Concrete mathematics : a foundation for computer science

5.1 BASIC IDENTITIES 167
and this recurrence is satisfied also by the right-hand side of (5.19). By
induction, both sides must be equal; QED.
But there’s a neater proof. When r is an integer in the range 0 3 r 3 -m,
the binomial theorem tells us that both sides of (5.19) are (x+y)“‘+‘y~‘. And
since both sides are polynomials in r of degree m or less, agreement at m + 1
different values is enough (but just barely!) to prove equality in general.
It may seem foolish to have an identity where one sum equals another.
Neither side is in closed form. But sometimes one side turns out to be easier
to evaluate than the other. For example, if we set x = -1 and y = 1, we get
y(y)(-l,x =
k