# 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