Concrete mathematics : a foundation for computer science

550 ANSWERS TO EXERCISES
applied to l/( 1 - z), where S means “multiply by l/( 1 - z).” There are m!
terms
EosEk, s&s . . . SEk,,,
where 0 6 ki < j, and every such term evaluates to zrm/( 1 - zm) m+’ if r is the
number of places where ki < ki+r . Exactly (y) terms have a given value of r,
so the coefficient of zmn is x2;’ (~)(“‘,‘~‘) = (n+l)“’ by (6.37). (The fact
that operation Ek can be expressed with complex roots of unity seems to be
of no help in this problem.)
7.55 Suppose that Po(z)F(z) + ... + P,(z)Fiml(z) = Qo(z)G(z) + ... +
Qn(z)Gin)(z) = 0, where P,(z) and Qn(z) are nonzero. (a) Let H(z) =
F(z) + G (2). Then there are rational functions Rk,l (z) for 0 < 1 < m + n such
that Hck)(z) = Rk,o(z)FCo)(z) + ... + Rk,mpl(~)F’mp’)(~) + Rk,,,(z)Gcol(z) +
. . + Rk,m+n-1 (z) G(n-‘i(z). The m+n+l vectors (Rk,O(z),...,Rk,m+n-l(~))
are linearly dependent in the (m + n)-dimensional vector space whose components
are rational functions; hence there are rational functions S(z), not
all zero, such that SO(Z)H~~~(Z) + ... + S,+,(Z)H~~+~~(Z) = 0. (b) Similarly,
let H(z) = F(z) G (2). There are rational Rk,l(z) for 0 $ 1 < mn
with H’k’(~) = XL;’ I;