Concrete mathematics : a foundation for computer science

202 BINOMIAL COEFFICIENTS
Table 202 General convolution identities, valid for integer n 3 0.
(5.62)
(5.63)
(5.64)
= (tn+ r+s)ntnT++rS+S. (5.65)
We have learned that it’s generally a good idea to look at special cases of
general results. What happens, for example, if we set t = l? The generalized
binomial ‘BI (z) is very simple-it’s just
B,(z) = X2” = &;
k>O
therefore IB1 (z) doesn’t give us anything we didn’t already know from Vandermonde’s
convolution. But El (z) is an important function,
&(z) = x(k+,)k-l; = l+z+;~~+$r~+$~+... (5.66)
k>O
that we haven’t seen before; it satisfies the basic identity Ah! This is the
iterated power
&(z) = ,=Q)
function
(5.67) E(1n.z) = zLz’.
that I’ve often
This function, first studied by Eisenstein [75], arises in many applications.
wondered about.
The special cases t = 2 and t = -1 of the generalized binomial are of zztrzr,,
particular interest, because their coefficients occur again and again in problems
that have a recursive structure. Therefore it’s useful to display these