Concrete mathematics : a foundation for computer science

524 ANSWERS TO EXERCISES
Since ~06jn/m
k>n/m
Incidentally, the functions ‘B,,,(zm) and L2i+‘z~B1+II,(L2~+‘~)‘~m are the m+l
complex roots of the equation w”‘+’ - wm = z”l.
5.88 Use the facts that Jr(e it - e nt) dt/t = Inn and (1 - ee’)/t $ 1.
(We have (“,) = O(kmx ‘) a,s k + 00, by (5.83); so this bound implies that
Stirling’s series tk sk (i) converges when x > -1. Hermite [155] showed that
the sum is In r( 1 + x).)
5.89 Adding this to (5.19) gives ~~‘(x+y)~+’ on both sides, by the binomial
theorem. Differentiation gives
and we can replace k by k + m + 1 and apply (5.15) to get
& (m;:: k) (-‘I; ‘) (-X)m+l+ky--l-k-n
In hypergeometric form, this reduces to
which is the special case (a, b, c, z) = (n + 1, m + 1 + r, m + 2, -x/y) of the
reflection law (5.101). (Thus (5.105) is related to reflection and to the formula
in exercise 52.)
5.90 If r is a nonnegative integer, the sum is finite, and the derivation in
the text is valid as long as :none of the terms of the sum for 0 < k < r has
zero in the denominator. Otherwise the sum is infinite, and the kth term
(k ml- ‘) / ( k -i-l) is approximately k” ’(-s - l)!/(-r - l)! by (5.83). So we
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