As an illustration of the kind of manipulations necessary to develop and work with the special functions identified previously, let's expand somewhat our discussion of Legendre's equation and Legendre polynomials. The development and manipulations of the other special functions are handled in a similar manner (especially the various polynomial relationships - Hermite and Laguerre polynomials, for example).

Solution via Power Series

Recall that Legendre’s equation is given by

Since eqn. (8.11) is analytic around x₀ = 0, we can use the standard power series method to determine y(x). For this case , let

and upon substitution of this form and its appropriate derivative relationships into the original equation, we get the recurrence relation

where a₀ and a₁ are arbitrary constants and . Therefore, we can write the solution to Legendre's equation as

where

and

These series converge for .

Standard Form for Legendre Polynomials

where

Now, instead of writing all the non-vanishing coefficients in terms of a₀ or a₁, let’s write them in terms of the coefficient of the highest power of x (i.e. a_n). In particular, choosing a_n as

gives

for all n where the domain of interest is .

To put the desired polynomials into final form, note that using eqn. (8.17) with m = n-2 gives

and, after some manipulation, this can be written as

Performing similar manipulations (i.e. some more magic) with m = n-4, eqn. (8.17) can also be written as

This procedure can be continued to develop a general relationship for, or

and

where M = n/2 or M = (n-1)/2, which whichever is an integer.

Some Low Order Legendre Polynomials

Putting specific values into eqns. (8.20) and (8.21) gives (recall that ):

Some Important Relationships

we can develop the low order polynomials as follows:

and

To illustrate the use of eqn. (8.23), let's develop an explicit expression for P₃(x). To do this we simply let n = 2 in the recurrence relationship, or

Thus,

Since P_n(x) is simply a polynomial of order n, we can easily find first or higher-order derivative information. For example, is given by

A recurrence formula, however, is very handy for computer implementation. Using eqn. (8.24), we can generate this same result with

or as before.

where is the Kronecker delta function.

Now multiply eqn. (8.28) by P_n(x) and eqn. (8.29) by P_m(x) and subtract the resultant expressions giving

but

Therefore, the above expression reduces to

Focusing again on the left hand side of this last expression, we see that

Therefore,

Finally, integrating this expression over the domain of interest gives

Note that the first part of this expression vanishes because the term evaluated at the limits goes identically to zero. Thus, since , the above expression reduces to

This is a statement of orthogonality for .

Developing a general expression for the normalization (i.e. for the case where m = n) is not very straightforward at all and there are a number of approaches that can be used (all of which are tedious). The approach chosen here starts with the Generating Function for Legendre polynomials,

Squaring both sides of eqn. (8.32) gives

Now integrating this expression gives

where the last equality is a result of the orthogonality relationship in eqn. (8.31).

Working on the left hand side of this expression, we have with

that

But for t² < 1, the term containing the natural log function can be rewritten in terms of an infinite series expansion as

Therefore, the integral becomes

Finally, we have the result

and equating like coefficients, we see that

which is the desired normalization for the orthogonality relation for Legendre polynomials when m = n.

Application Notes

The primary purpose of the above developments is simply to demonstrate several important relations for a particular set of orthogonal polynomials. Similar manipulations can be performed for the other orthogonal functions (Hermite polynomials, Laguerre polynomials, etc.) and the reader is encouraged to seek out further details as needed for a particular application. Note that the choice of the specific orthogonal polynomial for a given application is often dictated by the domain of interest.

For Legendre polynomials, for example, the functions are orthogonal over an interval and this range makes them particularly suitable for problems involving spherical coordinates. In particular, Legendre polynomials are used extensively where the directional dependence of some quantity is treated explicitly - such as particle transport problems. Often, one of the direction variables, say , ranges from 0 to (i.e. ) and a simple change of variables, , has varying between ; the domain of interest for Legendre polynomials.

For example, say some quantity, , is a function of the direction variable . Then,

and one can write in terms of Legendre polynomials, or

where, in practice, the infinite series is truncated to a finite number of terms, giving an approximate relationship for . The first part of eqn. (8.35) is just a Generalized Fourier Series (or sometimes called a Fourier-Legendre series) representation for the function . The truncation to a finite number of terms represents the usual approximation made in most practical applications.

The expansion coefficients in eqn. (8.35) can be found by multiplying both sides of the expression by and integrating to give

Finally, one simply uses the orthogonality property of the Legendre polynomials and solves for a_n, which gives

or

One computes and stores the a_n's given basic information about , and then, when needed, is reconstructed using eqn. (8.35).