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

Abstract

This paper offers a new and complete description of subnormal solutions of certain non-homogeneous second order periodic linear differential equations first studied by Gundersen and Steinbart in 1994. We have established a previously unknown relation that the general solutions (\textit{i.e.}, whether subnormal or not) of the DEs can be solved explicitly in terms of classical special functions, namely the Bessel, Lommel and Struve functions, which are important because of their numerous physical applications. In particular, we show that the subnormal solutions are written explicitly in terms of the degenerate Lommel functions $S_{\mu, \nu}(\zeta)$ and several classical special polynomials related to the Bessel functions. In fact, we solve an equivalent problem 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 finite order of growth in $\mathbf{C}$. We achieve this goal by proving new properties and identities for these functions. A number of semi-classical quantization-type results are obtained as consequences. Thus our results not only recover and extend the result of Gundersen and Steinbart \cite{GS94}, but the new identities and properties found for the Lommel functions are of independent interest in a wider context.