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This paper offers a new and complete description of subnormal solutions ofcertain non-homogeneous second order periodic linear differential equationsfirst studied by Gundersen and Steinbart in 1994. We have established apreviously unknown relation that the general solutions (\textit{i.e.}, whethersubnormal or not) of the DEs can be solved explicitly in terms of classicalspecial functions, namely the Bessel, Lommel and Struve functions, which areimportant because of their numerous physical applications. In particular, weshow that the subnormal solutions are written explicitly in terms of thedegenerate Lommel functions $S_{\mu, \nu}(\zeta)$ and several classical specialpolynomials related to the Bessel functions. In fact, we solve an equivalentproblem in special functions that each branch of the Lommel function $S_{\mu,\nu}(\zeta)$ degenerates if and only if $S_{\mu, \nu}({\rm e}^z)$ has finiteorder of growth in $\mathbf{C}$. We achieve this goal by proving new propertiesand identities for these functions. A number of semi-classicalquantization-type results are obtained as consequences. Thus our results notonly recover and extend the result of Gundersen and Steinbart \cite{GS94}, butthe new identities and properties found for the Lommel functions are ofindependent interest in a wider context.

On subnormal solutions of periodic non-homogeneous linear differential equations, special functions and special polynomials