Concrete mathematics : a foundation for computer science

5.1 BASIC IDENTITIES 159
Those of us who tend not to discover such slick proofs, or who are otherwise
into tedium, might prefer to derive (5.8) by a straightforward manipulation
of the definition. If k > 0,
(r- l)k (r- l)k-’
(‘*‘)+(;I:) = k!+ (k- l)!
= (T-l)lf=l(r-k) + (r-l)k-‘k
k! k!
= (r-l)Er = f = r
k! k! 0k ’
Again, the cases for k < 0 are easy to handle.
We’ve just seen three rather different proofs of the addition formula. This
is not surprising; binomial coefficients have many useful properties, several of
which are bound to lead to proofs of an identity at hand.
The addition formula is essentially a recurrence for the numbers of Pascal’s
triangle, so we’ll see that it is especially useful for proving other identities
by induction. We can also get a new identity immediately by unfolding the
recurrence. For example,
(Z) = (;) + (Z)
= (D+(i)+(f)
= (;)+(;)+(;)+(i)
= (I)++++,
Since (!,) = 0, that term disappears and we can stop. This method yields
the general formula
,5-,(‘:“) = (a) + (‘7’) +...+ (“n”)
= (r’:“)) integer n. (5.9)
Notice that we don’t need the lower limit k 3 0 on the index of summation,
because the terms with k < 0 are zero.
This formula expresses one binomial coefficient as the sum of others whose
upper and lower indices stay the same distance apart. We found it by repeatedly
expanding the binomial coefficient with the smallest lower index: first