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This question was posed in sci.math.research group on Google:

http://groups.google.com/group/sci.math.research/browse_thread/thread/48ca096ec5f08a39?hl=en

After latexing the mind-bending plain text of the discussion it looks like this:

assuming . The author's conjecture was that for large :

Summing by parts is quite a standard device. Though, like with integration by parts the difficult part often is use it at the right moment. Whittakker and Watson ascribe it's systematic introduction to Abel. Probably the best account of it is given in "Concrete Mathematics". The authors introduce "definite sums" which are effectively the sums with an omitted last term:

The cryptic is added solely to enhance the analogy with definite integrals.

The general formula is the following:

Where is the difference operator and is the shift operator. The formula is easily proved by evaluating

Part of the sum (*) on the right prompts for the binomial formula. Hence, it would be good to pull it out of the sum. Let's try:

For

Putting it all together:

This gives an equation for :