18 Orthogonal Polynomials Classical Orthogonal Polynomials 18.7 Interrelations and Limit Relations 18.9 Recurrence Relations and Derivatives

§18.8 Differential Equations

ⓘ

Notes:: For Table 18.8.1, items 1, 2, 4, 8–10, 12–13, see Szegő (1975, (4.2.4), (4.24.2), (4.7.5), (5.1.2), (5.5.2)), respectively; item 3 is the special case $\alpha=\beta$ of item 2; items 5, 6, 7 are the special cases $\alpha=\beta=-\frac{1}{2}$ , $\alpha=\beta=\frac{1}{2}$ , $\alpha=\beta=0$ , respectively, of Row 2. Item 11 can be easily reduced to item 10.
Referenced by:: Erratum (V1.2.0) Section 18.8
Permalink:: http://dlmf.nist.gov/18.8
See also:: Annotations for Ch.18

See Table 18.8.1 and also §22.6 of Abramowitz and Stegun (1964).

Table 18.8.1: Classical OP’s: differential equations

A(x)f^{\prime\prime}(x)+B(x)f^{\prime}(x)+C(x)f(x)+\lambda_{n}f(x)=0

#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	$\lambda_{n}$
1	$P^{(\alpha,\beta)}_{n}\left(x\right)$	$1-x^{2}$	$\beta-\alpha-(\alpha+\beta+2)x$	0	$n(n+\alpha+\beta+1)$
2	$\begin{gathered}\textstyle\left(\sin\frac{1}{2}x\right)^{\alpha+\frac{1}{2}}% \left(\cos\frac{1}{2}x\right)^{\beta+\frac{1}{2}}\\ \times P^{(\alpha,\beta)}_{n}\left(\cos x\right)\end{gathered}$	$1$	$0$	$\dfrac{\tfrac{1}{4}-\alpha^{2}}{4{\sin}^{2}\frac{1}{2}x}+\dfrac{\frac{1}{4}-% \beta^{2}}{4{\cos}^{2}\frac{1}{2}x}$	$\left(n+\frac{1}{2}(\alpha+\beta+1)\right)^{2}$
3	$(\sin x)^{\alpha+\frac{1}{2}}P^{(\alpha,\alpha)}_{n}\left(\cos x\right)$	$1$	$0$	$(\frac{1}{4}-\alpha^{2})/{\sin}^{2}x$	$(n+\alpha+\frac{1}{2})^{2}$
4	$C^{(\lambda)}_{n}\left(x\right)$	$1-x^{2}$	$-(2\lambda+1)x$	$0$	$n(n+2\lambda)$
5	$T_{n}\left(x\right)$	$1-x^{2}$	$-x$	$0$	$n^{2}$
6	$U_{n}\left(x\right)$	$1-x^{2}$	$-3x$	$0$	$n(n+2)$
7	$P_{n}\left(x\right)$	$1-x^{2}$	$-2x$	$0$	$n(n+1)$
8	$L^{(\alpha)}_{n}\left(x\right)$	$x$	$\alpha+1-x$	$0$	$n$
9	${\mathrm{e}}^{-\frac{1}{2}x^{2}}x^{\alpha+\frac{1}{2}}L^{(\alpha)}_{n}\left(x^% {2}\right)$	$1$	$0$	$-x^{2}+(\frac{1}{4}-\alpha^{2})x^{-2}$	$4n+2\alpha+2$
10	${\mathrm{e}}^{-\frac{1}{2}x}x^{\frac{1}{2}\alpha}L^{(\alpha)}_{n}\left(x\right)$	$x$	$1$	$-\frac{1}{4}x-\frac{1}{4}\alpha^{2}x^{-1}$	$n+\frac{1}{2}(\alpha+1)$
11	${\mathrm{e}}^{-n^{-1}x}x^{\ell+1}L^{(2\ell+1)}_{n-\ell-1}\left(2n^{-1}x\right)$	$1$	$0$	$\frac{2}{x}-\frac{\ell(\ell+1)}{x^{2}}$	$-\frac{1}{n^{2}}$
12	$H_{n}\left(x\right)$	$1$	$-2x$	$0$	$2n$
13	${\mathrm{e}}^{-\frac{1}{2}x^{2}}H_{n}\left(x\right)$	$1$	$0$	$-x^{2}$	$2n+1$
14	$\mathit{He}_{n}\left(x\right)$	$1$	$-x$	$0$	$n$

ⓘ

Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\sin\NVar{z}$ : sine function, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Keywords:: Chebyshev polynomials, Hermite polynomials, Jacobi polynomials, Laguerre polynomials, Legendre polynomials, classical orthogonal polynomials, differential equation, differential equations, ultraspherical polynomials
A&S Ref:: 22.6.1, 22.6.4, 22.6.5, 22.6.8, 22.6.15, 22.6.18, 22.6.19, 22.6.20, 22.6.21
Referenced by:: §1.18(v), item 1., §18.36(vi), §18.39(ii), §18.39(ii), §18.39(ii), §18.39(ii), §18.8, §18.8, §18.8, Erratum (V1.2.0) Section 18.8
Permalink:: http://dlmf.nist.gov/18.8.T1
Addition (effective with 1.2.0):: Line numbers and rows 10 and 11 were added to this table.
See also:: Annotations for §18.8 and Ch.18

Item 11 of Table 18.8.1 yields (18.39.36) for $Z=1$ .

18.7 Interrelations and Limit Relations 18.9 Recurrence Relations and Derivatives

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov