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# Legendre polynomials and Rodrigues' formula

Instead of solving the recursion relation (1-54) for the coefficients in the Legendre polynomials, it is easier to use the following trick:

Consider the function

 fn(x)=(x2-1)n. (20)

By differentiation we see that it satisfies the following first order differential equation,

 (1-x2)f'n+2nxfn=0. (21)

Differentiating this equation we get the second order differential eq. for fn,

 (1-x2)fn''-2(n-1)xfn'+2nfn=0. (22)

We wish to differentiate this n times by use of Leibniz's formula,

 (23)

Applying this to (22) we easily get

 (1-x2)fn(n+2)-2xfn(n+1)+n(n+1)fn(n)=0, (24)

which is exactly Lergendre's differential equation (1-49). This equation is therefore satisfied by the polynomials

 (25)

The Legendre polynomials Pn(x) are normalized by the requirement Pn(1)=1. Using

 (26)

we get

 (27)

This is Rodrigues' formula for the Legendre function. By means of the binominal formula we get

 (28)

The summation starts at r=0 and end when n-2r is 0 (n=even) or 1 (n=odd). In evaluating Pn it is most easy to use (27) directly. Thus, .

Next: The Gamma function and Up: No Title Previous: The Green's function
Mette Lund
2000-04-27