Concrete mathematics : a foundation for computer science

234 BINOMIAL COEFFICIENTS
37 Show that an analog of the binomial theorem holds for factorial powers.
That is, prove the identities
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for all nonnegative integers n.
Show that all nonnegative integers n can be represented uniquely in the
formn = (y)+(:)+(i) w here
a, b, and c are integers with 0 6 a < b < c.
(This is called the binomial number system.)
Show that if xy = ax -t by then xnyn = xE=:=, (‘“;~,~“) (anbnpkxk +
an- kbnyk) for all n > 0. Find a similar formula for the more general
product xmyn.
40 Find a closed form for
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integers m,n 3 0.
Evaluate tk (L)k!/(n + 1 + k)! when n is a nonnegative integer.
Find the indefinite sum 2 (( -1 )“/(t)) 6x, and use it to compute the sum
xL=,(-l)“/(L) in closed form.
43 Prove the triple-binomial identity (5.28). Hint: First replace (iz:) by
Ej (m&-j> (!I’
44 Use identity (5.32) to find closed forms for the double sums
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~(-l)“k(i~k) (3) (L) (m’~~~-k) and
jF,ll)j+k(;) (l;) (bk) (:)/(;x) ’
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given integers m 3 a 3 0 and n 3 b 3 0.
Find a closed form for tks,, (234-k.
Evaluate the following s’um in closed form, when n is a positive integer:
Hint: Generating functions win again.