where and are the principal curvatures.
The formula was obtained by O. Rodrigues (1815).
A representation of orthogonal polynomials in terms of a weight function using differentiation. If a weight function satisfies a Pearson differential equation
and if, moreover, at the end points of the orthogonality interval the following conditions hold:
then the orthogonal polynomial can be represented by a Rodrigues formula:
where is a constant. Rodrigues' formula holds only for orthogonal polynomials and for polynomials obtained from the latter by linear transformations of the argument. Originally, this formula was established by O. Rodrigues  for the Legendre polynomials.
|||O. Rodrigues, "Mémoire sur l'attraction des spheroides" Correspondence sur l'Ecole Polytechnique , 3 (1816) pp. 361–385|
|[a1]||G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 1–4 , Chelsea, reprint (1972)|
|[a2]||M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) pp. 145|
|[a3]||G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975)|
|[a4]||T.S. Chihara, "An introduction to orthogonal polynomials" , Gordon & Breach (1978)|
Rodrigues formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rodrigues_formula&oldid=12580