Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.10 HYPERGEOMETRIC FUNCTIONS

by making appropriate changes of the independent and dependent variables,

any second-order differential equation with three regular singularities and an

ordinary point at infinity can be transformed into the hypergeometric equation

(18.136) with the singularities at =−1, 1 and∞. As we discuss below, this allows

Legendre functions, associated Legendre functions and Chebyshev functions, for

example, to be written as particular cases ofhypergeometric functions, which are

the solutions to (18.136).

Since the pointx= 0 is a regular singularity of (18.136), we may find at least

one solution in the form of a Frobenius series (see section 16.3):

y(x)=

∑∞

n=0

anxn+σ. (18.137)

Substituting this series into (18.136) and dividing through byxσ−^1 ,weobtain

∑∞

n=0

{(1−x)(n+σ)(n+σ−1) + [c−(a+b+1)x](n+σ)−abx}anxn=0.

(18.138)

Settingx= 0, so that only then= 0 term remains, we obtain the indicial equation

σ(σ−1) +cσ= 0, which has the rootsσ= 0 andσ=1−c. Thus, provided

cis not an integer, one can obtain two linearly independent solutions of the

hypergeometric equation in the form (18.137).

Forσ= 0 the corresponding solution is a simple power series. Substituting

σ= 0 into (18.138) and demanding that the coefficient ofxnvanishes, we find the

recurrence relation

n[(n−1) +c]an−[(n−1)(a+b+n−1) +ab]an− 1 =0,

(18.139)

which, on simplifying and replacingnbyn+ 1, yields the recurrence relation

an+1=

(a+n)(b+n)

(n+1)(c+n)

an. (18.140)

It is conventional to make the simple choicea 0 = 1. Thus, providedcis not a

negative integer or zero, we may write the solution as follows:

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

ab

c

x

1!

+

a(a+1)b(b+1)

c(c+1)

x^2

2!

+··· (18.141)

=

Γ(c)

Γ(a)Γ(b)

∑∞

n=0

Γ(a+n)Γ(b+n)

Γ(c+n)

xn

n!

, (18.142)

whereF(a, b, c;x) is known as thehypergeometric functionorhypergeometric

series, and in the second equality we have used the property (18.154) of the