Formula:6496

Digital Library of Mathematical Functions ID 18.18.E9

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "TeX input is invalid."): {\displaystyle \LegendrepolyP{n}@{\cos@@{\theta_{1}}\cos@@{\theta_{2}}+\sin@@{\theta_{1}}\sin@@{\theta_{2}}\cos@@{\phi}}={\LegendrepolyP{n}@{\cos@@{\theta_{1}}}\LegendrepolyP{n}@{\cos@@{\theta_{2}}}+2\sum_{\ell=1}^{n}\frac{(n-\ell)!\;(n+\ell)!}{2^{2\ell}(n!)^{2}}(\sin@@{\theta_{1}})^{\ell}\JacobipolyP{\ell}{\ell}{n-\ell}@{\cos@@{\theta_{1}}}(\sin@@{\theta_{2}})^{\ell}\JacobipolyP{\ell}{\ell}{n-\ell}@{\cos@@{\theta_{2}}}\cos@{\ell\phi}}.}

Symbols List

• ${\displaystyle{\displaystyle P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)}}$: Jacobi polynomial
• ${\displaystyle{\displaystyle P_{\NVar{n}}\left(\NVar{x}\right)}}$: Legendre polynomial
• ${\displaystyle{\displaystyle\cos\NVar{z}}}$: cosine function
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\sin\NVar{z}}}$: sine function
• ${\displaystyle{\displaystyle\ell}}$: nonnegative integer
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
 

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