Rodrigues' formula

Formula for the Legendre polynomials

In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) generates the Legendre polynomials. It was independently introduced by Olinde Rodrigues (1816), Sir James Ivory (1824) and Carl Gustav Jacobi (1827). 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. Askey (2005) describes the history of the Rodrigues formula in detail.

Statement

Let ( P n ( x ) ) n = 0 ∞ {\displaystyle (P_{n}(x))_{n=0}^{\infty }} be a sequence of orthogonal polynomials defined on the interval [ a , b ] {\displaystyle [a,b]} satisfying the orthogonality condition

∫ a b P m ( x ) P n ( x ) w ( x ) d x = K n δ m , n , {\displaystyle \int _{a}^{b}P_{m}(x)P_{n}(x)w(x)\,dx=K_{n}\delta _{m,n},}
where w ( x ) {\displaystyle w(x)} is a suitable weight function, K n {\displaystyle K_{n}} is a constant depending on n {\displaystyle n} , and δ m , n {\displaystyle \delta _{m,n}} is the Kronecker delta. If the weight function w ( x ) {\displaystyle w(x)} satisfies the following differential equation (called Pearson's differential equation),
w ′ ( x ) w ( x ) = A ( x ) B ( x ) , {\displaystyle {\frac {w'(x)}{w(x)}}={\frac {A(x)}{B(x)}},}
where A ( x ) {\displaystyle A(x)} is a polynomial with degree at most 1 and B ( x ) {\displaystyle B(x)} is a polynomial with degree at most 2 and, further, the limits
lim x → a w ( x ) B ( x ) = 0 , lim x → b w ( x ) B ( x ) = 0. {\displaystyle \lim _{x\to a}w(x)B(x)=0,\qquad \lim _{x\to b}w(x)B(x)=0.}
Then it can be shown that P n ( x ) {\displaystyle P_{n}(x)} satisfies a relation of the form,
P n ( x ) = c n w ( x ) d n d x n [ B ( x ) n w ( x ) ] , {\displaystyle P_{n}(x)={\frac {c_{n}}{w(x)}}{\frac {d^{n}}{dx^{n}}}\!\left[B(x)^{n}w(x)\right],}
for some constants c n {\displaystyle c_{n}} . 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 {\displaystyle P_{n}} :

P n ( x ) = 1 2 n n ! d n d x n [ ( x 2 − 1 ) n ] . {\displaystyle P_{n}(x)={\frac {1}{2^{n}n!}}{\frac {d^{n}}{dx^{n}}}\!\left[(x^{2}-1)^{n}\right]\!.}

Laguerre polynomials are usually denoted L0, L1, ..., and the Rodrigues formula can be written as

L n ( x ) = e x n ! d n d x n [ e − x x n ] = 1 n ! ( d d x − 1 ) n x n , {\displaystyle L_{n}(x)={\frac {e^{x}}{n!}}{\frac {d^{n}}{dx^{n}}}\!\left[e^{-x}x^{n}\right]={\frac {1}{n!}}\left({\frac {d}{dx}}-1\right)^{n}x^{n},}

The Rodrigues formula for the Hermite polynomials can be written as

H n ( x ) = ( − 1 ) n e x 2 d n d x n [ e − x 2 ] = ( 2 x − d d x ) n ⋅ 1. {\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}\!\left[e^{-x^{2}}\right]=\left(2x-{\frac {d}{dx}}\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

  1. ^ "Rodrigues formula – Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2018-04-18.
  • Askey, Richard (2005), "The 1839 paper on permutations: its relation to the Rodrigues formula and further developments", in Altmann, Simón L.; Ortiz, Eduardo L. (eds.), Mathematics and social utopias in France: Olinde Rodrigues and his times, History of mathematics, vol. 28, Providence, R.I.: American Mathematical Society, pp. 105–118, ISBN 978-0-8218-3860-0
  • Ivory, James (1824), "On the Figure Requisite to Maintain the Equilibrium of a Homogeneous Fluid Mass That Revolves Upon an Axis", Philosophical Transactions of the Royal Society of London, 114, The Royal Society: 85–150, doi:10.1098/rstl.1824.0008, JSTOR 107707
  • Jacobi, C. G. J. (1827), "Ueber eine besondere Gattung algebraischer Functionen, die aus der Entwicklung der Function (1 − 2xz + z2)1/2 entstehen.", Journal für die Reine und Angewandte Mathematik (in German), 2: 223–226, doi:10.1515/crll.1827.2.223, ISSN 0075-4102, S2CID 120291793
  • O'Connor, John J.; Robertson, Edmund F., "Olinde Rodrigues", MacTutor History of Mathematics Archive, University of St Andrews
  • Rodrigues, Olinde (1816), "De l'attraction des sphéroïdes", Correspondence sur l'École Impériale Polytechnique, (Thesis for the Faculty of Science of the University of Paris), 3 (3): 361–385