Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SPECIAL FUNCTIONS

Differentiating this formktimes with respect tohgives

∑∞

n=k

Hn

(n−k)!

hn−k=

∂kG

∂hk

=ex

2 ∂k

∂hk

e−(x−h)

2

=(−1)kex

2 ∂k

∂xk

e−(x−h)

2

.

Relabelling the summation on the LHS using the new indexm=n−k,weobtain

∑∞

m=0

Hm+k

m!

hm=(−1)kex

2 ∂k

∂xk

e−(x−h)

2

.

Settingh= 0 in this equation, we find

Hk(x)=(−1)kex

2 dk

dxk

(e−x

2

),

which is the Rodrigues’ formula (18.130) for the Hermite polynomials.

The generating function (18.133) is also useful for determining special values

of the Hermite polynomials. In particular, it is straightforward to show that

H 2 n(0) = (−1)n(2n)!/n!andH 2 n+1(0) = 0.

Recurrence relations

The two most useful recurrence relations satisfied by the Hermite polynomials

are given by

Hn+1(x)=2xHn(x)− 2 nHn− 1 (x), (18.134)

Hn′(x)=2nHn− 1 (x). (18.135)

The first relation provides a simple iterative way of evaluating thenth Hermite

polynomials at some pointx=x 0 , given the values ofH 0 (x)andH 1 (x)atthat

point. For proofs of these recurrence relations, see exercise 18.5.

18.10 Hypergeometric functions

The hypergeometric equation has the form

x(1−x)y′′+[c−(a+b+1)x]y′−aby=0, (18.136)

and has three regular singular points, atx=0, 1 ,∞, but no essential singularities.

The parametersa,bandcare given real numbers.

In our discussions of Legendre functions, associated Legendre functions and

Chebyshev functions in sections 18.1, 18.2 and 18.4, respectively, it was noted that

in each case the corresponding second-order differential equation had three regular

singular points, atx=− 1 , 1 ,∞, and no essential singularities. The hypergeometric

equation can, in fact, be considered as the ‘canonical form’ for second-order

differential equations with this number of singularities. It may be shown§that,

§See, for example, J. Mathews and R. L. Walker,Mathematical Methods of Physics, 2nd edn (Reading

MA: Addision–Wesley, 1971).