Concrete mathematics : a foundation for computer science

220 BINOMIAL COEFFICIENTS
A similar trick works with lower parameters, but in this case things shift
down instead of up:
=
x (bl - 1) a!. . . c& zk
k>O (b, -l)i;bi...b;k!
We can now combine all these operations and make a mathematical “pun” Ever hear the one
by expressing the same quantity in two different ways. Namely, we have about the brothers
altl, . . . . a,+1
(9+a,)...(4+a,)F � = al...a,F
bl, . . . . b,
who named their
cattle ranch Focus,
because it’s where
the sons raise meat?
and
(8 + b, - 1). . . (4 + b, -- l)F
== (bl-l)...(bn-1)F ,,“I”“‘~+),
I ...I n
where F = F(al , . . . , a,; bl , . . . , b,;z). And (5.106) tells us that the top line
is the derivative of the bottom line. Therefore the general hypergeometric
function F satisfies the differential equation
D(9 + bl - 1). . . (9 + b,, - l)F = (4 + al). . . (9 + a,)F, (5.107)
where D is the operator 2.
This cries out for an example. Let’s find the differential equation satisfied
by the standard a-over-1 hypergeometric series F(z) = F(a, b; c; z). According
to (5.107), we have
D(9+c-1)F = (i?+a)(4+b)F.
What does this mean in ordinary notation ? Well, (4 + c - l)F is zF’(z) +
(c - 1 )F(z), and the derivative of this gives the left-hand side,
F’(z) + zF”(z) + (c - l)F’(z) .