Rodrigues formula

A formula relating the differential of the normal $ \mathbf n $ to a surface to the differential of the radius vector $ \mathbf r $ of the surface in the principal direction:

$$ d \mathbf n = - k _ {1} d \mathbf r \ \ \textrm{ or } \ \ d \mathbf n = - k _ {2} d \mathbf r , $$

where $ k _ {1} $ and $ k _ {2} $ 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 $ h ( x) $ satisfies a Pearson differential equation

$$ \frac{h ^ \prime ( x) }{h ( x) } = \ \frac{p _ {0} + p _ {1} x }{q _ {0} + q _ {1} x + q _ {2} x ^ {2} } \equiv \ \frac{A ( x) }{B ( x) } ,\ \ x \in ( a , b ) , $$

and if, moreover, at the end points of the orthogonality interval the following conditions hold:

$$ \lim\limits _ {x \downarrow a } h ( x) B ( x) = \ \lim\limits _ {x \uparrow b } h ( x) B ( x) = 0 , $$

then the orthogonal polynomial $ P _ {n} ( x) $ can be represented by a Rodrigues formula:

$$ P _ {n} ( x) = \ c _ {n} \frac{[ h ( x) B ^ {n} ( x) ] ^ {(n)} }{h ( x) } , $$

where $ c _ {n} $ 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)