Formula:6699

Digital Library of Mathematical Functions ID 18.30.E5

${\displaystyle{\displaystyle\frac{(-1)^{n}{\left(\alpha+\beta+c+1\right)_{n}}n!\,P^{(\alpha,\beta)}_{n}\left(x;c\right)}{{\left(\alpha+\beta+2c+1\right)_{n}}{\left(\beta+c+1\right)_{n}}}=\sum_{\ell=0}^{n}\frac{{\left(-n\right)_{\ell}}{\left(n+\alpha+\beta+2c+1\right)_{\ell}}}{{\left(c+1\right)_{\ell}}{\left(\beta+c+1\right)_{\ell}}}\left(\tfrac{1}{2}x+\tfrac{1}{2}\right)^{\ell}\*{{}_{4}F_{3}}\left({\ell-n,n+\ell+\alpha+\beta+2c+1,\beta+c,c\atop\beta+\ell+c+1,\ell+c+1,\alpha+\beta+2c};1\right),}}$

Symbols List

• ${\displaystyle{\displaystyle{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_{q}};\NVar{z}\right)}}$: alternatively F q p ( a ; b ; z ) or F q p ( a b ; z ) generalized hypergeometric function
• ${\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}$:
• ${\displaystyle{\displaystyle P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)}}$: associated Jacobi polynomial
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\ell}}$: nonnegative integer
• ${\displaystyle{\displaystyle x}}$: real variable
• ${\displaystyle{\displaystyle n}}$: continuous nonnegative real
 

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