Asymptotic behaviour of $\sum\limits_{j=0}^\infty\frac{\Gamma(\alpha+n+j)}{\Gamma(\alpha+\beta+n+j)}\frac{(-\lambda)^j}{j!}$? What is the asymptotic behaviour (or, maybe, some "good" lower and upper bounds in terms of $n$) of the sum
$$\sum\limits_{j=0}^\infty\frac{\Gamma(\alpha+n+j)}{\Gamma(\alpha+\beta+n+j)}\frac{(-\lambda)^j}{j!}, \quad \textrm{as } n\to\infty, $$
where $\lambda,\alpha, \beta$ are some positive constants?  
 A: Using Kummer's transformation for the confluent hypergeometric function $M$ (https://dlmf.nist.gov/13.2.E39) and its leading-order asymptotics for large second parameter (https://dlmf.nist.gov/13.8.E2), it is found that
\begin{align*}
& \sum\limits_{j = 0}^\infty  \frac{\Gamma (\alpha  + n + j)}{\Gamma (\alpha  + \beta  + n + j)}\frac{( - \lambda )^j }{j!}  = \frac{\Gamma (\alpha  + n)}{\Gamma (\alpha  + \beta  + n)}M(\alpha  + n,\alpha  + \beta  + n, - \lambda ) \\ &= \frac{\Gamma (\alpha  + n)}{\Gamma (\alpha  + \beta  + n)}e^{ - \lambda } M(\beta ,\alpha  + \beta  + n,\lambda ) \\ & = \frac{e^{ - \lambda } }{(\alpha  + \beta  + n)^\beta }\left( 1 + \mathcal{O}\!\left( \frac{1}{\alpha  + \beta  + n} \right) \right) = \frac{e^{ - \lambda } }{n^\beta }\left( 1 + \mathcal{O}\!\left( \frac{1}{n} \right) \right),
\end{align*}
as $n\to+\infty$ with $\lambda$, $\alpha$ and $\beta$ being fixed. You may obtain more precise estimates using https://dlmf.nist.gov/13.8.i.
Addendum: Using the standard integral representation of $M$ (https://dlmf.nist.gov/13.4, note the difference between $M$ and $\mathbf{M}$), one has
\begin{align*}
\frac{\Gamma (\alpha  + n)}{\Gamma (\alpha  + \beta  + n)}e^{ - \lambda } M(\beta ,\alpha  + \beta  + n,\lambda ) & = \frac{e^{ - \lambda } }{\Gamma (\beta )}\int_0^1 e^{\lambda t} t^{\beta  - 1} (1 - t)^{\alpha  + n - 1} dt \\ & = \frac{e^{ - \lambda } }{\Gamma (\beta )}\int_0^{+\infty } e^{ - nt} t^{\beta  - 1} F(\alpha ,\beta ,\lambda ,t)dt ,
\end{align*}
where
$$
F(\alpha ,\beta ,\lambda ,t) = e^{ - \alpha t} \left( \frac{1 - e^{ - t} }{t} \right)^{\beta  - 1} e^{\lambda (1 - e^{ - t} )} .
$$
The $F$ has a power series
$$
F(\alpha ,\beta ,\lambda ,t) = 1 + \sum\limits_{k = 1}^\infty  a_k (\alpha ,\beta ,\lambda )t^k  
$$
for $|t|<2\pi$. The coefficients may be expressed in terms of Nörlund and Touchard polynomials. Using Watson's lemma (https://dlmf.nist.gov/2.3.ii) we obtain the asymptotic expansion
$$
\frac{\Gamma (\alpha  + n)}{\Gamma (\alpha  + \beta  + n)}e^{ - \lambda } M(\beta ,\alpha  + \beta  + n,\lambda ) \sim \frac{e^{ - \lambda } }{n^\beta  }\left( 1 + \sum\limits_{k = 1}^\infty  \frac{a_k (\alpha ,\beta ,\lambda )(\beta )_k }{n^k} \right),
$$
as $n\to+\infty$ with $\lambda$, $\alpha$ and $\beta$ being fixed. Here $(\beta)_k = \Gamma(\beta +k)/\Gamma(\beta)$ is the Pochhammer symbol.
A: Further comments:
$$\sum\limits_{j=0}^\infty\frac{\Gamma(\alpha+n+j)}{\Gamma(\alpha+\beta+n+j)}\frac{(-\lambda)^j}{j!}=\frac{\Gamma(a + n)}{\Gamma(a+b+n)} \;_1F_1(a+n;a+b+n;-\lambda)
$$
If you substitute constants in wolfram alpha, you get a potential correspondence to a combination of Barnes-G functions and associated Laguerre Polynomials
$$
\frac{\Gamma(a + n)}{\Gamma(a+b+n)} \;_1F_1(a+n;a+b+n;-\lambda) \approx \frac{G(n+1+a)\Gamma(1-n-a)\Gamma(n+a+b)G(n+a+b)}{G(a+n)\Gamma(b)G(1+n+a+b)} L^{n+a+b-1}_{-n-a}(-\lambda)
$$
Where approx means, (I don't know the constraints on the arguments where this is true), this might help you consider the asymptotics further?
Derived from:

