I have the following infinite sum
$$ H(z) = \sum_{i = 0}^{\infty} \left(\frac{C}{C+i}\right)^n \ln\left(\frac{C}{C+i}\right) \frac{z^i}{i!}$$
Without the logarithm term, I am able to write it as a generalized hypergeometric function, but with the logarithm, I have no idea. Can it be written in terms of special functions?
Here $C > 0, n > 1$, $z \quad \epsilon \quad \mathbb{C} $
Without the logarithm:
$$ G(z) = \sum_{i = 0}^{\infty} \left(\frac{C}{C+i}\right)^n \frac{z^i}{i!} = _n F_n(C, C, ..., C; C+1, C+1, ..., C+1; z) $$