Rodrigues’ Formula

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Section 5.3 Rodrigues’ Formula

Legendre polynomials can be found from Rodrigues’ formula

\begin{equation} P_\ell(z) = \frac{1}{2^\ell \ell!}\,\frac{d^\ell}{dz^\ell}\left(z^2-1\right)^\ell\tag{5.3.1} \end{equation}

Rodrigues’ Formula can be used to generate the polynomials quickly. To do this, write

\begin{equation} \left(z^2-1\right)^\ell = (z-1)^\ell (z+1)^\ell = a^\ell b^\ell\tag{5.3.2} \end{equation}

and use the product rule

\begin{align} \frac{d^{\ell}}{dz^{\ell}}\amp\left(z^2-1\right)^\ell = \left(\frac{d^{\ell} a}{dz^{\ell}}\right) \,b + \ell \left(\frac{d^{\ell-1}a}{dz^{\ell-1}}\right) \left(\frac{db}{dz}\right)\notag\\ \amp\qquad + \frac{\ell(\ell-1)}{2!} \left(\frac{d^{\ell-2} a}{dz^{\ell-2}}\right) \left(\frac{d^2 b}{dz^2}\right) + ... + a \left(\frac{d^{\ell} b}{dz^{\ell}}\right)\tag{5.3.3} \end{align}

where the coefficients in the \(i^{\text{th}}\) term in the product is the binomial coefficient

\begin{equation} \binom{\ell}{i} = \binom{\ell}{\ell-i} = \frac{\ell!}{(\ell-i)!\,i!}\tag{5.3.4} \end{equation}

Rodrigues’ formula can be used to prove many of the properties of the Legendre polynomials found in Section 5.2. It is also useful for calculating formulas involving Legendre polynomials inside integrals.