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) §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) §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

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