636 Chapter^8
In the next theorem we consider some of the most elementary properties
of the hypergeometric function.
Theorem 8.51 The hypergeometric function F( a, b, c; z) has the follow-
ing properties:
- F(::,b,c;z) == F(b,a)c;z}.
2. ddzmF(a,b,c;z)== (a(';;~bmF(a+m,b+m,c+m;z),m==l,2, ....
- (a - b)F(a, b,c; z) ==a 1a + 1, b, c; z) - bF(a, b + 1,c; z).
- F(a, b,c; z) == F(a, b + l,c + 1; z) - :~~~:~ zF(a + 1, b + l,c + 2; z).
- (z - z^2 )F" + [c - (a+ b + l)z]F' - abF == 0.
6. If lzl < 1 and Rec> Reb > 0, then
F(a b c· z) = r(c) [1 tb-^1 (1 -t)c-b-^1 (1 -tz)-a dt
' ' ' r(b)r(c - b) lo
7. If c =fa 0, -1, -2, ... and Re(c-a - b) > 0, then F(a,b,c;I) is an
analytic function of each of the parameters a, b, and c.8. F(a, b,c; 1) = r(c)r(c-a - b)/r(c-a)r(c-b), Re(c-a - b) > 0.
Proofs (1) It follows at once from the definition.
(2) To prove this property it suffices to differentiate (8.22-2) with respect
to z. We obtain
!:_F(a,b,c;z) = f (a)n(b)n zn-1 == f (a)n+1(b)n+l Zn
dz n=l (n - l)!(c)n n=O n!(c)n+l-~~~(a-1-I)n(b+I)n n_abF( lb l. )
- L....., '( l) z - a+ , + , c + 1, z
c n=O n. c+ n c
(8.22-3)Hence a second differentiation yields
d^2 (a)2(b)2
dz 2 F( a, b, c; z) = (ch F( a + 2, b + 2, c + 2; z)and so on. Thus by repeated application of (8.22-3) we arrive at the
required formula.
(3) Again, by using the series representation for F, we have