Table of Integrals, Series, and Products, 8th Edition

7.41–7.42 Laguerre polynomials

7.411

1. 

${\displaystyle {\int }_0^t{L}_n(x)\text{d}x=}{L}_n(t)-L_{n+1}(t)/(n+1)$

  MO 110

2. 

$\underset{0}{\overset{t}{{\displaystyle \int }}}{L}_n^{\alpha }(x)\text{d}x=L_n^{\alpha }(t)-L_{n+1}^{\alpha }(t)\left(\begin{array}{c}n+\alpha \\ n\end{array}\right)+\left(\begin{array}{c}n+1+\alpha \\ n+1\end{array}\right)$

  EH II 189(16)a

3. 

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

  EH II 189(15)a

4. 

${\displaystyle {\int }_0^t{L}_m(x)L_n\text{(}t-x\text{)d}x=}{L}_{m+n}(t)-L_{m+n+1}(t)$

  EH II 191(31)

5. 

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

  MO 110

7.412

1. 

${\displaystyle {\int }_0^1{(1-x)}^{\mu -1}{x}^{\alpha }{L}_n^{\alpha }(ax)\text{d}x}=\frac{\mathrm{\Gamma }(\alpha +n+1)\mathrm{\Gamma }(\mu )}{\mathrm{\Gamma }(\alpha +\mu +n+1)}{L}_n^{\alpha +\mu }(a)$

  EH II 191(30)a, ...