I'v come across the need to evaluate the sum: $$S(m,M)=\sum_{t=0}^{m}\frac{\Gamma(m+t+\frac{1}{2})(-M)^{-t}}{m!(m-t)!}$$ For non-negative integer $m$ and real and positive $M$. This sum can be given by the confluent hypergeometric function: $$S(m,M)=\frac{\pi(-M)^{-m}}{m!}\frac{_1F_1(-m, \frac{1}{2}-2m;-M)}{\Gamma(\frac{1}{2}-2m)}$$
Holding $M$ fixed and inspecting $S$ as $m$ is increased from $0$ to $M$ and beyond, $S(m,M)$ is positive and monotonically decaying, until some critical $m_c < M$ after which two phenomenon occur:
- $S(m,M)$ begins alternating signs as $m$ changes, and
- $|S(m,M)|$ starts to diverge.
For large values of $M$ (the interesting limit for me), this empirically occurs consistently at $m_c\approx0.75M$. Indeed,according to this page $_1F_1(-m, \frac{1}{2}-2m;x)$ has one real negative root for odd $m$, and solving for it numerically yields the root around $M = 1.33m$.
Why would the two points above occur, and can one deduce why this happens specifically at roughly $\frac{M}{m_c}\approx\frac{4}{3}$?
