The following summation turned up in the course of my research:
$$S_n=\sum_{k=0}^n {n \choose k}\lambda^k P(k,t)$$
where $P(k,t)=\frac{1}{\Gamma(k)}\int_{0}^t e^{-x}x^{k-1}dx$ is the lower incomplete regularized gamma function, and $t>0$ a constant. I would like to characterize $S_n$ in terms of "simple" functions (i.e., like $\operatorname{erf}$, not hypergeometric functions) of $\lambda$, $n$, and $t$ -- ideally I'd like to obtain tight upper and lower bounds.
I approach this as follows: When $n<t$, $P(k,t)$ is very close to unity for all $k=1,\ldots,n$, so $S_n$ is very close to $(1+\lambda)^n$. For $n>t+c\sqrt{t}$, $P(k,t)$ gets very close to zero very fast (the order is exponential in $c$ by 8.11.10 in DLMF , so to obtain upper and lower bounds on $S_n$ in this regime I cut off the summation at $k=t$ for lower bound on $S_n$ and at $k=t+c\sqrt{t}$ for the upper bound:
$$\sum_{k=0}^t {n \choose k}\lambda^k \leq S_n \leq\sum_{k=0}^{t+c\sqrt{t}} {n \choose k}\lambda^k\tag{1}$$
The approach is motivated by the behavior or lower incomplete regularized gamma function depicted here as an example with $k$ on the x-axis and $t=50$:
What does the community think of my approach? Is it correct? Could I do better?
If it is correct, how do I find expressions for the partial sums in $(1)$? Is it possible to do using the aforementioned simple functions of $\lambda$, $n$, and $t$?
(this previous question pertains to this last point, however it is not completely relevant since it assumes that the partial sum is taken over $k=0,\ldots,\alpha n$ with $\alpha$ a constant, while here $\alpha=a/n$ with constant $a$. I asked the question about this in a comment, but thought that it might be better to write up where it came from.)
