The integration of Bessel $J_{n}(x)$ and exponential function $e^{ix}$ is given as
$$\int_0^u J_n(x) e^{i x}dx =-n + e^{-i u} \frac{P_{n-2}(u)}{u^{n-3}} J_0(u) + e^{-i u} \frac{Q_{n-1}(u)}{u^{n-2}} J_1(u),$$
where $P_n(u)$ and $Q_n(u)$ are certain $n$-order polynomials of $u$.
It is mentioned that the polynomials $P_n(u)$ and $Q_n(u)$ can be obtained through deriving the recurrence relation between the polynomials using the recurrence relations for the Bessel functions.
I'm afraid I don't know how to do this recurrence relation between the polynomials. Could someone please help me. Thanks.