Concrete mathematics : a foundation for computer science

5.5 HYPERGEOMETRIC FUNCTIONS 209
looks like in hypergeometric notation. We need to write the sum as an infinite
series that starts at k = 0, so we replace k by n - k:
r+n-k
E
(r+n-k)! = tk
n-k x k,O r! (n - k)! x .
/ k>O
This series is formally infinite but actually finite, because the (n - k)! in the
denominator will make tk = 0 when k > n. (We’ll see later that l/x! is
defined for all x, and that l/x! = 0 when x is a negative integer. But for now,
let’s blithely disregard such technicalities until we gain more hypergeometric
experience.) The term ratio is
tk+l (r+n-k-l)!r!(n-k)! n-k
- tk = r!(n-k-l)!(r+n-k)! = r+n-k
(k+ l)(k-n)(l)
= (k-n-r)(k+ 1)
Furthermore to = (“,“). Hence the parallel summation law is equivalent to
the hypergeometric identity
("n")r(:l+il) = (r+,,').
Dividing through by (“,“) g’Ives a slightly simpler version,
Let’s do another one. The term ratio of identity (5.16),
integer m,
(5.82)
is (k-m)/(r-m+k+l) =(k+l)(k-m)(l)/(k-m+r+l)(k+l), after
we replace k by m - k; hence (5.16) gives a closed form for
This is essentially the same as the hypergeometric function on the left of
(5.82), but with m in place of n and r + 1 in place of -r. Therefore identity
(5.16) could have been derived from (5.82), the hypergeometric version of
(5.9). (No wonder we found it easy to prove (5.16) by using (5.g).)
First derangements, Before we go further, we should think about degenerate cases, because
now degenerates. hypergeometrics are not defined when a lower parameter is zero or a negative