Concrete mathematics : a foundation for computer science

16 What is the general solution of the double recurrence
A n,O = % [n>ol ; Ao,k = 0, ifk>O;
A n.k = k&-l,k + A,- l,k-1 , integers k, n,
when k and n range over the set of all integers?
6 EXERCISES 297
17 Solve the following recurrences, assuming that I;/ is zero when n < 0 or
k < 0:
a IL1 = /n~l~+nl~~~l+[~~=k=Ol, for n, k > 0.
b /;I = (n-- k)lnkl/ + lLz:l + [n=k=Ol, for n, k 3 0.
c I;/ = k~n~l~+k~~~~~+[n=k=O], for n, k 3 0.
18 Prove that the Stirling polynomials satisfy
(x+l)~n(x+l) = (x-n)o,(x)+xo,-,(x)
19 Prove that the generalized Stirling numbers satisfy
~{x~k}[xe~+k](-l)k/(~‘+:) = 0, intewn>O.
$ [x~k]{x~~+k}i-lik/(~++:) = 0, integern>O.
20 Find a closed form for xz=, Hf’.
21 Show that if H, = an/bn, where a, and b, are integers, the denominator
b, is a multiple of 2L1snj. Hint: Consider the number 2L1snl -‘H, - i.
22 Prove that the infinite sum
converges for all complex numbers z, except when z is a negative integer;
and show that it equals H, when z is a nonnegative integer. (Therefore we
can use this formula to define harmonic numbers H, when z is complex.)
23 Equation (6.81) gives the coefficients of z/(e’ - 1), when expanded in
powers of z. What are the coefficients of z/(e’ + 1 )? Hint: Consider the
identity (e’+ l)(e’- 1) = ezZ- 1.