§18.8 Differential Equations

Notes:: For Table 18.8.1, Rows 2, 3, 5, 9–10, 11–12, see Szegö (1975, (4.2.4), (4.24.2), (4.7.5), (5.1.2), (5.5.2)), respectively; Row 4 is the special case $\alpha=\beta$ of Row 2; Rows 6, 7, 8 are the special cases $\alpha=\beta=-\frac{1}{2}$ , $\alpha=\beta=\frac{1}{2}$ , $\alpha=\beta=0$ , respectively, of Row 2.
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See Table 18.8.1 and also Table 22.6 of Abramowitz and Stegun (1964).

Table 18.8.1: Classical OP’s: differential equations $A(x)f^{{\prime\prime}}(x)+B(x)f^{{\prime}}(x)+C(x)f(x)+\lambda _{n}f(x)=0$ .

$f(x)$	$A(x)$	$B(x)$	$C(x)$	$\lambda _{n}$
$\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$	$1-x^{2}$	$\beta-\alpha-(\alpha+\beta+2)x$	0	$n(n+\alpha+\beta+1)$
$\begin{gathered}\textstyle\left(\mathop{\sin\/}\nolimits\frac{1}{2}x\right)^{{\alpha+\frac{1}{2}}}\left(\mathop{\cos\/}\nolimits\frac{1}{2}x\right)^{{\beta+\frac{1}{2}}}\\\times\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits x\right)\end{gathered}$	1	0	$\dfrac{\tfrac{1}{4}-\alpha^{2}}{4{\mathop{\sin\/}\nolimits^{{2}}}\frac{1}{2}x}+\dfrac{\frac{1}{4}-\beta^{2}}{4{\mathop{\cos\/}\nolimits^{{2}}}\frac{1}{2}x}$	$\left(n+\frac{1}{2}(\alpha+\beta+1)\right)^{2}$
$(\mathop{\sin\/}\nolimits x)^{{\alpha+\frac{1}{2}}}\mathop{P^{{(\alpha,\alpha)}}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits x\right)$	1	0	$(\frac{1}{4}-\alpha^{2})/{\mathop{\sin\/}\nolimits^{{2}}}x$	$(n+\alpha+\frac{1}{2})^{2}$
$\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$	$1-x^{2}$	$-(2\lambda+1)x$	0	$n(n+2\lambda)$
$\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$	$1-x^{2}$	$-x$	0	$n^{2}$
$\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$	$1-x^{2}$	$-3x$	0	$n(n+2)$
$\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$	$1-x^{2}$	$-2x$	0	$n(n+1)$
$\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$	$x$	$\alpha+1-x$	0	$n$
$e^{{-\frac{1}{2}x^{2}}}x^{{\alpha+\frac{1}{2}}}\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x^{2}\right)$	1	0	$-x^{2}+(\frac{1}{4}-\alpha^{2})x^{{-2}}$	$4n+2\alpha+2$
$\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$	1	$-2x$	0	$2n$
$e^{{-\frac{1}{2}x^{2}}}\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$	1	0	$-x^{2}$	$2n+1$
$\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$	1	$-x$	0	$n$