Concrete mathematics : a foundation for computer science

How do you proflounce
4 ?
(Dunno, but 7j$
calls it ?artheta’.)
5.6 HYPERGEOMETRIC TRANSFORMATIONS 219
Notice that when z = 1 this reduces to Vandermonde’s convolution, (5.93).
Differentiation seems to be useful, if this example is any indication; we
also found it helpful in Chapter 2, when summing x + 2x2 + . . . + nxn. Let’s
see what happens when a general hypergeometric series is differentiated with
respect to 2:
al (al+l)i;. . . a,(a,+l)kzk
= 2 b 1 (b,+l)“...b(b
n n +l)kk!
al . . . a,
bl . ..b. F (5.10’3)
The parameters move out and shift up.
It’s also possible to use differentiation to tweak just one of the parameters
while holding the rest of them fixed. For this we use the operator
which acts on a function by differentiating it and then
operator gives
which by itself isn’t too useful. But if we multiply
parameters, say al, and add 4F, we get
al(al+l)‘ak...akzk
= ’ by.J&,
k?O n .
= alF
Only one parameter has been shifted.
al+l, a2, . . . . a,
bl, . . . . b,
multiplying by z. This
F by one of its upper