The LommelS1(mu, nu, z) function is defined in terms of the hypergeometric function

FunctionAdvisor( definition, LommelS1);

$\left[\mathrm{LommelS1}\left(a\,b\,z\right)=\frac{z^{a+1}\mathrm{hypergeom}\left(\left[1\right]\,\left[\frac{3}{2}+\frac{b}{2}+\frac{a}{2}\,\frac{3}{2}-\frac{b}{2}+\frac{a}{2}\right]\,-\frac{z^2}{4}\right)}{\left(a-b+1\right)\left(a+b+1\right)}\,-a+b-1\ne 0\wedge a+b+1\ne 0\wedge \left(\frac{3}{2}-\frac{b}{2}+\frac{a}{2}\right)::\left(\neg {\mathbb{Z}}^{\left(0\,-\right)}\right)\wedge \left(\frac{3}{2}+\frac{b}{2}+\frac{a}{2}\right)::\left(\neg {\mathbb{Z}}^{\left(0\,-\right)}\right)\right]$

and LommelS2(mu, nu, z) is defined in terms of LommelS1(mu, nu, z) and Bessel functions.

LommelS2(mu,nu,z) = convert(LommelS2(mu,nu,z), LommelS1);

$\mathrm{LommelS2}\left(\mathrm{\mu }\,\mathrm{\nu }\,z\right)=\mathrm{LommelS1}\left(\mathrm{\mu }\,\mathrm{\nu }\,z\right)+2^{\mathrm{\mu }-1}\mathrm{\Gamma }\left(\frac{\mathrm{\mu }}{2}-\frac{\mathrm{\nu }}{2}+\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{\mathrm{\mu }}{2}+\frac{\mathrm{\nu }}{2}+\frac{1}{2}\right)\left(\sin \left(\frac{\left(\mathrm{\mu }-\mathrm{\nu }\right)\mathrm{\pi }}{2}\right)\mathrm{BesselJ}\left(\mathrm{\nu }\,z\right)-\cos \left(\frac{\left(\mathrm{\mu }-\mathrm{\nu }\right)\mathrm{\pi }}{2}\right)\mathrm{BesselY}\left(\mathrm{\nu }\,z\right)\right)$

These functions solve the non-homogeneous linear differential equation of second order.

z^2*diff(f(z),`$`(z,2))+z*diff(f(z),z)+(z^2-nu^2)*f(z) = z^(mu+1);

$z^2\left(\frac{{\ⅆ}^2}{\ⅆz^2}f\left(z\right)\right)+z\left(\frac{\ⅆ}{\ⅆz}f\left(z\right)\right)+\left(-{\mathrm{\nu }}^2+z^2\right)f\left(z\right)=z^{\mathrm{\mu }+1}$

The Lommel functions also solve the following third order linear homogeneous differential equation with polynomial coefficients.

FunctionAdvisor( DE, LommelS1(mu,nu,z));

$\left[f\left(z\right)=\mathrm{LommelS1}\left(\mathrm{\mu }\,\mathrm{\nu }\,z\right)\,\left[\frac{{\ⅆ}^3}{\ⅆz^3}f\left(z\right)=\frac{\left(\mathrm{\mu }-2\right)\left(\frac{{\ⅆ}^2}{\ⅆz^2}f\left(z\right)\right)}{z}+\frac{\left({\mathrm{\nu }}^2-z^2+\mathrm{\mu }\right)\left(\frac{\ⅆ}{\ⅆz}f\left(z\right)\right)}{z^2}+\frac{\left(\left(\mathrm{\mu }-1\right)z^2-{\mathrm{\nu }}^2\left(\mathrm{\mu }+1\right)\right)f\left(z\right)}{z^3}\right]\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{relate}\,\mathrm{AngerJ}\,\mathrm{LommelS1}\right)$

$\mathrm{AngerJ}\left(a\,z\right)=\frac{\sin \left(a\mathrm{\pi }\right)\left(\mathrm{LommelS1}\left(0\,a\,z\right)-a\mathrm{LommelS1}\left(\mathrm{-1}\,a\,z\right)\right)}{\mathrm{\pi }}$

$\mathrm{FunctionAdvisor}\left(\mathrm{relate}\,\mathrm{WeberE}\,\mathrm{LommelS1}\right)$

$\mathrm{WeberE}\left(a\,z\right)=\frac{-a\left(1-\cos \left(a\mathrm{\pi }\right)\right)\mathrm{LommelS1}\left(\mathrm{-1}\,a\,z\right)+\left(-1-\cos \left(a\mathrm{\pi }\right)\right)\mathrm{LommelS1}\left(0\,a\,z\right)}{\mathrm{\pi }}$

