x)y(x-1)
}</math>
<div align="right">Substitution(s): <math>{\displaystyle y(x)=\qCharlier{n}@{q^{-x}}{a}{q}}</math></div><br />
 
== Forward shift operator ==
 
<math id="KLS:14.23:10">{\displaystyle
\qCharlier{n}@{q^{-x-1}}{a}{q}-\qCharlier{n}@{q^{-x}}{a}{q}=-a^{-1}q^{-x}(1-q^n)\qCharlier{n-1}@{q^{-x}}{aq^{-1}}{q}
}</math><br />
<math id="KLS:14.23:11">{\displaystyle
\frac{\Delta \qCharlier{n}@{q^{-x}}{a}{q}}{\Delta q^{-x}}=-\frac{q(1-q^n)}{a(1-q)}\qCharlier{n-1}@{q^{-x}}{aq^{-1}}{q}
}</math>
 
== Backward shift operator ==
 
<math id="KLS:14.23:12">{\displaystyle
\qCharlier{n}@{q^{-x}}{a}{q}-a^{-1}q^{-x}(1-q^x)\qCharlier{n}@{q^{-x+1}}{a}{q}=\qCharlier{n+1}@{q^{-x}}{aq}{q}
}</math><br />
<math id="KLS:14.23:13">{\displaystyle
\frac{\nabla\left[w(x;a;q)\qCharlier{n}@{q^{-x}}{a}{q}\right]}{\nabla q^{-x}}
=\frac{1}{1-q}w(x;aq;q)\qCharlier{n+1}@{q^{-x}}{aq}{q}
}</math>
<div align="right">Substitution(s): <math>{\displaystyle w(x;a;q)=\frac{a^xq^{\binomial{x+1}{2}}}{\qPochhammer{q}{q}{x}}}</math></div><br />


:'''\bigqLaguerre{n}''' produces <math>{\displaystyle \bigqLaguerre{n}}</math><br />
== Rodrigues-type formula ==
:'''\bigqLaguerre{n}@{x}{a}{b}{q}''' produces <math>{\displaystyle \bigqLaguerre{n}@{x}{a}{b}{q}}</math><br />


These are defined by
<math id="KLS:14.23:15">{\displaystyle  
<math>{\displaystyle  
w(x;a;q)\qCharlier{n}@{q^{-x}}{a}{q}=(1-q)^n\left(\nabla_q\right)^n\left[w(x;aq^{-n};q)\right]
\bigqLaguerre{n}@{x}{a}{b}{q}:=\qHyperrphis{3}{2}@@{q^{-n},0,x}{aq,bq}{q}{q}.
}</math>
<div align="right">Substitution(s): <math>{\displaystyle w(x;a;q)=\frac{a^xq^{\binomial{x+1}{2}}}{\qPochhammer{q}{q}{x}}}</math></div><br />
<math id="KLS:14.23:16">{\displaystyle
\nabla_q:=\frac{\nabla}{\nabla q^{-x}}
}</math>
}</math>
<br />


== Symbols List ==
== Generating functions ==


<span class="plainlinks">[http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre <math>{\displaystyle P_{n}}</math>]</span> : big <math>{\displaystyle q}</math>-Laguerre polynomial : [http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre]<br />
<math id="KLS:14.23:17">{\displaystyle  
<span class="plainlinks">[http://dlmf.nist.gov/17.4#E1 <math>{\displaystyle {{}_{r}\phi_{s}}}</math>]</span> : basic hypergeometric (or <math>{\displaystyle q}</math>-hypergeometric) function : [http://dlmf.nist.gov/17.4#E1 http://dlmf.nist.gov/17.4#E1]
\frac{1}{\qPochhammer{t}{q}{\infty}}\,\qHyperrphis{1}{1}@@{q^{-x}}{0}{q}{-a^{-1}qt}
<br /><div id="drmf_foot">
=\sum_{n=0}^{\infty}\frac{\qCharlier{n}@{q^{-x}}{a}{q}}{\qPochhammer{q}{q}{n}}t^n
<div id="alignleft"> << [[Definition:bigqJacobiIVparam|Definition:bigqJacobiIVparam]] </div>
}</math><br />
<div id="aligncenter"> [[Main_Page|Main Page]] </div>
<math id="KLS:14.23:18">{\displaystyle  
<div id="alignright"> [[Definition:bigqLegendre|Definition:bigqLegendre]] >> </div>
\frac{1}{\qPochhammer{t}{q}{\infty}}\,\qHyperrphis{0}{1}@@{-}{-a^{-1}q}{q}{-a^{-1}q^{-x+1}t}
=\sum_{n=0}^{\infty}\frac{\qCharlier{n}@{q^{-x}}{a}{q}}{\qPochhammer{-a^{-1}q,q}{q}{n}}t^n
}</math>
 
== Limit relations ==
 
=== ''q''-Meixner polynomial to ''q''-Charlier polynomial ===
 
<math id="KLS:14.23:19">{\displaystyle
\qMeixner{n}@{x}{0}{c}{q}=\qCharlier{n}@{x}{c}{q}
}</math><br />
 
