§30.3 Eigenvalues

With $\mu =m=0,1,2,\mathrm{\dots }$, the spheroidal wave functions ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ are solutions of Equation (30.2.1) which are bounded on $(-1,1)$, or equivalently, which are of the form ${(1-x^2)}^{\frac{1}{2}m}g(x)$ where $g(z)$ is an entire function of $z$. These solutions exist only for eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$, $n=m,m+1,m+2,\mathrm{\dots }$, of the parameter $\lambda$.

The eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$ are analytic functions of the real variable ${\gamma }^2$ and satisfy

If $p$ is an even nonnegative integer, then the continued-fraction equation

where ${\alpha }_k$, ${\beta }_k$, ${\gamma }_k$ are defined by

has the solutions $\lambda ={\lambda }_{m+2j}^m\left({\gamma }^2\right)$, $j=0,1,2,\mathrm{\dots }$. If $p$ is an odd positive integer, then Equation (30.3.5) has the solutions $\lambda ={\lambda }_{m+2j+1}^m\left({\gamma }^2\right)$, $j=0,1,2,\mathrm{\dots }$. If $p=0$ or $p=1$, the finite continued-fraction on the left-hand side of (30.3.5) equals 0; if $p>1$ its last denominator is ${\beta }_0-\lambda$ or ${\beta }_1-\lambda$.

In equation (30.3.5) we can also use

For values of $r_n^m$ see Meixner et al. (1980, p. 109).

Further coefficients can be found with the Maple program SWF9; see §30.18(i).