$\mathrm{FunctionAdvisor}\left(\mathrm{relate}\,\mathrm{StruveH}\,\mathrm{LommelS1}\right)$

$\mathrm{StruveH}\left(a\,z\right)=\frac{2\mathrm{LommelS1}\left(a\,a\,z\right)}{\mathrm{\Gamma }\left(a+\frac{1}{2}\right)\sqrt{\mathrm{\pi }}{2}^a}$

$\mathrm{FunctionAdvisor}\left(\mathrm{relate}\,\mathrm{StruveL}\,\mathrm{LommelS1}\right)$

$\mathrm{StruveL}\left(a\,z\right)=\frac{-2\mathrm{I}\mathrm{LommelS1}\left(a\,a\,\mathrm{I}z\right)z^a}{\mathrm{\Gamma }\left(a+\frac{1}{2}\right)\sqrt{\mathrm{\pi }}{\left(2\mathrm{I}z\right)}^a}$

$\mathrm{LommelS1}\left(\mathrm{\mu }\,\mathrm{\nu }\,z\right)=\mathrm{convert}\left(\mathrm{LommelS1}\left(\mathrm{\mu }\,\mathrm{\nu }\,z\right)\,\mathrm{MeijerG}\right)$

$\mathrm{LommelS1}\left(\mathrm{\mu }\,\mathrm{\nu }\,z\right)=2^{\mathrm{\mu }-1}\mathrm{\Gamma }\left(\frac{\mathrm{\mu }}{2}+\frac{\mathrm{\nu }}{2}+\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{\mathrm{\mu }}{2}-\frac{\mathrm{\nu }}{2}+\frac{1}{2}\right)\mathrm{MeijerG}\left(\left[\left[\frac{\mathrm{\mu }}{2}+\frac{1}{2}\right]\,\left[\right]\right]\,\left[\left[\frac{\mathrm{\mu }}{2}+\frac{1}{2}\right]\,\left[\frac{\mathrm{\nu }}{2}\,-\frac{\mathrm{\nu }}{2}\right]\right]\,\frac{z^2}{4}\right)$

$\mathrm{LommelS2}\left(\mathrm{\mu }\,\mathrm{\nu }\,z\right)=\mathrm{convert}\left(\mathrm{LommelS2}\left(\mathrm{\mu }\,\mathrm{\nu }\,z\right)\,\mathrm{MeijerG}\right)$

$\mathrm{LommelS2}\left(\mathrm{\mu }\,\mathrm{\nu }\,z\right)=\frac{\mathrm{MeijerG}\left(\left[\left[\frac{\mathrm{\mu }}{2}+\frac{1}{2}\right]\,\left[\right]\right]\,\left[\left[\frac{\mathrm{\mu }}{2}+\frac{1}{2}\,\frac{\mathrm{\nu }}{2}\,-\frac{\mathrm{\nu }}{2}\right]\,\left[\right]\right]\,\frac{z^2}{4}\right)2^{\mathrm{\mu }}}{2\mathrm{\Gamma }\left(-\frac{\mathrm{\mu }}{2}+\frac{\mathrm{\nu }}{2}+\frac{1}{2}\right)\mathrm{\Gamma }\left(-\frac{\mathrm{\mu }}{2}-\frac{\mathrm{\nu }}{2}+\frac{1}{2}\right)}$

$\mathrm{with}\left(\mathrm{MathematicalFunctions}\,\mathrm{Series}\right)$

$\left[\mathrm{Series}\right]$

$\mathrm{Series}\left(\mathrm{LommelS1}\left(\mathrm{\mu }\,\mathrm{\nu }\,z\right)\,z\,4\right)$

$z^{\mathrm{\mu }}\left(\frac{1}{\left(1+\mathrm{\mu }-\mathrm{\nu }\right)\left(1+\mathrm{\mu }+\mathrm{\nu }\right)}z-\frac{1}{\left(1+\mathrm{\mu }-\mathrm{\nu }\right)\left(\mathrm{\mu }-\mathrm{\nu }+3\right)\left(1+\mathrm{\mu }+\mathrm{\nu }\right)\left(\mathrm{\mu }+\mathrm{\nu }+3\right)}{z}^3+O\left(z^5\right)\right),\left(\mathrm{\mu }+\mathrm{\nu }\right)::\left(\neg \left({\mathbb{Z}}^{-}\wedge \mathrm{odd}\right)\right)\wedge \left(\mathrm{\mu }-\mathrm{\nu }\right)::\left(\neg \left({\mathbb{Z}}^{-}\wedge \mathrm{odd}\right)\right)$

$\mathrm{Series}\left(\mathrm{LommelS2}\left(\mathrm{\mu }\,\mathrm{\nu }\,z\right)\,z\,4\right)$

