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

FunctionAdvisor( definition, LommelS1);

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);

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);

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

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

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.