How to simplify the multiplication of Bessel and modified Bessel function of the first kind

Question

I was wondering if anyone can help me with the procedure of simplifying the following formula: $$ e^{ - \beta x} \mathcal{T}_b \sum\limits_m {i^m J_m (kr)e^{im(\theta - \alpha )} } $$ using the following identity: $$ e^{ - \beta x} = e^{ - \beta b} \sum\limits_s {( - 1)^s I_s (\beta r)e^{is\theta } } $$ to the following formula in which the index of the modified Bessel function is modified to $m-s$: $$ e^{ - \beta b} \sum\limits_m {( - 1)^m U_m (r)e^{im\theta } } $$ where $U_m$ is equal to $$ \mathcal{T}_b \sum\limits_s {( - i)^s I_{m - s} (\beta r)J_s (kr)e^{ - is\alpha } } . $$ In these formulas $\theta$, $\alpha$, and $x$ are arbitrary real variables and $m$ and $s$ are integers varying from $-\infty$ to $+\infty$ and other variables are constant. Also $x = b + r \cos(\theta)$ and $y = r \sin(\theta)$.