Given the series $$1 + \sum_{k = 1}^{\infty} \frac{\beta(\beta - 1) ... (\beta - k + 1)}{k!} x^k$$ how can I find a differential equation for which this series is a solution?
I don't have any idea how to think about this. Any hint would be appreciated.
HINT: Call the series $f(x)$ and check that $$(1+x) \cdot f'(x) = \beta \cdot f(x)$$