1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions

= aF( a+ 1, b, c; z) - bF( a, b + 1, c; z)

since (a+ n)(a)n = a(a + l)n·

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Any one of the hypergeometric functions F( a+ 1, b, c; z ), F( a, b + 1, c; z)
and F(a,b,c+ 1,c;z) is said to be contiguous to F(a,b,c;z). Between F
and any two of the contiguous functions there is a relation similar to the
one proved above, with coefficients that are either constants or linear in
z. The number of such relations is (~) = 15. The reader will find the
remaining 14 contiguous relations listed in Exercises 8.10, problem 4.
(4) We have
a(c - b)
F(a,b+ l,c+ 1; z)- c(c+ l) zF(a + 1,b + 1,c + 2;z)

= l + f (a)n(b + l)n Zn_ a(c - b) [z + f (a+ l)n(b + l)n zn+ll

n=l n!(c+l)n c(c+l) n=l n!(c+2)n

(^1) [

a(b+l) a(c-b)] ~ (a)n+1(b+l)n b(c+n+l) n+l

= + - z+L,; z

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

= l + ab + ~ (a)n+1(b)n+i n+i = F( b. )

z L,; ( l)'( ) z a, , c, z

c n=l n +. c n+l

(5) To show that the hypergeometric function is a particular solution of
the second-order differential equation

(z - z^2 )w" + (c -(a+ b + l)z]w' -abw = 0 (8.22-4)

let

w = F(a b c· z) = ~ (a)n(b)n Zn

, , ' n=O L,; n. '( ) c n

Then, using for convenience the differential operator () = z( d/ dz), and

noting that (0 + k)zn = (n + k)zn (k any constant), we get

()(() + c _ l) w = ~ L,; n(n + c -'( ) l)(a)n(b)n z n

n=O n. C n

~ (a)n(b)n n

= ~ (n - l)!(c)n-1 z

= ~ (a)n+i(b)n+l zn+l

~ n!(c)n