Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.