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 n ( b n +l)kk! al . . . a, F bl . ..b.
How do you proflounce 4 ?
(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
(Dunno, but 7j$ calls it ?artheta’.) which acts on a function by differentiating it and then multiplying by z. This operator gives
which by itself isn’t too useful. But if we multiply F by one of its upper parameters, say al, and add 4F, we get
= ’ by.J&,
al(al+l)‘ak...akzk
k?O
= alF
Only one parameter has been shifted.
n
.
al+l, a2, . . . . a,
bl, . . . . b,
