The Lommel differential equation, named after Eugen von Lommel, is an inhomogeneous form of the Bessel differential equation:

z 2 d 2 y d z 2 + z d y d z + ( z 2 − ν 2 ) y = z μ + 1 . {\displaystyle z^{2}{\frac {d^{2}y}{dz^{2}}}+z{\frac {dy}{dz}}+(z^{2}-\nu ^{2})y=z^{\mu +1}.}

Solutions are given by the Lommel functions sμ,ν(z) and Sμ,ν(z), introduced by Eugen von Lommel (1880),

s μ , ν ( z ) = π 2 [ Y ν ( z ) ∫ 0 z x μ J ν ( x ) d x − J ν ( z ) ∫ 0 z x μ Y ν ( x ) d x ] , {\displaystyle s_{\mu ,\nu }(z)={\frac {\pi }{2}}\left[Y_{\nu }(z)\!\int _{0}^{z}\!\!x^{\mu }J_{\nu }(x)\,dx-J_{\nu }(z)\!\int _{0}^{z}\!\!x^{\mu }Y_{\nu }(x)\,dx\right],} S μ , ν ( z ) = s μ , ν ( z ) + 2 μ − 1 Γ ( μ + ν + 1 2 ) Γ ( μ − ν + 1 2 ) ( sin ⁡ [ ( μ − ν ) π 2 ] J ν ( z ) − cos ⁡ [ ( μ − ν ) π 2 ] Y ν ( z ) ) , {\displaystyle S_{\mu ,\nu }(z)=s_{\mu ,\nu }(z)+2^{\mu -1}\Gamma \left({\frac {\mu +\nu +1}{2}}\right)\Gamma \left({\frac {\mu -\nu +1}{2}}\right)\left(\sin \left[(\mu -\nu ){\frac {\pi }{2}}\right]J_{\nu }(z)-\cos \left[(\mu -\nu ){\frac {\pi }{2}}\right]Y_{\nu }(z)\right),}

where Jν(z) is a Bessel function of the first kind and Yν(z) a Bessel function of the second kind.

The s function can also be written as

s μ , ν ( z ) = z μ + 1 ( μ − ν + 1 ) ( μ + ν + 1 ) 1 F 2 ( 1 ; μ 2 − ν 2 + 3 2 , μ 2 + ν 2 + 3 2 ; − z 2 4 ) , {\displaystyle s_{\mu ,\nu }(z)={\frac {z^{\mu +1}}{(\mu -\nu +1)(\mu +\nu +1)}}{}_{1}F_{2}(1;{\frac {\mu }{2}}-{\frac {\nu }{2}}+{\frac {3}{2}},{\frac {\mu }{2}}+{\frac {\nu }{2}}+{\frac {3}{2}};-{\frac {z^{2}}{4}}),}

where pFq is a generalized hypergeometric function.