Concrete mathematics : a foundation for computer science

356 GENERATING FUNCTIONS
7.7 DIRICHLET GENERATING FUNCTIONS
There are many other possible ways to generate a sequence from a
series; any system of “kernel” functions K,(z) such that
g,, K,(z) = 0 ==+ g,, = 0 for all n
t
n
can be used, at least in principle. Ordinary generating functions use K,(z) =
zn, and exponential generating functions use K, (z) = 2*/n!; we could also try
falling factorial powers zc, 01: binomial coefficients zs/n! = (R) .
The most important alternative to gf’s and egf’s uses the kernel functions
1 /n”; it is intended for sequences (41 , 92, . . . ) that begin with n = 1 instead
of n = 0:
This is called a Dirichlet generating function (dgf), because the German
mathematician Gustav Lejeune Dirichlet (1805-1859) made much of it.
For example, the dgf of the constant sequence (1 , 1 , 1, . . . ) is
(7.86)
This is Riemann’s zeta function, which we have also called the generalized
harmonic number Hk’ when z > 1.
The product of Dirichlet generating functions corresponds to a special
kind of convolution:
Thus F(z) c(z) = H(z) is the dgf of the sequence
hn = x f d h/d. (7.87)
d\n
For example, we know from (4.55) that td,n p(d) = [n= 1 I; this is
the Dirichlet convolution of the Mobius sequence (u( 1) , p( 2)) u( 3)) . . . ) with
(l,l,l,...), hence
In other words, the dgf of (p(l), FL(~), p(3), . . .) is Lo’.
(7.88)