Concrete mathematics : a foundation for computer science

(Here t and T
aren’t necessarily
related as in
~w9~J
5 EXERCISES 231
8 Evaluate xk (L)(-l)k(l -k/n)“. What is the approximate value of this
sum, when n is very large? Hint: This sum is An f (0) for some function f.
9 Show that the generalized exponentials of (5.58) obey the law
&t(z) = &(tz)
1/t
, if t # 0,
where E(z) is an abbreviation for &I(Z).
10 Show that -2(ln(l -2) + z)/ z2 is a hypergeometric function.
11 Express the two functions
23 25 2'
sin2 = z--+--rlt
3! 5! .
arcsinz
1.23
= 2 + 23 + 1.3.25 2.4.5 + 1.3.5.27 2.4.6.7 +"'
in terms of hypergeometric series.
12 Which of the following functions of k is a “hypergeometric term,” in the
sense of (5.115)? Explain why or why not.
a nk.
Basics
b kn.
c (k! + (k+ 1)!)/2.
d Hk, that is, f + t +. . . + t.
e t(k)T(n - k)/T(n), when t and T are hypergeometric terms.
f (t(k) + T(k))/2, when t and T are hypergeometric terms.
g (at(k) + bt(k+l) + ct(k+2))/(a + bt(1) + ct(2)), when t is a
hypergeometric term.
13 Find relations between the superfactorial function P, = nl, k! of exercise
4.55, the hyperfactorial function Q,, = nL=, kk, and the product
Rn = I-I;==, (;>.
14 Prove identity (5.25) by negating the upper index in Vandermonde’s convolution
(5.22). Then show that another negation yields (5.26).
15 What is tk (L)"(-l)"? Hint: See (5.29).
16 Evaluate the sum
c (o:Uk) (b:bk) (c:k)(-li*
when a, b, c are nonnegative integers.
17 Find a simple relation between (2n;“2) and (2n;i’2).