18 Orthogonal PolynomialsClassical Orthogonal Polynomials18.7 Interrelations and Limit Relations18.9 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).

#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	${\lambda }_n$
1	$P_n^{(\alpha ,\beta )}\left(x\right)$	$1-x^2$	$\beta -\alpha -(\alpha +\beta +2)x$	0	$n(n+\alpha +\beta +1)$
2	$\begin{array}{c}{\left(\sin \frac{1}{2}x\right)}^{\alpha +\frac{1}{2}}{\left(\cos \frac{1}{2}x\right)}^{\beta +\frac{1}{2}}\\ \times P_n^{(\alpha ,\beta )}\left(\cos x\right)\end{array}$	$1$	$0$	${\displaystyle \frac{\frac{1}{4}-{\alpha }^2}{4{\sin }^2\frac{1}{2}x}}+{\displaystyle \frac{\frac{1}{4}-{\beta }^2}{4{\cos }^2\frac{1}{2}x}}$	${\left(n+\frac{1}{2}(\alpha +\beta +1)\right)}^2$
3	${(\sin x)}^{\alpha +\frac{1}{2}}{P}_n^{(\alpha ,\alpha )}\left(\cos x\right)$	$1$	$0$	$(\frac{1}{4}-{\alpha }^2)/{\sin }^{2}x$	${(n+\alpha +\frac{1}{2})}^2$
4	$C_n^{(\lambda )}\left(x\right)$	$1-x^2$	$-(2\lambda +1)x$	$0$	$n(n+2\lambda )$
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	$L_n^{(\alpha )}\left(x\right)$	$x$	$\alpha +1-x$	$0$	$n$
9	${\mathrm{e}}^{-\frac{1}{2}{x}^2}{x}^{\alpha +\frac{1}{2}}{L}_n^{(\alpha )}\left(x^2\right)$	$1$	$0$	$-x^2+(\frac{1}{4}-{\alpha }^2)x^{-2}$	$4n+2\alpha +2$
10	${\mathrm{e}}^{-\frac{1}{2}x}{x}^{\frac{1}{2}\alpha }{L}_n^{(\alpha )}\left(x\right)$	$x$	$1$	$-\frac{1}{4}x-\frac{1}{4}{\alpha }^2{x}^{-1}$	$n+\frac{1}{2}(\alpha +1)$
11	${\mathrm{e}}^{-n^{-1}x}{x}^{\mathrm{\ell }+1}{L}_{n-\mathrm{\ell }-1}^{(2\mathrm{\ell }+1)}\left(2n^{-1}x\right)$	$1$	$0$	$\frac{2}{x}-\frac{\mathrm{\ell }(\mathrm{\ell }+1)}{x^2}$	$-\frac{1}{n^2}$
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	${\mathrm{𝐻𝑒}}_n\left(x\right)$	$1$	$-x$	$0$	$n$

ⓘ

Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $U_n\left(x\right)$: Chebyshev polynomial of the second kind, $H_n\left(x\right)$: Hermite polynomial, ${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $\mathrm{\ell }$: 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 Relations18.9 Recurrence Relations and Derivatives

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