11.2 Properties of Legendre
polynomials
11.2.1 Generating function
Let
be a function of the two variables
and
that can be expressed as a Taylor’s series in
,
. The function
is then called a generating function of the functions
.
Example
11.1:
-
Show that
is a generating function of the polynomials
.
Solution:
-
Look at
|
(11.16) |
Example
11.2:
-
Show that
is the generating function for the Bessel
functions,
|
(11.17) |
Example
11.3:
-
(The case of most interest here)
|
(11.18) |
11.2.2 Rodrigues’ Formula
|
(11.19) |
11.2.3 A table of properties
-
is even or odd if
is even or odd.
-
.
-
.
-
.
-
.
-
.
Let us prove some of these relations,
first Rodrigues’ formula. We start from the simple
formula
|
(11.20) |
which is easily proven by explicit
differentiation. This is then differentiated
times,
We have thus proven that
satisfies Legendre’s equation. The
normalisation follows from the evaluation of the highest
coefficient,
|
(11.22) |
and thus we need to multiply the derivative
with
to get the properly normalised
.
Let’s use the generating function to
prove some of the other properties: 2.:
|
(11.23) |
has all coefficients one, so
. Similarly for 3.:
|
(11.24) |
Property 5. can be found by
differentiating the generating function with respect to
:
Equating terms with identical powers of
we find
|
(11.26) |
Proofs for the other properties can be found
using similar methods.