I have the following function:
$$ G(z) = \sum_{j = 0}^{\infty} \frac{\Gamma(1 + n) z^j}{j ! (\Lambda + jC)^{n+1}} $$
If $n \quad \epsilon \quad \mathbb{Z}^{+} $, the above function can be manipulated as:
$$ G(z) = \frac{\Gamma(1 + n)}{\Lambda^{n+1}} \sum_{j = 0}^{\infty} \left(\frac{\frac{\Lambda}{C}}{\frac{\Lambda}{C} + j}\right)^{n+1} \frac{z^j}{j!} $$
Which can be written as:
$$ G(z) = \frac{\Gamma(1 + n)}{\Lambda^{n+1}} \sum_{j = 0}^{\infty} \left(\frac{\frac{\Lambda}{C}(\frac{\Lambda}{C} + 1)(\frac{\Lambda}{C} + 2) \cdots (\frac{\Lambda}{C} + j - 1)}{(\frac{\Lambda}{C} + 1)(\frac{\Lambda}{C} + 2) \cdots (\frac{\Lambda}{C} + j - 1)(\frac{\Lambda}{C} + j)}\right)^{n+1} \frac{z^j}{j!} $$
This resembles the generalized hypergeometric function:
$$ G(z) = \frac{\Gamma(1 + n)}{\Lambda^{n+1}} {}_p F_p \left(\frac{\Lambda}{C}, \frac{\Lambda}{C}, \cdots \frac{\Lambda}{C}; \frac{\Lambda}{C}+1, \frac{\Lambda}{C}+1, \cdots, \frac{\Lambda}{C}+1; z\right) $$, where $p = n + 1$.
In all these expressions $n, \Lambda, C > 0$
I am wondering if there is a nice closed form of the first expression when $n$ is not an integer. As far as I understand, the generalized hyper-geometric functions are defined in this case when $n$ is an integer.
