Formula:6485

Digital Library of Mathematical Functions ID 18.17.E47

${\displaystyle{\displaystyle\int_{0}^{x}t^{\alpha}\frac{L^{(\alpha)}_{m}\left(t\right)}{L^{(\alpha)}_{m}\left(0\right)}(x-t)^{\beta}\frac{L^{(\beta)}_{n}\left(x-t\right)}{L^{(\beta)}_{n}\left(0\right)}\mathrm{d}t=\frac{\Gamma\left(\alpha+1\right)\Gamma\left(\beta+1\right)}{\Gamma\left(\alpha+\beta+2\right)}x^{\alpha+\beta+1}\frac{L^{(\alpha+\beta+1)}_{m+n}\left(x\right)}{L^{(\alpha+\beta+1)}_{m+n}\left(0\right)}.}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)}}$: Laguerre (or generalized Laguerre) polynomial
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle m}}$: nonnegative integer
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle x}}$: real variable
 

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