## 7.41–7.42 Laguerre polynomials

7.411

1.

${\int }_{0}^{t}{L}_{n}\left(x\right)\text{d}x={L}_{n}\left(t\right)-{L}_{n+1}\left(t\right)/\left(n+1\right)$ MO 110

2.

$\underset{0}{\overset{t}{\int }}{L}_{n}^{\alpha }\left(x\right)\text{d}x={L}_{n}^{\alpha }\left(t\right)-{L}_{n+1}^{\alpha }\left(t\right)\left(\begin{array}{c}n+\alpha \\ n\end{array}\right)+\left(\begin{array}{c}n+1+\alpha \\ n+1\end{array}\right)$ EH II 189(16)a

3.

$\underset{0}{\overset{t}{\int }}{L}_{n-1}^{\alpha +1}\left(x\right)\text{d}x=-{L}_{n}^{\alpha }\left(t\right)+\left(\begin{array}{c}n+\alpha \\ n\end{array}\right)$ EH II 189(15)a

4.

${\int }_{0}^{t}{L}_{m}\left(x\right){L}_{n}\text{(}t-x\text{)d}x={L}_{m+n}\left(t\right)-{L}_{m+n+1}\left(t\right)$ EH II 191(31)

5.

$\sum _{k=0}^{\infty }\begin{array}{cc}{\left[\underset{0}{\overset{t}{\int }}\frac{{L}_{k}\left(x\right)}{k!}dx\right]}^{2}={e}^{t}-1& \left[t\ge 0\right]\end{array}$

MO 110

7.412

1.

${\int }_{0}^{1}{\left(1-x\right)}^{\mu -1}{x}^{\alpha }{L}_{n}^{\alpha }\left(ax\right)\text{d}x=\frac{\mathrm{\Gamma }\left(\alpha +n+1\right)\mathrm{\Gamma }\left(\mu \right)}{\mathrm{\Gamma }\left(\alpha +\mu +n+1\right)}{L}_{n}^{\alpha +\mu }\left(a\right)$

EH II 191(30)a, ...

Get Table of Integrals, Series, and Products, 8th Edition now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.