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A '''Lommel polynomial''' ''R''<sub>''m'',&nu;</sub>(''z''), introduced by {{harvs|txt|authorlink=Eugen von Lommel|first=Eugen von|last= Lommel|year=1871}}, is a polynomial in 1/''z'' giving the recurrence relation
== broadly ==
:<math>\displaystyle J_{m+\nu}(z) = J_\nu(z)R_{m,\nu}(z) - J_{\nu-1}(z)R_{m-1,\nu+1}(z) </math>
where ''J''<sub>&nu;</sub>(''z'') is a [[Bessel function]] of the first kind.


They are given explicitly by
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:<math>R_{m,\nu} = \sum_{n=0}^{[m/2]}\frac{(-1)^m(m-n)!\Gamma(\nu+m-n)}{n!(m-2n)!\Gamma(\nu+n)}(z/2)^{2n-m}.</math>
 
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==See also==
== broadly ==
*[[Lommel function]]
*[[Neumann polynomial]]


==References==
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*{{Citation | last1=Erdélyi | first1=Arthur | author-link=Arthur Erdélyi | last2=Magnus | first2=Wilhelm | author2-link=Wilhelm Magnus | last3=Oberhettinger | first3=Fritz | last4=Tricomi | first4=Francesco G. | title=Higher transcendental functions. Vol II | publisher=McGraw-Hill Book Company, Inc., New York-Toronto-London | id={{MathSciNet | id = 0058756}} | year=1953 | url=http://apps.nrbook.com/bateman/Vol2.pdf}}
 
*{{springer|id=l/l060810|first=A. B. |last=Ivanov}}
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*{{citation|first=Eugen von |last=Lommel|title=Zur Theorie der Bessel'schen Functionen
 
|journal =Mathematische Annalen
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|publisher =Springer |place=Berlin / Heidelberg
 
|volume =4|issue= 1 |year= 1871
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|doi =10.1007/BF01443302
 
|pages =103–116}}
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[[Category:Polynomials]]
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