Approximation of a series containing Bessel functions

Question

I have this series: $$\displaystyle S=\sum_{k=0}^N\left(J_k(x)-J_k(y)\right)$$ where: $J_k(\dot{})$ is the Bessel function of order $k$ with $x\in\mathbb{R}$ and $y\in\mathbb{R}$. I have to calculate this series for $N\approx 4000$. Can I use some semplification in order to express the Bessel function in a way useful for speed numerical evaluation? Thanks in advance.

Answer 1

Since for every $k\geq 1$ we have: $$\frac{d}{dx} J_k(x) = \frac{J_{k-1}(x)-J_{k+1}(x)}{2} $$ it follows that: $$\begin{eqnarray*} S &=& J_0(x)-J_0(y)+\sum_{k=1}^{n}\int_{y}^{x}\frac{J_{k-1}(t)-J_{k+1}(t)}{2}\,dt\\&=&J_0(x)-J_0(y)+\frac{1}{2}\int_{y}^{x}\left(J_0(t)+J_1(t)-J_n(t)-J_{n+1}(t)\right)\,dt\end{eqnarray*} $$ and the latter is easier to evaluate since it involves only four different Bessel functions instead of $n$.