Recursive Definition
Expanding the Taylor series in equation (1) for the first two terms gives
for the first two Legendre Polynomials. To obtain further terms without resorting to direct expansion of the Taylor series, equation (1) is differentiated with respect to t on both sides and rearranged to obtain
Replacing the quotient of the square root with its definition in (1), and equating the coefficients of powers of t in the resulting expansion gives Bonnet’s recursion formula
This relation, along with the first two polynomials P0 and P1, allows the Legendre Polynomials to be generated recursively.
Explicit representations include
where the latter, which is immediate from the recursion formula, expresses the Legendre polynomials by simple monomials and involves the multiplicative formula of the binomial coefficient.
The first few Legendre polynomials are:
| n | |
| 0 | |
| 1 | |
| 2 | |
| 3 | |
| 4 | |
| 5 | |
| 6 | |
| 7 | |
| 8 | |
| 9 | |
| 10 |
The graphs of these polynomials (up to n = 5) are shown below:
Read more about this topic: Legendre Polynomials
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