Concrete mathematics : a foundation for computer science

Kummer was a
summer.
5.5 HYPERGEOMETRIC FUNCTIONS 213
The sum in Problem 2 and Problem 4 likewise yields F( 2,1 - n; 2 - m; 1).
(We need to replace k by k + 1 first.) And the “menacing” sum in Problem 6
turns out to be just F(n + 1, -n; 2; 1). Is there nothing more to sum, besides
disguised versions of Vandermonde’s powerful convolution?
Well, yes, Problem 3 is a bit different. It deals with a special case of the
general sum tk (“kk) zk considered in (5.74), and this leads to a closed-form
expression for
We also proved something new in (5.55), when we looked at the coefficients
of (1 - z)~( 1 + z)~:
F l-c-2n, - 2 n
(2n)! (c - 1 )!
-1 = (-l)n-
( C 1 >
n! (c+n-l)!’
integer n 3 0.
This is called Kummer’s formula when it’s generalized to complex numbers:
(5.94)
The summer of ‘36. (Ernst Kummer [187] proved this in 1836.)
It’s interesting to compare these two formulas. Replacing c by l -2n- a,
we find that the results are consistent if and only if
(5.95)
when n is a positive integer. Suppose, for example, that n = 3; then we
should have -6!/3! = limX+ 3x!/(2x)!. We know that (-3)! and (-6)! are
both infinite; but we might choose to ignore that difficulty and to imagine
that (-3)! = (-3)(-4)(-5)(-6)!,so that the two occurrences of (-6)! will
cancel. Such temptations must, however, be resisted, because they lead to
the wrong answer! The limit of x!/(2x)! as x + -3 is not (-3) (-4) (-5) but
rather -6!/3! = (-4)(-5)(-6), according to (5.95).
The right way to evaluate the limit in (5.95) is to use equation (5.87),
which relates negative-argument factorials to positive-argument Gamma functions.
If we replace x by -n + e and let e + 0, two applications of (5.87)
give
(-n-e)! F(n+e) sin(2n + 2e)rt
(-2n - 2e)! F(2n + 2e) = sin(n + e)rc