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The Legendre differential equation isFor a nonnegative integer $\ell$, the polynomial solution normalized by $P_{\ell }(1)=1$ is the Legendre polynomial $P_{\ell }$. For example, $P_0=1$, $P_1=x$ and $P_2=(3x^2-1)/2$ satisfy their respective equations by substitution.

Its self-adjoint form is $[(1-x^2)y^{\prime }{]}^{\prime }+\ell (\ell +1)y=0$. On $(-1,1)$, regular solutions of distinct integer degrees are orthogonal with weight one; the vanishing endpoint coefficient removes the boundary term when the equations are multiplied crosswise and subtracted. The singular endpoints $\pm 1$ require care: a second independent solution generally has logarithmic endpoint behavior rather than being another regular polynomial. Rodrigues formula for Legendre polynomials constructs the normalized family explicitly.

The polynomial $v$ has leading term $x^{2l}$. Its $l$th derivative has leading term $(2l)!x^l/l!$, and all other terms have lower degree. Thus the Rodrigues formula for Legendre polynomials defines a polynomial with nonzero leading coefficientIt therefore has degree exactly $l$, including the constant case $P_0=1$.

A Rodrigues formula constructs an orthogonal polynomial by differentiating a weight multiplied by a power. The Rodrigues formula for Legendre polynomials uses a derivative of $(x^2-1{)}^l$. For Generalized Laguerre polynomials,