Formula:6212

Digital Library of Mathematical Functions ID 18.5.E7

${\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(x\right)=\sum_{\ell=0}^{n}\frac{{\left(n+\alpha+\beta+1\right)_{\ell}}{\left(\alpha+\ell+1\right)_{n-\ell}}}{\ell!\;(n-\ell)!}\left(\frac{x-1}{2}\right)^{\ell}=\frac{{\left(\alpha+1\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-n,n+\alpha+\beta+1\atop\alpha+1};\frac{1-x}{2}\right),}}$

Symbols List

• ${\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}$: = F ( a , b ; c ; z ) notation for Gauss’ hypergeometric function
• ${\displaystyle{\displaystyle P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)}}$: Jacobi polynomial
• ${\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}$:
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\ell}}$: nonnegative integer
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle x}}$: real variable
 

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