# Laguerre form

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##### 4: 3.5 Quadrature
For the choice $q_{n}(x)=\frac{1}{\sqrt{h_{n}}}L^{(\alpha)}_{n}\left(x\right)$ the recurrence relation (3.5.30_5) takes the form
##### 6: 18.15 Asymptotic Approximations
###### §18.15(iv) Laguerre
18.15.14 $L^{(\alpha)}_{n}\left(x\right)=\frac{n^{\frac{1}{2}\alpha-\frac{1}{4}}e^{\frac% {1}{2}x}}{\pi^{\frac{1}{2}}x^{\frac{1}{2}\alpha+\frac{1}{4}}}\left(\cos\theta_% {n}^{(\alpha)}(x)\left(\sum_{m=0}^{M-1}\frac{a_{m}(x)}{n^{\frac{1}{2}m}}+O% \left(\frac{1}{n^{\frac{1}{2}M}}\right)\right)+\sin\theta_{n}^{(\alpha)}(x)% \left(\sum_{m=1}^{M-1}\frac{b_{m}(x)}{n^{\frac{1}{2}m}}+O\left(\frac{1}{n^{% \frac{1}{2}M}}\right)\right)\right),$
The asymptotic behavior of the classical OP’s as $x\to\pm\infty$ with the degree and parameters fixed is evident from their explicit polynomial forms; see, for example, (18.2.7) and the last two columns of Table 18.3.1. For asymptotic approximations of Jacobi, ultraspherical, and Laguerre polynomials in terms of Hermite polynomials, see López and Temme (1999a). These approximations apply when the parameters are large, namely $\alpha$ and $\beta$ (subject to restrictions) in the case of Jacobi polynomials, $\lambda$ in the case of ultraspherical polynomials, and $|\alpha|+|x|$ in the case of Laguerre polynomials. …
##### 7: Bibliography T
• N. M. Temme (1985) Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. Quart. Appl. Math. 43 (1), pp. 103–123.
• N. M. Temme (1986) Laguerre polynomials: Asymptotics for large degree. Technical report Technical Report AM-R8610, CWI, Amsterdam, The Netherlands.
• N. M. Temme (1990a) Asymptotic estimates for Laguerre polynomials. Z. Angew. Math. Phys. 41 (1), pp. 114–126.
• F. G. Tricomi (1949) Sul comportamento asintotico dell’$n$-esimo polinomio di Laguerre nell’intorno dell’ascissa $4n$ . Comment. Math. Helv. 22, pp. 150–167.
• F. Tu and Y. Yang (2013) Algebraic transformations of hypergeometric functions and automorphic forms on Shimura curves. Trans. Amer. Math. Soc. 365 (12), pp. 6697–6729.
• ##### 8: 18.14 Inequalities
###### Laguerre
Let the maxima $x_{n,m}$, $m=0,1,\dots,n-1$, of $|L^{(\alpha)}_{n}\left(x\right)|$ in $[0,\infty)$ be arranged so that … The successive maxima of $|H_{n}\left(x\right)|$ form a decreasing sequence for $x\leq 0$, and an increasing sequence for $x\geq 0$.
##### 9: 9.17 Methods of Computation
However, in the case of $\mathrm{Ai}\left(z\right)$ and $\mathrm{Bi}\left(z\right)$ this accuracy can be increased considerably by use of the exponentially-improved forms of expansion supplied in §9.7(v). … For details, including the application of a generalized form of Gaussian quadrature, see Gordon (1969, Appendix A) and Schulten et al. (1979). … The second method is to apply generalized Gauss–Laguerre quadrature (§3.5(v)) to the integral (9.5.8). …
##### 10: Errata
• Subsection 33.14(iv)

Just below (33.14.9), the constraint described in the text “$\ell<(-\epsilon)^{-1/2}$ when $\epsilon<0$,” was removed. In Equation (33.14.13), the constraint $\epsilon_{1},\epsilon_{2}>0$ was added. In the line immediately below (33.14.13), it was clarified that $s\left(\epsilon,\ell;r\right)$ is $\exp\left(-r/n\right)$ times a polynomial in $r/n$, instead of simply a polynomial in $r$. In Equation (33.14.14), a second equality was added which relates $\phi_{n,\ell}(r)$ to Laguerre polynomials. A sentence was added immediately below (33.14.15) indicating that the functions $\phi_{n,\ell}$, $n=\ell,\ell+1,\ldots$, do not form a complete orthonormal system.