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I am trying to find an asymptotic behavior, for large real $t$, of the following sum \begin{align} Q(t)=\sum_{0\le n\le t}e^{-(t-n)}\frac{t-n}{1+n}L_n^{(1)}(t-n) \end{align} where $L_n^{(\alpha)}$ is a generalized Laguerre polynomial.

When plotted, $Q(t)$ is roughly of order $t^{-3/4}$, but the difference between other powers is not that evident.

There is an asymptotic formula for Laguerre polynomials, which states that for large $n$ and fixed $\alpha$ and $x>0$ we have \begin{align} L_n^{(\alpha)}(x) = \frac{n^{\frac{\alpha}{2}-\frac{1}{4}}}{\sqrt{\pi}} \frac{e^{\frac{x}{2}}}{x^{\frac{\alpha}{2}+\frac{1}{4}}} \cos\left(2 \sqrt{nx}- \frac{\pi}{2}\left(\alpha+\frac{1}{2} \right) \right)+O\left(n^{\frac{\alpha}{2}-\frac{3}{4}}\right) \end{align}

I tried to use it to formally compute the asymptotic behavior as follows \begin{align} |Q(t)|&\le\sum_{0\le n\le t}e^{-(t-n)}\frac{t-n}{n+1}|L_n^{(1)}(t-n)|\\ &\approx\int_0^t e^{-(t-n)/2}(t-n)^{7/4}n^{-3/4}dn=\mathcal O(t^{-3/4}) \end{align} where I used $n+1\approx n$, $|\cos(x)|\le1$ and replaced the summation with integration.

Unfortunately, this approximation for Laguerre polynomials fails for small $n$, so I cannot use it as an uniform bound for all the summands, additionally in my case $x$ is not fixed. Is there a way to fix this argument or produce a different one to show the right asymptotic? Or maybe my hypothesis is wrong and $Q(t)$ behaves in an entirely different way?

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Just a small remark: for a good bound you cannot disregard the sign change of $L_n(t-n)$; your sum is oscillatory. –  Andrei MF Aug 5 '13 at 10:52
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