$z^{-\mathrm{\nu }}\left(\frac{4^{\mathrm{\nu }}\mathrm{\Gamma }\left(\frac{\mathrm{\mu }}{2}-\frac{\mathrm{\nu }}{2}+\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{\mathrm{\mu }}{2}+\frac{\mathrm{\nu }}{2}+\frac{1}{2}\right)\csc \left(\mathrm{\pi }\mathrm{\nu }\right)\cos \left(\frac{\left(\mathrm{\mu }-\mathrm{\nu }\right)\mathrm{\pi }}{2}\right)2^{\mathrm{\mu }-\mathrm{\nu }}}{2\mathrm{\Gamma }\left(-\mathrm{\nu }+1\right)}-\frac{1}{8}\frac{4^{\mathrm{\nu }}\mathrm{\Gamma }\left(\frac{\mathrm{\mu }}{2}-\frac{\mathrm{\nu }}{2}+\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{\mathrm{\mu }}{2}+\frac{\mathrm{\nu }}{2}+\frac{1}{2}\right)\csc \left(\mathrm{\pi }\mathrm{\nu }\right)\cos \left(\frac{\left(\mathrm{\mu }-\mathrm{\nu }\right)\mathrm{\pi }}{2}\right)2^{\mathrm{\mu }-\mathrm{\nu }}}{\mathrm{\Gamma }\left(-\mathrm{\nu }+2\right)}{z}^2+O\left(z^4\right)\right)+z^{\mathrm{\nu }}\left(-\frac{\mathrm{\Gamma }\left(\frac{\mathrm{\mu }}{2}-\frac{\mathrm{\nu }}{2}+\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{\mathrm{\mu }}{2}+\frac{\mathrm{\nu }}{2}+\frac{1}{2}\right)\csc \left(\mathrm{\pi }\mathrm{\nu }\right)\cos \left(\frac{\mathrm{\pi }\left(\mathrm{\mu }+\mathrm{\nu }\right)}{2}\right)2^{\mathrm{\mu }-\mathrm{\nu }}}{2\mathrm{\Gamma }\left(\mathrm{\nu }+1\right)}+\frac{1}{8}\frac{\mathrm{\Gamma }\left(\frac{\mathrm{\mu }}{2}-\frac{\mathrm{\nu }}{2}+\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{\mathrm{\mu }}{2}+\frac{\mathrm{\nu }}{2}+\frac{1}{2}\right)\csc \left(\mathrm{\pi }\mathrm{\nu }\right)\cos \left(\frac{\mathrm{\pi }\left(\mathrm{\mu }+\mathrm{\nu }\right)}{2}\right)2^{\mathrm{\mu }-\mathrm{\nu }}}{\mathrm{\Gamma }\left(\mathrm{\nu }+2\right)}{z}^2+O\left(z^4\right)\right)+z^{\mathrm{\mu }}\left(\frac{1}{\left(1+\mathrm{\mu }-\mathrm{\nu }\right)\left(1+\mathrm{\mu }+\mathrm{\nu }\right)}z-\frac{1}{\left(1+\mathrm{\mu }-\mathrm{\nu }\right)\left(\mathrm{\mu }-\mathrm{\nu }+3\right)\left(1+\mathrm{\mu }+\mathrm{\nu }\right)\left(\mathrm{\mu }+\mathrm{\nu }+3\right)}{z}^3+O\left(z^5\right)\right),\mathrm{\nu }::\left(\neg \mathbb{Z}\right)\wedge \left(\frac{\mathrm{\mu }}{2}-\frac{\mathrm{\nu }}{2}+\frac{1}{2}\right)::\left(\neg {\mathbb{Z}}^{\left(0\,-\right)}\right)\wedge \left(\frac{\mathrm{\mu }}{2}+\frac{\mathrm{\nu }}{2}+\frac{1}{2}\right)::\left(\neg {\mathbb{Z}}^{\left(0\,-\right)}\right)\wedge \left(\frac{\mathrm{\mu }}{2}-\frac{\mathrm{\nu }}{2}+\frac{3}{2}\right)::\left(\neg {\mathbb{Z}}^{\left(0\,-\right)}\right)\wedge \left(\frac{\mathrm{\mu }}{2}+\frac{\mathrm{\nu }}{2}+\frac{3}{2}\right)::\left(\neg {\mathbb{Z}}^{\left(0\,-\right)}\right)$

Abramowitz, M., and Stegun, I., eds. Handbook of Mathematical Functions. New York: Dover publications.

Gradshteyn, and Ryzhik. Table of Integrals, Series and Products. 5th ed. Academic Press.

Luke, Y. The Special Functions and Their Approximations. Vol. 1 Chap. 6.