Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SPECIAL FUNCTIONS

where in the second equality we have used the expression (18.165) relating the gamma and
beta functions. Using the definition (18.162) of the beta function, we then find

F(a, b, c;x)=

Γ(c)

Γ(a)Γ(b)Γ(c−b)

∑∞

n=0

Γ(a+n)

xn

n!

∫ 1

0

tb+n−^1 (1−t)c−b−^1 dt

=

Γ(c)

Γ(b)Γ(c−b)

∫ 1

0

dt tb−^1 (1−t)c−b−^1

∑∞

n=0

Γ(a+n)

Γ(a)

(tx)n

n!

,

where in the second equality we have rearranged the expression and reversed the order
of integration and summation. Finally, one recognises the sum overnas being equal to
(1−tx)−a, and so we obtain the final result (18.144).

The integral representation may be used to prove a wide variety of properties of

the hypergeometric functions. As a simple example, on settingx= 1 in (18.144),

and using properties of the beta function discussed in section 18.12.2, one quickly

finds that, providedcis not a negative integer or zero andc>a+b,

F(a, b, c;1) =

Γ(c)Γ(c−a−b)

Γ(c−a)Γ(c−b)

.

Relationships between hypergeometric functions

There exist a great many relationships between hypergeometric functions with

different arguments. These are most easily derived by making use of the integral

representation (18.144) or the series form (18.141). It is not feasible to list all the

relationships here, so we simply note two useful examples, which read

F(a, b, c;x)=(1−x)c−a−bF(c−a, c−b, c;x), (18.145)

F′(a, b, c;x)=

ab

c

F(a+1,b+1,c+1;x), (18.146)

where the prime in the second relation denotesd/dx. The first result follows

straightforwardly from the integral representation using the substitutiont=

(1−u)/(1−ux), whereas the second result may be proved more easily from the

series expansion.

In addition to the above results, one may also derive relationships between

F(a, b, c;x) and any two of the six ‘contiguous functions’F(a± 1 ,b,c;x),F(a, b±

1 ,c;x)andF(a, b, c±1;x). These ‘contiguous relations’ serve as the recurrence

relations for the hypergeometric functions. An example of such a relationship is

(c−a)F(a− 1 ,b,c;x)+(2a−c−ax+bx)F(a, b, c;x)+a(x−1)F(a+1,b,c;x)=0.

Repeated application of such relationships allows one to expressF(a+l, b+m, c+

n;x), wherel, m, nare integers (withc+nnot equalling a negative integer or zero),

as a linear combination ofF(a, b, c;x) and one of its contiguous functions.