Concrete mathematics : a foundation for computer science

216 BINOMIAL COEFFICIENTS
But look at the lower parameter ‘- 2m’. Negative integers are verboten, so
this identity is undefined!
It’s high time to look at such limiting cases carefully, as promised earlier,
because degenerate hypergeometrics can often be evaluated by approaching
them from nearby nondegenerate points. We must be careful when we do this,
because different results can be obtained if we take limits in different ways.
For example, here are two limits that turn out to be quite different when one
of the upper parameters is increased by c:
hFO F
-lSE, -3
-2+e
-= a,,(l + (4;;k;i + (--1+4(4-3)(-2)
(--2+El(-l+EI2!
+ (-l+~l(~)(l+~l( -3)1-2)(-l)
(-2+E)(-l+E)(E)3! )
FzF(I:';zll) := lii(l+#$+O+O)
:= q+o+o zz -;
Similarly, we have defined (1;) = 0 = lime-c (-2’) ; this is not the same
as lime.+7 (1;::) = 1. The proper way to treat (5.98) as a limit is to realize
that the upper parameter -m is being used to make all terms of the series
tkaO (2c:kk)2k zero for k > m; this means that we want to make the following
more precise statement:
(2mm) liiF(y2;,“,12) = 22m, integerm>O. (5.99)
Each term of this limit is well defined, because the denominator factor (-2m)’
does not become zero until k. > 2m. Therefore this limit gives us exactly the
sum (5.20) we began with.
5.6 HYPERGEOMETRIC TRANSFORMATIONS
It should be clear by now that a database of known hypergeometric
closed forms is a useful tool for doing sums of binomial coefficients. We
simply convert any given sum into its canonical hypergeometric form, then
look it up in the table. If it’s there, fine, we’ve got the answer. If not, we can
add it to the database if the sum turns out to be expressible in closed form.
We might also include entries in the table that say, “This sum does not have a
simple closed form in general.” For example, the sum xkSrn (L) corresponds