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"



Simple variation on factorial --- upper bound
Edit. Of course, I would find the mistake after posting. k ranges over the even numbers from 2 to 2h, so the result is not as nice. I will try rescuing the approach. End Edit


Each term is close to (m- k+ sqrt(d) + 1/2)(m - k- sqrt(d) + 1/2), where k ranges from 1 to h=(m-n)/2. (h may be off by 1.) Your product is close to n! squared times the product of two binomial coefficients: m+ a choose h and m- b choose h, where a anb are close to sqrt(d). Letting a and b range over floor(sqrt(d)) and ceil(sqrt(d)) are a start if h is not too small.
For finer estimates, let a and b be appropriate real numbers.


Gerhard "Email Me About System Design" Paseman, 2011.06.21 


