Concrete mathematics : a foundation for computer science

222 BINOMIAL COEFFICIENTS
One use of this differential theory is to find and prove new transformations.
For example, we can readily verify that both of the hypergeometrics
satisfy the differential equation
~(1 -z)F"(z) + (afb +- ;)(l -2z)F'(z) -4abF(z) = 0;
hence Gauss’s identity [116, equation 1021
(5.110)
must be true. In particular, ICaution: We can’t
use (5.110) safely
when Izl > l/Z,
F( ,:4;:; 1;) = F(o+4;IT-11’) ’
2
whenever both infinite sums converge.
(5.111) unless both sides
are polynomials;
see exercise 53.)
Every new identity for hypergeometrics has consequences for binomial
coefficients, and this one is no exception. Let’s consider the sum
&(m,k)(m+r+l) (q)“, integersm>n>O.
The terms are nonzero for 0 < k < m - n, and with a little delicate limittaking
as before we can express this sum as the hypergeometric
n-m, -n-m-lfae
liio m F
0 n ( -m+ 6
The value of OL doesn’t affect the limit, since the nonpositive upper parameter
n - m cuts the sum off early. We can set OL = 2, so that (5.111) applies.
The limit can now be evaluated because the right-hand side is a special case
of (5.92). The result can be expressed in simplified form,
gm,k)(m+,+l) (G)
= ((m+nn1’2)2nPm[m+n is even], ~~~~o, (5.112)
as shown in exercise 54. For example, when m = 5 and n = 2 we get
(z)(i) - ($($/2 + (:)(;)/4 -- (z)(i)/8 = 10 - 24 + 21 - 7 = 0; when m = 4
and n = 2, both sides give z.