Consider the following integral $$\int_0^\pi e^{a\cos\theta} \sin^n\theta\ \mathrm{d}\theta, $$ where $a$ is a real number and $n$ is an integer.
Is it possible to relate the integral given above to any special function?
My guess is that it should be some kind of Bessel function. But I am not able to solve it.
As uniquesolution commented $$J_n=\int_0^\pi e^{a\cos(\theta)} \sin^n(\theta) \,d\theta=\sqrt{\pi } \Gamma \left(\frac{n+1}{2}\right) \, _0\tilde{F}_1\left(;\frac{n}{2}+1;\frac{a^2}{4}\right)$$ where appears the regularized confluent hypergeometric function.
However, for specific values of $n$, the expressions look "nicer" $$\left( \begin{array}{cc} n & J_n \\ 0 & \pi I_0(a) \\ 1 & \frac{2 \sinh (a)}{a} \\ 2 & \frac{\pi }{a}I_1(a)\\ 3 & \frac{4 a \cosh (a)-4 \sinh (a)}{a^3} \\ 4 & \frac{3 \pi }{a^2}I_2(a) \\ 5 & \frac{16 \left(\left(a^2+3\right) \sinh (a)-3 a \cosh (a)\right)}{a^5} \\ 6 & \frac{15 \pi }{a^3} I_3(a) \\ 7 & \frac{96 \left(a \left(a^2+15\right) \cosh (a)-3 \left(2 a^2+5\right) \sinh (a)\right)}{a^7} \\ 8 & \frac{105 \pi }{a^4}I_4(a) \\ 9 & \frac{768 \left(\left(a^4+45 a^2+105\right) \sinh (a)-5 a \left(2 a^2+21\right) \cosh (a)\right)}{a^9} \\ 10 & \frac{945 \pi }{a^5}I_5(a) \end{array} \right)$$ and, if $n$ is even,
$$J _{2n}=\sqrt{\pi }\, \left(\frac 2a\right)^n \Gamma \left(n+\frac{1}{2}\right) I_n(a)=\pi \frac{(2n-1)!!}{a^n}I_n(a)$$ where you see the Bessel functions you suspected.
For odd values of $n$, I did not find anything simple.