Concrete mathematics : a foundation for computer science

180 BINOMIAL COEFFICIENTS
Problem 4: A sum involving two binomial coefficients.
Our next task is to find: a closed form for
integers m > n 3 0.
Wait a minute. Where’s the second binomial coefficient promised in the title
of this problem? And why should we try to simplify a sum we’ve already
simplified? (This is the sum S from Problem 2.)
Well, this is a sum that’s easier to simplify if we view the summand
as a product of two binomial coefficients, and then use one of the general
identities found in Table 169. The second binomial coefficient materializes
when we rewrite k as (y):
And identity (5.26) is the one to apply, since its index of summation appears
in both upper indices and with opposite signs.
But our sum isn’t quite in the correct form yet. The upper limit of
summation should be m - 1:) if we’re to have a perfect match with (5.26). No
problem; the terms for n