State and prove Rodrigues’s formula.
Answers
Step-by-step explanation:
a formula relating the differential of the normal  to a surface to the differential of the radius vector  of the surface in the principal direction:

where  and  are the principal curvatures.
The formula was obtained by O. Rodrigues (1815).
A.B. Ivanov
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 [1] for the Legendre polynomials.
References
[1]O. Rodrigues, "Mémoire sur l'attraction des spheroides" Correspondence sur l'Ecole Polytechnique , 3 (1816) pp. 361–385
P.K. Suetin
Comments
For part 1) see also [a1], [a2]. For part 2) see also [a3], [a4].
References
[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)