Let $\alpha,\:\beta$ be positive constants and $I_0(x)$ the modified Bessel function. Any idea about how to calculate the following integral ?
$$
\begin{equation}
f(\alpha,\beta)=\int_0^1dx\:I_0(\alpha \sqrt{1-x})\:e^{\beta x}.
\end{equation}
$$
I've tried expanding the exponential but then the coefficients are given by some $_p F_q$ function whose summation is perhaps more difficult than the original problem.
Best Answer
We have \begin{align*} \int_0^1 {I_0 (\alpha \sqrt {1 - x} )e^{\beta x} dx} & = 2e^\beta \int_0^1 {I_0 (\alpha t)e^{ - \beta t^2 } tdt} \\ & = 2e^\beta \sum\limits_{n = 0}^\infty {\frac{1}{{n!^2 }}\left( {\frac{\alpha }{2}} \right)^{2n} \int_0^1 {e^{ - \beta t^2 } t^{2n + 1} dt} } \\ & = \frac{{e^\beta }}{\beta }\sum\limits_{n = 0}^\infty {\frac{{P(n + 1,\beta )}}{{n!}}\left( {\frac{\alpha }{{2\sqrt \beta }}} \right)^{2n} } , \end{align*} where $P$ is the normalised lower incomplete gamma function. It may be written in terms of the Marcum $Q$-function as $$ \frac{1}{\beta }\exp \left( {\frac{{\alpha ^2 }}{{4\beta }} + \beta } \right)\left( {1 - Q_1 \!\left( {\frac{\alpha }{{\sqrt {2\beta } }},\sqrt {2\beta } } \right)} \right). $$