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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,