# 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  for the Legendre polynomials.

How to Cite This Entry:
Rodrigues formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rodrigues_formula&oldid=51224
This article was adapted from an original article by A.B. Ivanov, P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article