=== ''q''-Krawtchouk polynomial to ''q''-Charlier polynomial ===
 
<math id="KLS:14.23:20">{\displaystyle
\lim_{N\rightarrow\infty}\qKrawtchouk{n}@{q^{-x}}{a^{-1}q^{-N}}{N}{q}=\qCharlier{n}@{q^{-x}}{a}{q}
}</math><br />
 
=== ''q''-Charlier polynomial to Stieltjes-Wigert polynomial ===
 
<math id="KLS:14.23:21">{\displaystyle  
\lim_{a\rightarrow\infty}\qCharlier{n}@{ax}{a}{q}=\qPochhammer{q}{q}{n}\StieltjesWigert{n}@{x}{q}
}</math><br />
 
=== ''q''-Charlier polynomial to Charlier polynomial ===
 
<math id="KLS:14.23:22">{\displaystyle  
\lim_{q\rightarrow 1}\qCharlier{n}@{q^{-x}}{a(1-q)}{q}=\Charlier{n}@{x}{a}
}</math>
 
== Remark ==
 
<math id="KLS:14.23:23">{\displaystyle
\frac{\qCharlier{n}@{-x}{-q^{-\alpha}}{q}}{\qPochhammer{q}{q}{n}}=\qLaguerre[\alpha]{n}@{x}{q}
}</math>
<div id="drmf_foot">
<div id="alignleft"> << [[q-Bessel|q-Bessel]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#q-Charlier|q-Charlier]] </div>
<div id="alignright"> [[Al-Salam-Carlitz I|Al-Salam-Carlitz I]] >> </div>
</div>
</div>

Revision as of 00:32, 6 March 2017

q-Charlier

Basic hypergeometric representation

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

Orthogonality relation(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{x=0}^{\infty}\frac{a^x}{\qPochhammer{q}{q}{x}}q^{\binomial{x}{2}}\qCharlier{m}@{q^{-x}}{a}{q}\qCharlier{n}@{q^{-x}}{a}{q} {}=q^{-n}\qPochhammer{-a}{q}{\infty}\qPochhammer{-a^{-1}q,q}{q}{n}\,\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 q^{2n+1}(1-q^{-x})\qCharlier{n}@@{q^{-x}}{a}{q} {}=a\qCharlier{n+1}@@{q^{-x}}{a}{q}-\left[a+q(1-q^n)(a+q^n)\right]\qCharlier{n}@@{q^{-x}}{a}{q} {}+q(1-q^n)(a+q^n)\qCharlier{n-1}@@{q^{-x}}{a}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qCharlier{n}@@{q^{-x}}{a}{q}:=\qCharlier{n}@{q^{-x}}{a}{q} }}

Monic recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicqCharlier{n}@@{x}{a}{q}=\monicqCharlier{n+1}@@{x}{a}{q}+\left[1+q^{-2n-1}\left\{a+q(1-q^n)(a+q^n)\right\}\right]\monicqCharlier{n}@@{x}{a}{q} {}+aq^{-4n+1}(1-q^n)(a+q^n)\monicqCharlier{n-1}@@{x}{a}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qCharlier{n}@{q^{-x}}{a}{q}=\frac{(-1)^nq^{n^2}}{a^n}\monicqCharlier{n}@@{q^{-x}}{a}{q} }}

q-Difference equation

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

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


Forward shift operator

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

Backward shift operator

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


Rodrigues-type formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;a;q)\qCharlier{n}@{q^{-x}}{a}{q}=(1-q)^n\left(\nabla_q\right)^n\left[w(x;aq^{-n};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{a^xq^{\binomial{x+1}{2}}}{\qPochhammer{q}{q}{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 \frac{1}{\qPochhammer{t}{q}{\infty}}\,\qHyperrphis{1}{1}@@{q^{-x}}{0}{q}{-a^{-1}qt} =\sum_{n=0}^{\infty}\frac{\qCharlier{n}@{q^{-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 \frac{1}{\qPochhammer{t}{q}{\infty}}\,\qHyperrphis{0}{1}@@{-}{-a^{-1}q}{q}{-a^{-1}q^{-x+1}t} =\sum_{n=0}^{\infty}\frac{\qCharlier{n}@{q^{-x}}{a}{q}}{\qPochhammer{-a^{-1}q,q}{q}{n}}t^n }}

Limit relations

q-Meixner polynomial to q-Charlier polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qMeixner{n}@{x}{0}{c}{q}=\qCharlier{n}@{x}{c}{q} }}

q-Krawtchouk polynomial to q-Charlier polynomial

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

q-Charlier polynomial to Stieltjes-Wigert polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{a\rightarrow\infty}\qCharlier{n}@{ax}{a}{q}=\qPochhammer{q}{q}{n}\StieltjesWigert{n}@{x}{q} }}

q-Charlier polynomial to Charlier polynomial

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

Remark

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\qCharlier{n}@{-x}{-q^{-\alpha}}{q}}{\qPochhammer{q}{q}{n}}=\qLaguerre[\alpha]{n}@{x}{q} }}

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