Legendre Polynomials - Recursive Definition

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:

The graphs of these polynomials (up to n = 5) are shown below: