Rodrigues' formula explained

In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) generates the Legendre polynomials. It was 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

be a sequence of orthogonal polynomials defined on the interval satisfying the orthogonality condition $$\backslash int\_a^b\; P\_m(x)\; P\_n(x)\; w(x)\; \backslash ,\; dx\; =\; K\_n\; \backslash delta\_,$$where is a suitable weight function, is a constant depending on , and is the Kronecker delta. If the weight function satisfies the following differential equation (called Pearson's differential equation),$$\backslash frac\; =\; \backslash frac,$$where is a polynomial with degree at most 1 and is a polynomial with degree at most 2 and, further, the limits$$\backslash lim\_\; w(x)\; B(x)\; =\; 0,\; \backslash qquad\; \backslash lim\_\; w(x)\; B(x)\; =\; 0.$$Then it can be shown that satisfies a relation of the form,$$P\_n(x)\; =\; \backslash frac\; \backslash frac\; \backslash !\backslash left[B(x)^n\; w(x)\backslash right],$$for some constants . This relation is called Rodrigues' type formula, or just Rodrigues' formula.^[1]

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

Rodrigues stated his formula for Legendre polynomials

:$$P\_n(x)\; =\; \backslash frac\; \backslash frac\; \backslash !\backslash left[(x^2\; -1)^n\; \backslash right]\backslash !.$$

Laguerre polynomials are usually denoted L₀, L₁, ..., and the Rodrigues formula can be written as$$L\_n(x)\; =\; \backslash frac\; \backslash frac\; \backslash !\backslash left[e^\{-x\}\; x^n\backslash right]\; =\; \backslash frac\; \backslash left(\backslash frac\; -1\; \backslash right)\; ^n\; x^n,$$

The Rodrigues formula for the Hermite polynomials can be written as$$H\_n(x)=(-1)^n\; e^\; \backslash frac\; \backslash !\backslash left[e^\{-x^2\}\backslash right]\; =\; \backslash left(2x-\backslash frac\; \backslash right)^n\backslash cdot\; 1.$$

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.

Notes and References

1. Web site: Rodrigues formula – Encyclopedia of Mathematics . www.encyclopediaofmath.org. en. 2018-04-18.