Lommel polynomial

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

J m + ν ( z ) = J ν ( z ) R m , ν ( z ) − J ν − 1 ( z ) R m − 1 , ν + 1 ( z ) {\displaystyle \displaystyle J_{m+\nu }(z)=J_{\nu }(z)R_{m,\nu }(z)-J_{\nu -1}(z)R_{m-1,\nu +1}(z)} {\displaystyle \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.[1]

They are given explicitly by

R m , ν ( z ) = ∑ n = 0 [ m / 2 ] ( − 1 ) n ( m − n ) ! Γ ( ν + m − n ) n ! ( m − 2 n ) ! Γ ( ν + n ) ( z / 2 ) 2 n − m . {\displaystyle 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}.} {\displaystyle 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

  1. Eugen von Lommel (1871). "Zur Theorie der Bessel'schen Functionen". Mathematische Annalen. 4 (1). Berlin / Heidelberg: Springer: 103–116. doi:10.1007/BF01443302.