Formula:6471

Digital Library of Mathematical Functions ID 18.17.E33

${\displaystyle{\displaystyle\int_{-1}^{1}e^{-(x+1)z}P^{(\alpha,\beta)}_{n}\left(x\right)(1-x)^{\alpha}(1+x)^{\beta}\mathrm{d}x=\frac{(-1)^{n}2^{\alpha+\beta+n+1}\Gamma\left(\alpha+n+1\right)\Gamma\left(\beta+n+1\right)}{\Gamma\left(\alpha+\beta+2n+2\right)n!}z^{n}{{}_{1}F_{1}}\left({\beta+n+1\atop\alpha+\beta+2n+2};-2z\right),}}$

Constraint(s)

${\displaystyle{\displaystyle z\in\mathbb{C}}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)}}$: Jacobi polynomial
• ${\displaystyle{\displaystyle{{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)}}$: = M ( a , b , z ) notation for the Kummer confluent hypergeometric function
• ${\displaystyle{\displaystyle\mathbb{C}}}$:
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\in}}$:
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle x}}$: real variable
 

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