Concrete mathematics : a foundation for computer science

5.1 BASIC IDENTITIES 161
Let’s look at this derivation blow by blow. The key step is in the second line,
where we apply the symmetry law (5.4) to replace (“,‘“) by (“‘,‘“). We’re
allowed to do this only when m + k 3 0, so our first step restricts the range
of k by discarding the terms with k < -m. (This is legal because those terms
are zero.) Now we’re almost ready to apply (5.10); the third line sets this up,
replacing k by k - m and tidying up the range of summation. This step, like
the first, merely plays around with t-notation. Now k appears by itself in
the upper index and the limits of summation are in the proper form, so the
fourth line applies (5.10). One more use of symmetry finishes the job.
Certain sums that we did in Chapters 1 and 2 were actually special cases
of (5.10), or disguised versions of this identity. For example, the case m = 1
gives the sum of the nonnegative integers up through n:
(3 + (;) +...f (y) = O+l +...+n = (n:l)n = (“:‘).
And the general case is equivalent to Chapter 2’s rule
Obk