Bessel equation of order \(p\) is \[ x^2 y'' + x y' + (x^2 - p^2) y = 0 \tag{36} \] Assuming \(y = \sum_{m=0} a_m x^{r+m}\), find the two roots of the indicial equation (the two possible values of \(r\)). For both of them, solve the recurrence relation for the \(a_i\). You should find, for the largest root, \(J_p(x) = \sum_{k=0}^\infty \frac{(-1)^k}{k!\,\Gamma(k+p+1)} \left(\frac{x}{2}\right)^{2k+p}\), and for the smallest \(J_{-p}(x) = \sum_{k=0}^\infty \frac{(-1)^k}{k!\,\Gamma(k-p+1)} \left(\frac{x}{2}\right)^{2k-p}\) (for the normalization, assume that \(a_0 = \frac{1}{2^n n!}\)). These are Bessel function of the first kind.
The normalization \(a_0 = \frac{1}{2^n n!}\) in the source uses an undefined \(n\). The stated series have leading coefficient \(a_0 = \frac{1}{2^p\,\Gamma(p+1)}\) for \(J_p\) and \(a_0 = \frac{1}{2^{-p}\,\Gamma(1-p)}\) for \(J_{-p}\); the source’s formula is the case \(p = n \in \mathbb{N}_0\) of the first.