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where $m <n$ and $n,a,b,m$ are positive.

Any suggestion on how to approximate this sum asymptotically will be appreciated. So far I could do it in quite a rough way, by rewriting the numerator and then expanding log function (since $b +2j \geq b$):

$\log(nb+ a -j) \approx \log (nb+a) -\frac{j}{nb+a}$

Ultimately I need to express it as $O(h(n,a,b,m))$

EDIT: $t=O(n), b=O(n), \ a=o(n), m=O(a)$. Hope it helps.

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I think the method, and even the answer, will depend quite a bit on the relative sizes of $n$, $a$, $b$, $t$, and $m$. Can you say anything about their sizes? –  Greg Martin Jan 22 '12 at 7:16
    
I second Greg's request. Regarding the Edit, are we supposed to understand that $a$, $b$, $n$ and $t$ all converge to infinity? –  Did Jan 22 '12 at 9:01
    
$n$ can be taken arbitrarily large, $m<<n, \ b \leq C n$ for some $C$. –  sigma.z.1980 Jan 22 '12 at 22:45
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