DLMF:18.18.E8

MaRDI QIDQ6495

• Label: DLMF:18.18.E8

Digital Library of Mathematical Functions ID 18.18.E8

${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}\cos\phi\right)=\sum_{\ell=0}^{n}2^{2\ell}(n-\ell)!\frac{2\lambda+2\ell-1}{2\lambda-1}\frac{({\left(\lambda\right)_{\ell}})^{2}}{{\left(2\lambda\right)_{n+\ell}}}(\sin\theta_{1})^{\ell}C^{(\lambda+\ell)}_{n-\ell}\left(\cos\theta_{1}\right)(\sin\theta_{2})^{\ell}C^{(\lambda+\ell)}_{n-\ell}\left(\cos\theta_{2}\right)C^{(\lambda-\frac{1}{2})}_{\ell}\left(\cos\phi\right),}}$

Constraint(s)

${\displaystyle{\displaystyle\lambda\neq\frac{1}{2}}}$

Symbols List

• ${\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}$:
• ${\displaystyle{\displaystyle\cos\NVar{z}}}$: cosine function
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\sin\NVar{z}}}$: sine function
• ${\displaystyle{\displaystyle C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)}}$: ultraspherical (or Gegenbauer) polynomial
• ${\displaystyle{\displaystyle\ell}}$: nonnegative integer
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
 

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