I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $p\ge0$, $q\ge0$ are real, and $n,a,b$ are integers. I want to compute the following integral:
$$\int_0^{2 \pi} e^{p \cos (a \tau) + q \cos (b \tau)} \cos (n \tau) \frac{d \tau}{2 \pi}$$
This is a generalization of a Bessel integral, in that for $q = 0$, $a=1$ I know that:
$$\int_0^{2 \pi} e^{p \cos (\tau)}\cos (n \tau) \frac{d \tau}{2 \pi} =I_n(p)$$
where $I_n(p)$ is the modified Bessel function of the first kind.
