 Frobenius series for Bessel functions

For the Bessel differential equation $x^2{y}^{\prime \prime }+x{y}^{\prime }+(x^2-{\nu }^2)y=0$, substitute $y=x^s{\sum }_{n\ge 0}{a}_n{x}^n$. The indicial equation is $s^2-{\nu }^2=0$, and coefficients obey $[(n+s{)}^2-{\nu }^2]a_n=-a_{n-2}$. For $s=\nu \ge 0$, $a_1=0$ and all odd coefficients vanish. Iteration gives $a_{2m}=(-1{)}^m{a}_0\Gamma (\nu +1)/[4^{m}m!\Gamma (m+\nu +1)]$. 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 $\nu \ge 0$, 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.