Let $b>a>0$. Consider the sum $$F(\lambda):=\sum_{k=0}^{\infty}\frac{\Gamma{(a k)}}{\Gamma{(b k)}}{\lambda}^{-k}.$$
The asymptotic behavior of $F(\lambda)$ as $\lambda\to 0^{+}$ and as $\lambda \to+\infty$ are discussed here https://mathoverflow.net/questions/445799/the-asymptotic-behavior-of-f-lambda-sum-k-0-infty-frac-gammaa-k/445855#445855
Is it possible to compute the sum $F(\lambda)$ or approximate it with some remainder term (that obviously depends on whether $\lambda$ is small or large) ?