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

TL;DR

This paper characterizes subnormal solutions of certain periodic linear differential equations, expressing them explicitly via classical special functions like Bessel and Lommel functions, and establishes new identities with broad implications.

Contribution

It provides a complete description of subnormal solutions, linking them explicitly to classical special functions and deriving new properties and identities for these functions.

Findings

Subnormal solutions are explicitly expressed in terms of Lommel and Bessel functions.

New identities and properties for Lommel functions are established.

The results extend previous work and have applications in physics and special function theory.

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.