Formula:KLS:09.08:21: Difference between revisions

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<div id="drmf_head">
<div id="drmf_head">
<div id="alignleft"> << [[Formula:KLS:09.08:20|Formula:KLS:09.08:20]] </div>
<div id="alignleft"> << [[Formula:KLS:09.08:20|Formula:KLS:09.08:20]] </div>
<div id="aligncenter"> [[Jacobi#KLS:09.08:21|formula in Jacobi]] </div>
<div id="aligncenter"> [[Jacobi:_Special_cases#KLS:09.08:21|formula in Jacobi: Special cases]] </div>
<div id="alignright"> [[Formula:KLS:09.08:22|Formula:KLS:09.08:22]] >> </div>
<div id="alignright"> [[Formula:KLS:09.08:24|Formula:KLS:09.08:24]] >> </div>
</div>
</div>


<br /><div align="center"><math>{\displaystyle  
<br /><div align="center"><math>{\displaystyle  
\lim_{\alpha\rightarrow-\infty}
\frac{h_n}{h_0}=
\frac{\Jacobi{\alpha}{a-\alpha}{n}@{1+\alpha x}}{\Jacobi{\alpha}{a-\alpha}{n}@{1}}=\BesselPoly{n}@{x}{a}
\frac\lambda{\lambda+n} \frac{\pochhammer{2\lambda}{n}}{n!}  
h_0
}</math></div>
}</math></div>
== Substitution(s) ==
<div align="left"><math>{\displaystyle \frac{h_n}{h_0}
=\frac{\cpi^\frac12 \EulerGamma@{\lambda+\frac12}}{\EulerGamma@{\lambda+1}}
\frac{h_n}{h_0 (\Ultra{\lambda}{n}@{1})^2}
=
\frac\lambda{\lambda+n} \frac{n!}{\pochhammer{2\lambda}{n}}}</math></div><br />


== Proof ==
== Proof ==
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== Symbols List ==
== Symbols List ==


<span class="plainlinks">[http://dlmf.nist.gov/18.3#T1.t1.r3 <math>{\displaystyle P^{(\alpha,\beta)}_{n}}</math>]</span> : Jacobi polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r3 http://dlmf.nist.gov/18.3#T1.t1.r3]<br />
<span class="plainlinks">[http://dlmf.nist.gov/5.2#iii <math>{\displaystyle (a)_n}</math>]</span> : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]<br />
<span class="plainlinks">[http://dlmf.nist.gov/18.34#E1 <math>{\displaystyle y_{n}}</math>]</span> : Bessel polynomial : [http://dlmf.nist.gov/18.34#E1 http://dlmf.nist.gov/18.34#E1]
<span class="plainlinks">[http://dlmf.nist.gov/5.19.E4 <math>{\displaystyle \pi}</math>]</span> : ratio of a circle's circumference to its diameter : [http://dlmf.nist.gov/5.19.E4 http://dlmf.nist.gov/5.19.E4]<br />
<span class="plainlinks">[http://dlmf.nist.gov/5.2#E1 <math>{\displaystyle \Gamma}</math>]</span> : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1]<br />
<span class="plainlinks">[http://dlmf.nist.gov/18.3#T1.t1.r5 <math>{\displaystyle C^{\mu}_{n}}</math>]</span> : ultraspherical/Gegenbauer polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r5 http://dlmf.nist.gov/18.3#T1.t1.r5]
<br />
<br />


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<br /><div id="drmf_foot">
<br /><div id="drmf_foot">
<div id="alignleft"> << [[Formula:KLS:09.08:20|Formula:KLS:09.08:20]] </div>
<div id="alignleft"> << [[Formula:KLS:09.08:20|Formula:KLS:09.08:20]] </div>
<div id="aligncenter"> [[Jacobi#KLS:09.08:21|formula in Jacobi]] </div>
<div id="aligncenter"> [[Jacobi:_Special_cases#KLS:09.08:21|formula in Jacobi: Special cases]] </div>
<div id="alignright"> [[Formula:KLS:09.08:22|Formula:KLS:09.08:22]] >> </div>
<div id="alignright"> [[Formula:KLS:09.08:24|Formula:KLS:09.08:24]] >> </div>
</div>
</div>

Latest revision as of 07:35, 22 December 2019


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{h_n}{h_0}= \frac\lambda{\lambda+n} \frac{\pochhammer{2\lambda}{n}}{n!} h_0 }}

Substitution(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{h_n}{h_0} =\frac{\cpi^\frac12 \EulerGamma@{\lambda+\frac12}}{\EulerGamma@{\lambda+1}} \frac{h_n}{h_0 (\Ultra{\lambda}{n}@{1})^2} = \frac\lambda{\lambda+n} \frac{n!}{\pochhammer{2\lambda}{n}}}}


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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (a)_n}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \pi}}  : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Gamma}}  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle C^{\mu}_{n}}}  : ultraspherical/Gegenbauer polynomial : http://dlmf.nist.gov/18.3#T1.t1.r5

Bibliography

Equation in Section 9.8 of KLS.

URL links

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


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