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I'm looking for an analytical solution or approximation to the following infinite series.

$$\sum_{i=0}^\infty t^i I_i(z)\;,\;t\neq0,$$

with $I_i(z)$ as the modified Bessel function of the first kind.

There exists an analytical solution for the following infinite series:

$$\sum_{i=-\infty}^\infty t^i I_i(z)=\mathbb{e}^{\frac{1}{2}z(t+\frac{1}{t})}.$$

which yields the following relation

$$\sum_{i=0}^\infty t^i I_i(z)=\mathbb{e}^{\frac{1}{2}z(t+\frac{1}{t})}-\sum_{i=1}^{\infty} t^{-i} I_i(z)$$

using the even parity of the modified Bessel function.

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