The inverse Laplace transform of the modified Lommel functions

Particular solutions of the inhomogeneous Bessel differential equation x(2)f(mu,nu)('') (x) + xf'mu,nu (x) - (nu(2) - x(2))f(mu,nu)(x) = x(mu+1) are the Lommel functions, usually denoted as s(mu,nu) (x), S-mu,S-nu (x), and S-mu,S-nu (x). The inhomogeneous modified Bessel differential equation x(2)g ''(mu,nu) (x) + xg'(mu,nu) (x) - (nu(2) + x(2)) g(mu,nu) (x) = x(mu+1) has the particular solutions t(mu,nu) (x) = -i(1-mu)S(mu,nu) (ix), T-mu,T-nu (x) = -i(1-mu)S(mu,nu) (ix), and T-mu,T-nu (x) = -i(1-mu)S(mu,nu) (ix). For the modified Lommel functionsT(mu,nu) (x) and T-mu,T-nu (x) as well as T-mu,T-nu (root x) and T-mu,T-nu (root x), we give the inverse Laplace transform for mu = -1 and mu = 0. For mu < -1, the inverse Laplace transform can be obtained from a recurrence relation.