Lommel Polynomial

Author

Eric W. Weisstein
January 13, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/LommelPolynomial.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LommelPolynomial.html.

©2023 Wolfram Research, Inc. except for portions noted otherwise

Definition

LommelR1[m_,n_,z_]:=((Gamma[n+m])/(Gamma[n](((z/2))^m)))HypergeometricPFQ[{(1-m)/2,-m/2},{n,-m,1-n-m},-(z^2)]

FullSimplify[LommelR1[m,n,z]]

LommelR2[m_,n_,z_]:=((π z)/(2Sin[n π]))(BesselJ[n+m,z]BesselJ[-n+1,z]+(((-1))^m)BesselJ[-n-m,z]BesselJ[n-1,z])

Checks

Table[
With[{
m=Random[Integer,{0,5}],
n=Random[Complex,5I{-1,1},30],
z=Random[Complex,5I{-1,1},30]
},
LommelR1[m,n,z]-LommelR2[m,n,z]
],
{1000}]//Chop//Union//Timing

Table

Table[Subscript[R,1][m,ν][z]==FullSimplify@LommelR1[m,ν,z],{m,0,5}]//Column//TraditionalForm

Table[Subscript[R,2][m,ν][z]==FullSimplify@LommelR2[m,ν,z],{m,0,5}]//Column//TraditionalForm

Recurrence relations

(Subscript[J, m+ν])(z)==(Subscript[J, ν])(z) (Subscript[R, m,ν])(z)-(Subscript[J, m+ν])(z) (Subscript[R, m-1,ν+1])(z)

BesselJ[m+ν,z]==-BesselJ[m+ν,z] (Subscript[R, -1+m,1+ν])[z]+BesselJ[ν,z] (Subscript[R, m,ν])[z]/.m->1

for m==1, this equation turns into the usual recurrence formula for Bessel functions, hence