Concrete mathematics : a foundation for computer science

7.6 EXPONENTIAL GENERATING FUNCTIONS 351
this is the egf of (g-1, g2,. . . ). Thus differentiation on egf’s corresponds to the
left-shift operation (G(z) ~ go)/z on ordinary gf’s. (We used this left-shift
property of egf’s when we studied hypergeometric series, (5.106).) Integration
of an egf gives
g,,;dt = (7.73)
this is a right shift, the egf of (0, go, 91). . .).
The most interesting operation on egf’s, as on ordinary gf’s, is multiplication.
If i(z) and G(z) are egf’s for (f,,) and (gn), then i(z)G(z) = A(z) is
the egf for a sequence (hn) called the binomial convolution of (f,,) and (g,,):
Binomial coefficients appear here because (z) = n!/k! (n ~ k)!, hence
in other words, (h,/n!) is the ordinary convolution of (f,,/n!) and (g,,/n!).
Binomial convolutions occur frequently in applications. For example, we
defined the Bernoulli numbers in (6.79) by the implicit recurrence
Bi = [m=O], for all m 3 0;
this can be rewritten as a binomial convolution, if we substitute n for m + 1
and add the term ES, to both sides:
Bk = B,+[n=l], for all n 3 0.
We can now relate this recurrence to power series (as promised in Chapter 6)
by introducing the egf for Bernoulli numbers, B(z) = EnSo B,,z’/n!. The
left-hand side of (7.75) is the binomial convolution of (B,,) with the constant
sequence (1 , 1 , 1, . ); hence the egf of the left-hand side is B( z)e’. The egf
of the right-hand side is Ena (B, + [n=l])z”/n! = B(z) + z. Therefore we
must have B(z) = z/(e’ ~ 1); we have proved equation (6.81), which appears
also in Table 337 a:s equation (7.44).