# Laguerre polynomials

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##### 1: 18.3 Definitions
###### §18.3 Definitions
The classical OP’s comprise the Jacobi, Laguerre and Hermite polynomials. … For explicit power series coefficients up to $n=12$ for these polynomials and for coefficients up to $n=6$ for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). …
##### 3: 18.36 Miscellaneous Polynomials
###### §18.36(v) Non-Classical LaguerrePolynomials$L^{(-k)}_{n}\left(x\right)$, $k=1,2\dots$
For the Laguerre polynomials $L^{(\alpha)}_{n}\left(x\right)$ this requires, omitting all strictly positive factors, … The resulting EOP’s, $\hat{L}^{(k)}_{n}\left(x\right)$, $n=1,2,\dots$ satisfy … The $y(x)=\hat{L}^{(k)}_{n}\left(x\right)$ satisfy a second order Sturm–Liouville eigenvalue problem of the type illustrated in Table 18.8.1, as satisfied by classical OP’s, but now with rational, rather than polynomial coefficients: …
##### 5: 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). …
##### 6: 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 $|L^{(\alpha)}_{n}\left(x_{n,0}\right)|<|L^{(\alpha)}_{n}\left(x_{n,1}\right)|<% \cdots<|L^{(\alpha)}_{n}\left(x_{n,n-1}\right)|.$
##### 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
###### Laguerre
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 $\frac{L^{(\alpha)}_{n}\left(\lambda x\right)}{L^{(\alpha)}_{n}\left(0\right)}=% \sum_{\ell=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{\ell}\lambda^{\ell}(1-\lambda)^{n-% \ell}\frac{L^{(\alpha)}_{\ell}\left(x\right)}{L^{(\alpha)}_{\ell}\left(0\right% )}.$
##### 10: 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: $L^{(\alpha)}_{n}\left(x;q\right)$.