# Laguerre polynomials

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##### 1: 18.3 Definitions
###### §18.3 Definitions
Table 18.3.1 provides the definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and normalization (§§18.2(i) and 18.2(iii)). … For exact values of the coefficients of the Jacobi polynomials $P^{(\alpha,\beta)}_{n}\left(x\right)$, the ultraspherical polynomials $C^{(\lambda)}_{n}\left(x\right)$, the Chebyshev polynomials $T_{n}\left(x\right)$ and $U_{n}\left(x\right)$, the Legendre polynomials $P_{n}\left(x\right)$, the Laguerre polynomials $L_{n}\left(x\right)$, and the Hermite polynomials $H_{n}\left(x\right)$, see Abramowitz and Stegun (1964, pp. 793–801). …
##### 2: 18.4 Graphics Figure 18.4.5: Laguerre polynomials L n ⁡ ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify Figure 18.4.6: Laguerre polynomials L 3 ( α ) ⁡ ( x ) , α = 0 , 1 , 2 , 3 , 4 . Magnify Figure 18.4.8: Laguerre polynomials L 3 ( α ) ⁡ ( x ) , 0 ≤ α ≤ 3 , 0 ≤ x ≤ 10 . Magnify 3D Help Figure 18.4.9: Laguerre polynomials L 4 ( α ) ⁡ ( x ) , 0 ≤ α ≤ 3 , 0 ≤ x ≤ 10 . Magnify 3D Help
##### 3: 18.14 Inequalities
###### Laguerre
18.14.12 $(L^{(\alpha)}_{n}\left(x\right))^{2}\geq L^{(\alpha)}_{n-1}\left(x\right)L^{(% \alpha)}_{n+1}\left(x\right),$ $0\leq x<\infty$, $\alpha\geq 0$.
###### Laguerre
18.14.24 $|L^{(\alpha)}_{n}\left(x_{n,0}\right)|<|L^{(\alpha)}_{n}\left(x_{n,1}\right)|<% \cdots<|L^{(\alpha)}_{n}\left(x_{n,n-1}\right)|.$
##### 4: 18.41 Tables
For $P_{n}\left(x\right)$ ($=\mathsf{P}_{n}\left(x\right)$) see §14.33. Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates $T_{n}\left(x\right)$, $U_{n}\left(x\right)$, $L_{n}\left(x\right)$, and $H_{n}\left(x\right)$ for $n=0(1)12$. The ranges of $x$ are $0.2(.2)1$ for $T_{n}\left(x\right)$ and $U_{n}\left(x\right)$, and $0.5,1,3,5,10$ for $L_{n}\left(x\right)$ and $H_{n}\left(x\right)$. … For $P_{n}\left(x\right)$, $L_{n}\left(x\right)$, and $H_{n}\left(x\right)$ see §3.5(v). …
##### 5: 18.9 Recurrence Relations and Derivatives
18.9.13 $L^{(\alpha)}_{n}\left(x\right)=L^{(\alpha+1)}_{n}\left(x\right)-L^{(\alpha+1)}% _{n-1}\left(x\right),$
18.9.14 $xL^{(\alpha+1)}_{n}\left(x\right)=-(n+1)L^{(\alpha)}_{n+1}\left(x\right)+(n+% \alpha+1)L^{(\alpha)}_{n}\left(x\right).$
###### Laguerre
18.9.23 $\frac{\mathrm{d}}{\mathrm{d}x}L^{(\alpha)}_{n}\left(x\right)=-L^{(\alpha+1)}_{% n-1}\left(x\right),$
18.9.24 $\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{-x}x^{\alpha}L^{(\alpha)}_{n}\left(x% \right)\right)=(n+1)e^{-x}x^{\alpha-1}L^{(\alpha-1)}_{n+1}\left(x\right).$
##### 7: 18.6 Symmetry, Special Values, and Limits to Monomials
###### §18.6(ii) Limits to Monomials
18.6.5 $\lim_{\alpha\to\infty}\frac{L^{(\alpha)}_{n}\left(\alpha x\right)}{L^{(\alpha)% }_{n}\left(0\right)}=(1-x)^{n}.$
##### 8: 18.18 Sums
18.18.10 $L^{(\alpha_{1}+\dots+\alpha_{r}+r-1)}_{n}\left(x_{1}+\dots+x_{r}\right)=\sum_{% m_{1}+\dots+m_{r}=n}L^{(\alpha_{1})}_{m_{1}}\left(x_{1}\right)\cdots L^{(% \alpha_{r})}_{m_{r}}\left(x_{r}\right).$
18.18.12 $\frac{L^{(\alpha)}_{n}\left(\lambda x\right)}{L^{(\alpha)}_{n}\left(0\right)}=% \sum_{\ell=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{\ell}\lambda^{\ell}(1-\lambda)^{n-% \ell}\frac{L^{(\alpha)}_{\ell}\left(x\right)}{L^{(\alpha)}_{\ell}\left(0\right% )}.$
##### 9: 18.1 Notation
###### Classical OP’s
• Laguerre: $L^{(\alpha)}_{n}\left(x\right)$ and $L_{n}\left(x\right)=L^{(0)}_{n}\left(x\right)$. ($L^{(\alpha)}_{n}\left(x\right)$ with $\alpha\neq 0$ is also called Generalized Laguerre.)

• $q$-Laguerre: $L^{(\alpha)}_{n}\left(x;q\right)$.