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I have a sum of the form

$\sum^n_{i=0} \frac{{t}^{i} z^{i}}{i!}K_{a+i}(z), \quad\quad z\in\mathbb{R},z>0$

where $K$ is the modified Bessel function of the second type and $a$ is an integer, $t\in\mathbb{R}$ and $t>0$.

I know that if the sum is infinite, this can be solved using the multiplication theorem, but any hints for solving the finite case?

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