# Simple variation on factorial --- upper bound

Suppose $d$ is a constant $< n^2$, and $m > n$. I have a kind of factorial function where I subtract $d$ from each pair of terms.

Is there any simple upper bound on $$P = ( m (m-1) - d) \times ( (m-2) (m-3) - d ) \times \cdots \times \times ((n+1) n - d)$$

I only need an polynomially tight upper bound. If it makes it easier, you can assume $m = cn$ for $c$ close to 2.

The best I can come up with is $$P \leq \Bigl( \frac{m (m-1) - d}{m (m-1)} \Bigr)^{(m-n)/2} m!/n!$$ but I don't think this is tight.

Thanks!

EDIT: Previous version incorrectly stated "exponentially tight" vs "polynomially tight"

• You could pretend it is the product of m-n terms, half of them like m+k+sqrt(d) +1/2, the rest with a -sqrt(d) instead of +. Gerhard "Email Me About System Design" Paseman, 2011.06.21 Jun 21, 2011 at 18:14
• What do you mean by an “exponentially tight” upper bound? Your bound (as well as the trivial bound $m!/n!$) is tight up to a factor of $2^{O(m)}$, isn’t it? Jun 21, 2011 at 18:17
• @Emil, this was an error. I meant "polynomially tight" Jun 21, 2011 at 18:27