Definition:monicAlSalamCarlitzII and Formula:KLS:14.02:17: Difference between pages

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{{DISPLAYTITLE:Formula:KLS:14.02:17}}
<div id="drmf_head">
<div id="drmf_head">
<div id="alignleft"> << [[Definition:monicAlSalamCarlitzI|Definition:monicAlSalamCarlitzI]] </div>
<div id="alignleft"> << [[Formula:KLS:14.02:14|Formula:KLS:14.02:14]] </div>
<div id="aligncenter"> [[Main_Page|Main Page]] </div>
<div id="aligncenter"> [[q-Racah#KLS:14.02:17|formula in q-Racah]] </div>
<div id="alignright"> [[Definition:monicAlSalamChihara|Definition:monicAlSalamChihara]] >> </div>
<div id="alignright"> [[Formula:KLS:14.02:20|Formula:KLS:14.02:20]] >> </div>
</div>
</div>


The LaTeX DLMF and DRMF macro '''\monicAlSalamCarlitzII''' represents the monic <math>q</math> Al-Salam Carlitz II polynomial.
<br /><div align="center"><math>{\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)
}</math></div>


This macro is in the category of polynomials.
== Substitution(s) ==


In math mode, this macro can be called in the following ways:
<div align="left"><math>{\displaystyle D(x)=\frac{q(1-q^x)(1-\delta q^x)(\beta-\gamma q^x)(\alpha-\gamma\delta q^x)}
{(1-\gamma\delta q^{2x})(1-\gamma\delta q^{2x+1})}}</math> &<br />
<math>{\displaystyle B(x)=\frac{(1-\alpha q^{x+1})(1-\beta\delta q^{x+1})(1-\gamma q^{x+1})(1-\gamma\delta q^{x+1})}
{(1-\gamma\delta q^{2x+1})(1-\gamma\delta q^{2x+2})}}</math> &<br />
<math>{\displaystyle y(x)=\qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}}</math> &<br />
<math>{\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1}
=\lambda(x)=q^{-x}+cq^{x-N}
=q^{-x}+q^{x+\gamma+\delta+1}
=2a\cos@@{\theta}}</math> &<br />
<math>{\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}</math> &<br />
<math>{\displaystyle \mu(x):=q^{-x}+\gamma\delta q^{x+1}}</math> &<br />
<math>{\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1}
=\lambda(x)=q^{-x}+cq^{x-N}
=q^{-x}+q^{x+\gamma+\delta+1}
=2a\cos@@{\theta}}</math> &<br />
<math>{\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}</math></div><br />


:'''\monicAlSalamCarlitzII{a}{n}''' produces <math>{\displaystyle \monicAlSalamCarlitzII{a}{n}}</math><br />
== Proof ==
:'''\monicAlSalamCarlitzII{a}{n}@{x}{q}''' produces <math>{\displaystyle \monicAlSalamCarlitzII{a}{n}@{x}{q}}</math><br />
:'''\monicAlSalamCarlitzII{a}{n}@@{x}{q}''' produces <math>{\displaystyle \monicAlSalamCarlitzII{a}{n}@@{x}{q}}</math><br />


These are defined by
We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.
<math>
 
{\displaystyle
== Symbols List ==
\AlSalamCarlitzII{a}{n}@{x}{q}=:\monicAlSalamCarlitzII{a}{n}@{x}{q}.
 
}</math>
& : logical and<br />
<span class="plainlinks">[http://dlmf.nist.gov/18.28#E19 <math>{\displaystyle R_{n}}</math>]</span> : <math>{\displaystyle q}</math>-Racah polynomial : [http://dlmf.nist.gov/18.28#E19 http://dlmf.nist.gov/18.28#E19]<br />
<span class="plainlinks">[http://dlmf.nist.gov/4.14#E2 <math>{\displaystyle \mathrm{cos}}</math>]</span> : cosine function : [http://dlmf.nist.gov/4.14#E2 http://dlmf.nist.gov/4.14#E2]
<br />
<br />


== Symbols List ==
== Bibliography==
 
<span class="plainlinks">[http://homepage.tudelft.nl/11r49/askey/contents.html Equation in Section 14.2]</span> of [[Bibliography#KLS|'''KLS''']].
 
== URL links ==
 
We ask users to provide relevant URL links in this space.


<span class="plainlinks">[http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzII <math>{\displaystyle {\widehat V}^{(n)}_{\alpha}}</math>]</span> : monic Al-Salam-Carlitz II polynomial : [http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzII http://drmf.wmflabs.org/wiki/Definition:monicAlSalamCarlitzII]<br />
<span class="plainlinks">[http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII <math>{\displaystyle V^{(n)}_{\alpha}}</math>]</span> : Al-Salam-Carlitz II polynomial : [http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII]
<br /><div id="drmf_foot">
<br /><div id="drmf_foot">
<div id="alignleft"> << [[Definition:monicAlSalamCarlitzI|Definition:monicAlSalamCarlitzI]] </div>
<div id="alignleft"> << [[Formula:KLS:14.02:14|Formula:KLS:14.02:14]] </div>
<div id="aligncenter"> [[Main_Page|Main Page]] </div>
<div id="aligncenter"> [[q-Racah#KLS:14.02:17|formula in q-Racah]] </div>
<div id="alignright"> [[Definition:monicAlSalamChihara|Definition:monicAlSalamChihara]] >> </div>
<div id="alignright"> [[Formula:KLS:14.02:20|Formula:KLS:14.02:20]] >> </div>
</div>
</div>

Revision as of 00:33, 6 March 2017


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)=\frac{q(1-q^x)(1-\delta q^x)(\beta-\gamma q^x)(\alpha-\gamma\delta q^x)} {(1-\gamma\delta q^{2x})(1-\gamma\delta q^{2x+1})}}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle B(x)=\frac{(1-\alpha q^{x+1})(1-\beta\delta q^{x+1})(1-\gamma q^{x+1})(1-\gamma\delta q^{x+1})} {(1-\gamma\delta q^{2x+1})(1-\gamma\delta q^{2x+2})}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}}} &
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} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+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 \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

& : logical and
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle R_{n}}}  : Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle q}} -Racah polynomial : http://dlmf.nist.gov/18.28#E19
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mathrm{cos}}}  : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.2 of KLS.

URL links

We ask users to provide relevant URL links in this space.


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