Concrete mathematics : a foundation for computer science

176 BINOMIAL COEFFICIENTS
sum on the upper index. Unfortunately those identities don’t work here. But
if the k were instead m - k, we could use absorption identity (5.6):
i--k)(~I~~~) = (m-n)(mmlE).
So here’s the key: We’ll rewrite k as m - (m - k) and split the sum S
into two sums:
where
m-k-l
m-n-l
) = f(m-(m-kl)(~~~~:>
k=O
=
m-k-l
m-n-l ) -f(m-ki(~I~~~)
k=O
= mg (,“I:::) -f(m-nJ(;g
k=O
= mA- (m-n)B,
The sums A and B that remain are none other than our old friends in
which the upper index varies while the lower index stays fixed. Let’s do B
first, because it looks simpler. A little bit of massaging is enough to make the
summand match the left side of (5.10):
In the last step we’ve included the terms with 0 6 k < m - n in the sum;
they’re all zero, because the upper index is less than the lower. Now we sum
on the upper index, using (5.10), and get