= Frobenius resonance in the spherical Bessel equation
{c}
{title2=$\sigma=\ell,-\ell-1$}
For $x^2y''+2xy'+[x^2-\ell(\ell+1)]y=0$ with integer $\ell\geq0$, the <Frobenius method> gives $\sigma=\ell$ or $-\ell-1$. The general coefficient equation is $[(j+\sigma)(j+\sigma+1)-\ell(\ell+1)]a_j=-a_{j-2}$, taking negative-index coefficients as zero. The regular branch has $a_1=0$ and $a_j=-a_{j-2}/[j(j+2\ell+1)]$. The singular branch has $a_j=-a_{j-2}/[j(j-2\ell-1)]$ except at $j=2\ell+1$. That resonant equation is compatible because its earlier odd coefficient vanishes; the newly free coefficient adds a multiple of the regular solution. Setting it to zero gives a pure even series times $x^{-\ell-1}$, independent of the regular branch, with no logarithmic term. For $\ell=0$ the resonance occurs already at $j=1$.
The normalization $a_0=2^\ell\ell!/(2\ell+1)!$ on the regular branch gives the <Spherical Bessel function>
$$
j_\ell(x)=2^\ell x^\ell\sum_{s=0}^\infty\frac{(-1)^s(s+\ell)!}{s!(2s+2\ell+1)!}x^{2s}.
$$
In particular $j_0(x)=\sin x/x$, with removable value one at zero.
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