Special functions related to $\int_0^{2\pi} e^{(\cos{t})^2 + k \cos{t}} dt$

Question

I'm interested in whether there are any closed-form representations of $$ \int_0^{2\pi} e^{(\cos{t})^2 + k \cos{t}} dt \quad \text{ or } \quad \int_0^{2\pi} e^{(q+\cos{t})^2} dt $$ in terms of other special functions, where $k$ and $q$ are real constants. Without the quadratic term I know the first integral is a (scaled) modified Bessel function, but it seems like the quadratic case is much less common, and indeed, Mathematica yields no leads on possible functions involved. Suggestions on possible avenues of approach are appreciated!

Answer 1

Expanding the comments above: $$ e^{\cos^2\theta} = \sqrt{e}\,I_0\left(\tfrac{1}{2}\right)+2\sqrt{e}\sum_{n\geq 1}I_n\left(\tfrac{1}{2}\right)\cos(2n\theta)\tag{1} $$ $$ e^{k\cos\theta} = I_0(k)+2\sum_{n\geq 1}I_n(k)\cos(n\theta)\tag{2} $$ hence by the orthogonality relations in $L^2(0,2\pi)$ $$ \int_{0}^{2\pi}\exp\left(\cos^2\theta+k\cos\theta\right)\,d\theta = 2\pi\sqrt{e}\,I_0\left(\tfrac{1}{2}\right)I_0(k)+4\pi\sqrt{e}\sum_{n\geq 1}I_n\left(\tfrac{1}{2}\right)I_{2n}(k).\tag{3}$$