The modified Bessel function of the first kind is defined as
$$I_q(\rho)=\sum_{m=0}^{\infty} \frac{\left(\frac{\rho}{2}\right)^{2m+q}}{m!\Gamma(m+q+1)}$$ where $\rho \in \mathbb{C}\setminus \{0\}$ and $q \in \mathbb{C}$. Suppose that $q,\rho \in \mathbb{R}$ and that $\rho>0$, $q \geq 0$. I need help to prove the behavior $$I_q(\rho )\approx \frac{e^{\rho}}{\sqrt{2 \pi \rho}}\,\,\,\text{as}\,\,\, \rho \to \infty$$
I tried using Stirling's approximation formula for the gamma function, but the general term inside the sum becomes so hard that I don't know to evaluate the series. I just need ideas on how to approach the calculation of this limit. Thank you.
Observation: In the above only the principal argument is considered and I'm trying to avoid (if possible) using integral representation of modified Bessel functions to prove the behavior