Rodrigues' formula

Let ${\displaystyle \{P_{n}(x)\}_{n=0}^{\infty }}$ be a sequence of orthogonal polynomials satisfying the orthogonality condition

${\displaystyle \int _{a}^{b}P_{m}(x)P_{n}(x)w(x)dx=K_{m,n}\delta _{m,n},}$

where, ${\displaystyle w(x)}$ is a suitable weight function, ${\displaystyle K_{m,n}}$ are constants and ${\displaystyle \delta _{m,n}}$ is the Kronecker delta. If the weight function ${\displaystyle w(x)}$ satisfies the following differential equation (called Pearson's differential equation),

${\displaystyle {\frac {w'(x)}{w(x)}}={\frac {A(x)}{B(x)}},}$

where ${\displaystyle A(x)}$ is a polynomial with degree at most 1 and ${\displaystyle B(x)}$ is a polynomial with degree at most 2 and, further, the limits

${\displaystyle \lim _{x\rightarrow a}w(x)B(x)=0,\qquad \lim _{x\rightarrow b}w(x)B(x)=0,}$

then, it can be shown that ${\displaystyle P_{n}(x)}$ satisfies a recurrence relation of the form,

${\displaystyle P_{n}(x)={\frac {c_{n}}{w(x)}}{\frac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}\left[B(x)^{n}w(x)\right],}$

for some constants ${\displaystyle c_{n}}$. 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 ${\displaystyle P_{n}}$:

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

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

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

The Rodrigues formula for the Hermite polynomial can be written as

${\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}e^{-x^{2}}=\left(2x-{\frac {d}{dx}}\right)^{n}\cdot 1}$.

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.

• 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, 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, The Royal Society, 114: 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