Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SPECIAL FUNCTIONS

gamma function.§It is straightforward to show that the hypergeometric series

converges in the range|x|<1. It also converges atx=1ifc>a+band

atx=−1ifc>a+b−1. We also note thatF(a, b, c;x) is symmetric in the

parametersaandb,i.e.F(a, b, c;x)=F(b, a, c;x).

The hypergeometric functiony(x)=F(a, b, c;x) is clearly not the general

solution to the hypergeometric equation (18.136), since we must also consider the

second root of the indicial equation. Substitutingσ=1−cinto (18.138) and

demanding that the coefficient ofxnvanishes, we find that we must have

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

which, on comparing with (18.139) and replacingnbyn+ 1, yields the recurrence

relation

an+1=

(a−c+1+n)(b−c+1+n)

(n+ 1)(2−c+n)

an.

We see that this recurrence relation has the same form as (18.140) if one makes

the replacementsa→a−c+1,b→b−c+ 1 andc→ 2 −c. Thus, providedc,

a−bandc−a−bare all non-integers, the general solution to the hypergeometric

equation, valid for|x|<1, may be written as

y(x)=AF(a, b, c;x)+Bx^1 −cF(a−c+1,b−c+1, 2 −c;x),

(18.143)

whereAandBare arbitrary constants to be fixed by the boundary conditions on

the solution. If the solution is to be regular atx=0,onerequiresB=0.

18.10.1 Properties of hypergeometric functions

Since the hypergeometric equation is so general in nature, it is not feasible to

present a comprehensive account of the hypergeometric functions. Nevertheless,

we outline here some of their most important properties.

Special cases

As mentioned above, the general nature of the hypergeometric equation allows us

to write a large number of elementary functions in terms of the hypergeometric

functionsF(a, b, c;x). Such identifications can be made from the series expansion

(18.142) directly, or by transformation of the hypergeometric equation into a more

familiar equation, the solutions to which are already known. Some particular

examples of well known special cases of the hypergeometric function are as

follows:

§We note that it is also common to denote the hypergeometric function by 2 F 1 (a, b, c;x). This

slightly odd-looking notation is meant to signify that, in the coefficient of each power ofx,there

are two parameters (aandb) in the numerator and one parameter (c) in the denominator.