Taylor coefficients of certain "deformed" Bessel functions

Question

Related links / discussions.

• (1) How to compute $\frac{1}{2\pi}\int_{-\pi}^\pi (\cos y + x)^{2k}dy$ and similar integrals
• (2) Answer to (1) https://math.stackexchange.com/a/4244473/168758

Let $I_0$ and $I_1$ be the Bessel functions of the first and second kind respectively, and for fixed $x \in \mathbb R$, define functions $(a,b) \mapsto F_0(a,b;x)$ and $(a,b) \mapsto F_1(a,b;x)$ by \begin{eqnarray*} \begin{split} F_0(a,b;x) &:= e^{(a+b)x}I_0(\sqrt{a^2+b^2}),\\ F_1(a,b;x) &:= e^{(a+b)x}\frac{a}{\sqrt{a^2+b^2}}I_1(\sqrt{a^2+b^2}). \end{split} \end{eqnarray*}

Question. In the Taylor expansion of $F_0$ and $F_1$ respectively, what is the coeficient of $a^k b^l$ ?