Formula:KLS:09.08:32

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C n λ ⁡ ( cos ⁡ θ ) = ∑ k = 0 n ( λ ) k ⁢ ( λ ) n - k k ! ⁢ ( n - k ) ! ⁢ e i ⁢ ( n - 2 ⁢ k ) ⁢ θ ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝜃 superscript subscript 𝑘 0 𝑛 Pochhammer-symbol 𝜆 𝑘 Pochhammer-symbol 𝜆 𝑛 𝑘 𝑘 𝑛 𝑘 imaginary-unit 𝑛 2 𝑘 𝜃 {\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(\cos\theta% \right)=\sum_{k=0}^{n}\frac{{\left(\lambda\right)_{k}}{\left(\lambda\right)_{n% -k}}}{k!(n-k)!}{\mathrm{e}^{\mathrm{i}(n-2k)\theta}}}}}

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
Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
e e {\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i

Bibliography

Equation in Section 9.8 of KLS.

URL links

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