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| {{for|the 3-dimensional rotation formula|Rodrigues' rotation formula}}
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| In mathematics, '''Rodrigues's formula''' (formerly called the '''Ivory–Jacobi formula''') is a formula for the [[Legendre polynomials]] independently introduced by {{harvs|txt|authorlink=Olinde Rodrigues|first=Olinde|last=Rodrigues|year=1816}}, {{harvs|txt|authorlink=James Ivory (mathematician)|first=Sir James|last= Ivory|year=1824}} and {{harvs|txt|authorlink=Carl Gustav Jacob Jacobi|first=Carl Gustav|last=Jacobi|year=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]]. {{harvtxt|Askey|2005}} describes the history of the Rodrigues formula in detail.
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| ==Statement==
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| Rodrigues stated his formula for [[Legendre polynomials]] <math>P_n</math>:
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| :<math>P_n(x) = {1 \over 2^n n!} {d^n \over dx^n } \left[ (x^2 -1)^n \right]. </math>
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| A similar formula holds for many other sequences of orthogonal functions arising from [[Sturm-Liouville equation]]s, and these are also called the Rodrigues formula for that case, especially when the resulting sequence is polynomial.
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| ==References==
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| *{{Citation | last1=Askey | first1=Richard |authorlink=Richard Askey| editor1-last=Altmann | editor1-first=Simón L. | editor2-last=Ortiz | editor2-first=Eduardo L. | title=Mathematics and social utopias in France: Olinde Rodrigues and his times | url=http://books.google.com/books?id=oTyJYUx8Jr4C&pg=PA105 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series= History of mathematics | isbn=978-0-8218-3860-0 | year=2005 | volume=28 | chapter=The 1839 paper on permutations: its relation to the Rodrigues formula and further developments | pages=105–118}}
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| *{{citation
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| | title = On the Figure Requisite to Maintain the Equilibrium of a Homogeneous Fluid Mass That Revolves Upon an Axis
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| | last=Ivory|first= James
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| | journal = Philosophical Transactions of the Royal Society of London
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| | volume = 114
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| |year=1824
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| | pages = 85–150
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| | jstor = 107707
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| | publisher = The Royal Society
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| }}
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| *{{Citation | last1=Jacobi | first1=C. G. J. | title=Ueber eine besondere Gattung algebraischer Functionen, die aus der Entwicklung der Function (1 − 2''xz'' + ''z''<sup>2</sup>)<sup>1/2</sup> entstehen. | language=German | doi=10.1515/crll.1827.2.223 | year=1827 | journal=Journal für Reine und Angewandte Mathematik | issn=0075-4102 | volume=2 | pages=223–226}}
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| *{{MacTutor|id=Rodrigues|title=Olinde Rodrigues}}
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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 = http://books.google.fr/books?id=dp4AAAAAYAAJ&hl=fr&pg=PA361#v=onepage&q&f=false}}
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| [[Category:Orthogonal polynomials]]
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