Login
Login

Lommel polynomial

{{Short description|Concept in mathematics}}

A Lommel polynomial Rm,ν(z) is a polynomial in 1/z giving the recurrence relation

:\displaystyle J_{m+\nu}(z) = J_\nu(z)R_{m,\nu}(z) - J_{\nu-1}(z)R_{m-1,\nu+1}(z)

where Jν(z) is a Bessel function of the first kind.{{cite journal |author=Eugen von Lommel |year=1871 |title=Zur Theorie der Bessel'schen Functionen |journal=Mathematische Annalen |volume=4 |issue=1 |pages=103–116 |publisher=Springer |place=Berlin / Heidelberg |doi=10.1007/BF01443302}}

They are given explicitly by

:R_{m,\nu}(z) = \sum_{n=0}^{[m/2]}\frac{(-1)^n(m-n)!\Gamma(\nu+m-n)}{n!(m-2n)!\Gamma(\nu+n)}(z/2)^{2n-m}.

See also

References

{{reflist}}

  • {{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 |mr=0058756 | year=1953 | url=http://apps.nrbook.com/bateman/Vol2.pdf}}
  • {{springer|id=l/l060810|first=A. B. |last=Ivanov}}

Category:Polynomials

Category:Special functions

{{polynomial-stub}}