Asymptotic behaviour of $\sum\limits_{j=0}^\infty\frac{\Gamma(\alpha+n+j)}{\Gamma(\alpha+\beta+n+j)}\frac{(-\lambda)^j}{j!}$?

Question

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?

Answer 1

Using Kummer's transformation for the confluent hypergeometric function $M$ and its leading-order asymptotics for large second parameter, 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)}\mathrm{e}^{ - \lambda } M(\beta ,\alpha + \beta + n,\lambda ) \\ & = \frac{\mathrm{e}^{ - \lambda } }{(\alpha + \beta + n)^\beta }\left( 1 + \mathcal{O}\!\left( \frac{1}{\alpha + \beta + n} \right) \right) = \frac{\mathrm{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 $\S13.8.(\mathrm{i})$.

Addendum: Using the standard integral representation of $M$ (note the difference between $M$ and $\mathbf{M}$), one has \begin{align*} \frac{\Gamma (\alpha + n)}{\Gamma (\alpha + \beta + n)}\mathrm{e}^{ - \lambda } M(\beta ,\alpha + \beta + n,\lambda ) & = \frac{\mathrm{e}^{ - \lambda } }{\Gamma (\beta )}\int_0^1 \mathrm{e}^{\lambda t} t^{\beta - 1} (1 - t)^{\alpha + n - 1} \mathrm{d}t \\ & = \frac{\mathrm{e}^{ - \lambda } }{\Gamma (\beta )}\int_0^{+\infty } \mathrm{e}^{ - nt} t^{\beta - 1} F(\alpha ,\beta ,\lambda ,t)\mathrm{d}t , \end{align*} where $$ F(\alpha ,\beta ,\lambda ,t) = \mathrm{e}^{ - \alpha t} \left( \frac{1 - \mathrm{e}^{ - t} }{t} \right)^{\beta - 1} \mathrm{e}^{\lambda (1 - \mathrm{e}^{ - t} )} . $$ 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 we obtain the asymptotic expansion $$ \frac{\Gamma (\alpha + n)}{\Gamma (\alpha + \beta + n)}\mathrm{e}^{ - \lambda } M(\beta ,\alpha + \beta + n,\lambda ) \sim \frac{\mathrm{e}^{ - \lambda } }{n^\beta }\left( 1 + \sum\limits_{k = 1}^\infty \frac{a_k (\alpha ,\beta ,\lambda )(\beta )_k }{n^k} \right), $$ as $n\to+\infty$ with $\lambda$, $\alpha$ and $\beta$ being fixed. Here $(\beta)_k = \Gamma(\beta +k)/\Gamma(\beta)$ is the Pochhammer symbol.

Answer 2

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?

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