1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 643

a~ (a+l)n n a (

= - L...J I( l) z =-cl> a+l,c+l;z)

c n=O n. c + n c.

Hence after m differentiations we have

~: «P(a, c; z) = ~:j: «P(a + m, c + m; z); m=l,2,. ..

(2) Starting with the right-hand side, we find that

c""' ""'( a - 1, c; z ) + Z'±' ""'( a, c + 1; z ) = c + L...J ~ '( (a -^1 ) )n z n

n=l n. c + 1 n-1

+ f (a)n zn+l

n=O n!( c + 1 )n

~ (a-1)

= c+ LJ n Zn

n=l n!(c + l)n-1

~ (a)n-1 n

+ LJ z

n=l (n - l)!(c + l)n-1

~ c(a)n n ( )

= c + L...J -i--( ) z = eel> a, c; z

n=1 n. c n

This is one of the (~) = 6 relations connecting cl> with any two of the

contiguous functions «P(a ± 1, c; z) and «P(a, c ± l; z). The other five are

listed in Exercises 8.10, problem 11.
(3) By (8.22-7) we have, mutatis mutandis,

(a)n =. r(c) 1

1

ta+n-1(1-t)c-a-1 dt

(c)n r(a)r(c - a) 0

(8.23-2)

provided that Rec> Rea> 0, n = 0, 1, 2, ... By using (8.23-2) in (8.23-1)

we get
00 1.

cI>(a c· z) = r(c) " zn 1 ta+n-1(1 -t)c-a-1 dt

' ' r( a )r( c - a) ~ nl 0

= r(c) 11 ta-1(1-ty-a-1 f (zt)n dt

r(a)r(c - a) o n=O n!

(8.23-3)