Formula:KLS:09.08:21: Difference between revisions

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<div id="drmf_head">
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<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>
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<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 08:35, 22 December 2019


h n h 0 = λ λ + n ⁢ ( 2 ⁢ λ ) n n ! ⁢ h 0 subscript ℎ 𝑛 subscript ℎ 0 𝜆 𝜆 𝑛 Pochhammer-symbol 2 𝜆 𝑛 𝑛 subscript ℎ 0 {\displaystyle{\displaystyle{\displaystyle\frac{h_{n}}{h_{0}}=\frac{\lambda}{% \lambda+n}\frac{{\left(2\lambda\right)_{n}}}{n!}h_{0}}}}

Substitution(s)

h n h 0 = π missing missing ⁢ 12 ⁢ Γ ⁡ ( λ + 1 2 ) Γ ⁡ ( λ + 1 ) ⁢ h n h 0 ⁢ ( C n λ ⁡ ( 1 ) ) 2 = λ λ + n ⁢ n ! ( 2 ⁢ λ ) n subscript ℎ 𝑛 subscript ℎ 0 missing missing 12 Euler-Gamma 𝜆 1 2 Euler-Gamma 𝜆 1 subscript ℎ 𝑛 subscript ℎ 0 superscript ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 1 2 𝜆 𝜆 𝑛 𝑛 Pochhammer-symbol 2 𝜆 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{h_{n}}{h_{0}}=\frac{{\pi^{% \frac{missing}{missing}}}12\Gamma\left(\lambda+\frac{1}{2}\right)}{\Gamma\left% (\lambda+1\right)}\frac{h_{n}}{h_{0}(C^{\lambda}_{n}\left(1\right))^{2}}=\frac% {\lambda}{\lambda+n}\frac{n!}{{\left(2\lambda\right)_{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

( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
π 𝜋 {\displaystyle{\displaystyle{\displaystyle\pi}}}  : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
Γ Γ {\displaystyle{\displaystyle{\displaystyle\Gamma}}}  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
C n μ subscript superscript 𝐶 𝜇 𝑛 {\displaystyle{\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

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