Neumann's addition theorem

Remarks

The proof uses the generating function $G(z, t) = \exp\left( \frac{z}{2}\left( t - \frac{1}{t} \right) \right) = \sum_{k \in \Z} J_k(z)\, t^k$. Since $G(x + y, t) = G(x, t)\, G(y, t)$, multiplying the two Laurent series in $t$ and comparing the coefficients of $t^n$ gives the formula. Example: with $n = 0$ and $y = -x$, one gets $1 = J_0(0) = \sum_{k} J_k(x) J_{-k}(-x) = \sum_k J_k(x)^2$, using $J_{-k}(-x) = J_k(x)$; so $J_0(x)^2 + 2\sum_{k \ge 1} J_k(x)^2 = 1$. This identity shows $\abs{J_0(x)} \le 1$ and $\abs{J_k(x)} \le \frac{1}{\sqrt{2}}$ for $k \ge 1$ and real $x$. The formula as written is for integer orders; for non-integer orders there are related addition theorems of Graf and Gegenbauer.

Going deeper

For a student

The theorem is a convolution identity: the sequence $(J_k(x + y))_k$ is the convolution of $(J_k(x))_k$ and $(J_k(y))_k$. In terms of Fourier series, it says that $e^{i(x+y)\sin\theta} = e^{ix\sin\theta} e^{iy\sin\theta}$, read coefficient by coefficient. Its consequence $\sum_k J_k(x)^2 = 1$ gives uniform bounds on Bessel functions of integer order on the real line.