= Frobenius series for Bessel functions
{c}
{title2=$a_{2m}=-a_{2m-2}/[4m(m+\nu)]$}
For the <Bessel differential equation> $x^2y''+xy'+(x^2-\nu^2)y=0$, substitute $y=x^s\sum_{n\geq0}a_nx^n$. The <indicial equation> is $s^2-\nu^2=0$, and coefficients obey $[(n+s)^2-\nu^2]a_n=-a_{n-2}$. For $s=\nu\geq0$, $a_1=0$ and all odd coefficients vanish. Iteration gives $a_{2m}=(-1)^ma_0\Gamma(\nu+1)/[4^mm!\Gamma(m+\nu+1)]$. Choosing $a_0=1/[2^\nu\Gamma(\nu+1)]$ gives the <Bessel function of the first kind> $J_\nu$.
For noninteger $\nu$, choosing the even-power series with $s=-\nu$ replaces $m+\nu$ by $m-\nu$ and yields $J_{-\nu}$ with its standard normalization. These solutions are independent. For integer $\nu\geq0$, the first solution has coefficients $(-1)^m/[2^{2m+\nu}m!(m+\nu)!]$ and defines an entire function: the term ratio tends to zero at every finite complex argument. For noninteger order, the even analytic series converges everywhere, while the prefactor $x^{\pm\nu}$ requires a branch off the positive real axis.
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