Rodrigues' Formula

In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) is a formula for the Legendre polynomials independently introduced by , and . The name "Rodrigues formula" was introduced by Heine in 1878, after Hermite pointed out in 1865 that Rodrigues was the first to discover it. The term is also used to describe similar formulas for other orthogonal polynomials. describes the history of the Rodrigues formula in detail.

Statement

Let $\_^\infty$ be a sequence of orthogonal polynomials satisfying the orthogonality condition
$\int_a^b P_m(x) P_n(x) w(x) \, dx = K_n \delta_,$
where $w(x)$ is a suitable weight function, $K_n$ is a constant depending on $n$, and $\delta_$ is the Kronecker delta. If the weight function $w(x)$ satisfies the following differential equation (called Pearson's differential equation),
$\frac = \frac,$
where $A(x)$ is a polynomial with degree at most 1 and $B(x)$ is a polynomial with degree at most 2 and, further, the limits
$\lim_ w(x) B(x) = 0, \qquad \lim_ w(x) B(x) = 0,$
then, it can be shown that $P_n(x)$ satisfies a recurrence relation of the form,
P_n(x) = \frac \frac\left B(x)^n w(x)\right
for some constants $c_n$. This relation is called ''Rodrigues' type formula'', or just ''Rodrigues' formula''.

The most known applications of Rodrigues' type formulas are the formulas for Legendre, Laguerre and Hermite polynomials:

Rodrigues stated his formula for Legendre polynomials $P_n$:
P_n(x) = \frac \frac \left (x^2 -1)^n \right

Laguerre polynomials are usually denoted ''L''₀, ''L''₁, ..., and the Rodrigues formula can be written as
$L_n(x) = \frac\frac\left(e^ x^n\right) = \frac \left( \frac -1 \right) ^n x^n,$

The Rodrigues formula for the Hermite polynomial can be written as
$H_n(x)=(-1)^n e^\frace^=\left (2x-\frac \right )^n \cdot 1 .$

Similar formulae hold for many other sequences of orthogonal functions arising from Sturm–Liouville equations, and these are also called the Rodrigues formula (or Rodrigues' type formula) for that case, especially when the resulting sequence is polynomial.

References

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*{{citation, first=Olinde, last= Rodrigues, authorlink=Olinde Rodrigues, series=(Thesis for the Faculty of Science of the University of Paris), title=De l'attraction des sphéroïdes, journal=Correspondence sur l'École Impériale Polytechnique, volume=3, issue=3, year=1816, pages= 361–385, url = https://books.google.com/books?id=dp4AAAAAYAAJ&pg=PA361

Orthogonal polynomials