Concrete mathematics : a foundation for computer science

The hypergeometric
database
should really be a
“knowledge base.”
to the hypergeometric
(~)(A2 l-1))
5.6 HYPERGEOMETRIC TRANSFORMATIONS 217
integers n 3 m 3 0;
this has a simple closed form only if m is near 0, in, or n.
But there’s more to the story, since hypergeometric functions also obey
identities of their own. This means that every closed form for hypergeometrics
leads to additional closed forms and to additional entries in the database. For
example, the identities in exercises 25 and 26 tell us how to transform one
hypergeometric into two others with similar but different parameters. These
can in turn be transformed again.
In 1793, J. F. PfafI discovered a surprising reflection law,
&F(a’cbl+) = F(a’;-blz), (5.101)
which is a transformation of another type. This is a formal identity in
power series, if the quantity (-z)“/( 1 - z)~+~ is replaced by the infinite series
(--z)k(l + (":")z+ (k+;+' ) z2 +. . .) when the left-hand side is expanded (see
exercise 50). We can use this law to derive new formulas from the identities
we already know, when z # 1.
For example, Kummer’s formula (5.94) can be combined with the reflection
law (5.101) if we choose the parameters so that both identities apply:
= k$$b-a)~, (5.102)
We can now set a = -n and go back from this equation to a new identity in
binomial coefficients that we might need some day:
= 2-,, (b/4! (b+n)!
b! (b/2+n)! ’
For example, when n = 3 this identity says that
4
4.5 4.5.6
l - 3 - +3
2(4 + b) 4(4 + b) (5 + b) - 8(4 + b)(5 + b)(6 + b)
(b+3)(b+2)(b+l)
= (b+6)(b+4)(b+2)
integer n 3 0. (5.103)