What is the value of this finite sum?

[Pere Callahan]

Hi,

I came across a somewhat unwieldy sum which I do not know how to manipulate any further. I suspect it might have something to do with a hypergeometric series, but I am not sufficiently familar with those series to be able to just see how it might be related to them.

The sum in question is

$$\sum_{1\leq r_1< r_2 < \dots < r_n \leq m}{\frac{ (k-1)!(k-2r_1)!(k-2r_2+1)! \dots (k-2r_{n-1}+n-2)!(k-2r_n+n-1)! }{ (k-2r_1+1)!(k-2r_2+2)! \dots (k-2r_{n-1}+n-1)!(k-2r_n+n)! }}$$

where k is some non-negative integer, $$0\leq n \leq k$$. m is defined by

$$m=\left\lfloor\frac{k+n}{2}\right\rfloor \geq n$$.

Do you know of any books where I could look up things like that?

Any help is greatly appreciated.

-Pere

 

[CRGreathouse]

Well, there's http://citeseer.ist.psu.edu/293266.html, but that's a pretty complicated book. Otherwise there are pages like
http://en.wikipedia.org/wiki/List_of_hypergeometric_identities
online that might have something similar.

 

[Pere Callahan]

Thanks. A=B seems to be an interesting book, I hadn't heard of it before.