Bessel function with shifted argument

Question

Is there any standard practice which may represents $J_m(a\pm kx)$ in terms of $J_m(kx)$ where $a$ is any constant and $m$ is integer $>-1$

Answer 1

it seems that it can work !

use the addition formula for bessel function :

where, $m>-1$

and the addition formula is :

1

2

the hint is to use the fourier expansion of $J_{m}(x)$ as follow :

$J_{m}(x)=\frac{1}{2\pi}$$\int_{-\pi}{\pi}$$e^{ixsin\xi}e^{-im\xi}d\xi$

where, $e^{ixsin\xi}=\sum{J_{m}{(x)}}z^{m}$