Formula:KLS:09.08:35: Difference between revisions

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Latest revision as of 08:35, 22 December 2019


C n λ ⁡ ( cos ⁡ θ ) = ( λ ) n n ! ⁢ ∑ k = 0 ∞ ( λ ) k ⁢ ( 1 - λ ) k ( 1 - λ - n ) k ⁢ k ! ⁢ cos ⁡ ( ( n - k + λ ) ⁢ θ + 1 2 ⁢ ( k - λ ) ⁢ π ) ( 2 ⁢ sin ⁡ θ ) k + λ ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝜃 Pochhammer-symbol 𝜆 𝑛 𝑛 superscript subscript 𝑘 0 Pochhammer-symbol 𝜆 𝑘 Pochhammer-symbol 1 𝜆 𝑘 Pochhammer-symbol 1 𝜆 𝑛 𝑘 𝑘 𝑛 𝑘 𝜆 𝜃 1 2 𝑘 𝜆 superscript 2 𝜃 𝑘 𝜆 {\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(\cos\theta% \right)=\frac{{\left(\lambda\right)_{n}}}{n!}\sum_{k=0}^{\infty}\frac{{\left(% \lambda\right)_{k}}{\left(1-\lambda\right)_{k}}}{{\left(1-\lambda-n\right)_{k}% }k!}\frac{\cos\left((n-k+\lambda)\theta+\frac{1}{2}(k-\lambda)\pi\right)}{(2% \sin\theta)^{k+\lambda}}}}}

Proof

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Symbols List

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
cos cos {\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}  : cosine function : http://dlmf.nist.gov/4.14#E2
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
sin sin {\displaystyle{\displaystyle{\displaystyle\mathrm{sin}}}}  : sine function : http://dlmf.nist.gov/4.14#E1

Bibliography

Equation in Section 9.8 of KLS.

URL links

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