Recurrence relations for Bessel functions

Remarks

For $J_\nu$ both relations follow by comparing coefficients in the power series $J_\nu(z) = \sum_{m \ge 0} \frac{(-1)^m}{m!\, \Gamma(m + \nu + 1)} \left( \frac{z}{2} \right)^{2m + \nu}$. Equivalent forms are $\left( z^\nu C_\nu \right)' = z^\nu C_{\nu-1}$ and $\left( z^{-\nu} C_\nu \right)' = -z^{-\nu} C_{\nu+1}$. For $Y_\nu$ they follow from the definition through $J_{\pm\nu}$ and a limit, and for the Hankel functions by linearity. Example: with $\nu = 0$ the second relation and $J_{-1} = -J_1$ give $J_0' = -J_1$. Example: from $J_{1/2}(x) = \sqrt{\frac{2}{\pi x}} \sin x$ and $J_{-1/2}(x) = \sqrt{\frac{2}{\pi x}} \cos x$, the first relation with $\nu = \frac{1}{2}$ gives $J_{3/2}(x) = \sqrt{\frac{2}{\pi x}} \left( \frac{\sin x}{x} - \cos x \right)$.

Going deeper

For a student

The first relation is a three-term recurrence in the order, so tables and software compute $J_n(x)$ for many $n$ from a few starting values. Forward use of it is unstable for $J_n$ when $n > x$, and one runs it backward instead (Miller's algorithm). The relations also show that Bessel functions of half-integer order are elementary, which gives the spherical Bessel functions. The modified Bessel functions satisfy the same relations with some signs changed.