The modified Lommel functions of the first and second kind give the solution to the Lommel differential equation with
a minus sign in front of the linear term, i.e.,

They are denoted and , respectively.

These functions will implemented in a future version of the Wolfram Language as LommelT1[m, n, z] and LommelT2[m,
n, z], respectively.

Not that while Rollinger (1964) uses the notation , is preferable so as not to conflict with the notation for
Lommel polynomials.

Rollinger, C. N. "Lommel Functions with Imagniary Argument." Quart. J. Appl. Math.21, 343-349, 1964.Shafer,
R. E. "Lommel Functions of Imaginary Arguments." Technical Report
UCRL-7806. Livermore, CA: Lawrence Livermore National Lab, 1964.Szymanski,
P. "On the Integral Representations of the Lommel Functions." Proc.
London Math. Soc.40, 71-82, 1936.Ziener, C. H. and
Schlemmer, H. P. "the Inverse Laplace Transform of the Modified Lommel
Functions." Integral Transforms and Special Functions24, 141-155,
2013.