On complex oscillation, function-theoretic quantization of non-homogeneous periodic ODEs and special functions

TL;DR

This paper establishes new conditions for quantizing certain complex periodic second-order non-homogeneous ODEs, linking non-oscillatory solutions to special functions and their zero distributions, with implications for special functions like Lommel functions.

Contribution

It provides necessary and sufficient conditions for the quantization of non-homogeneous periodic ODEs using complex oscillation theory, connecting solution behavior to special functions and zero distribution.

Findings

Non-oscillatory solutions correspond to degenerate polynomial/rational functions.

Finiteness of zeros in Lommel functions indicates quantization.

Degenerate solutions are linked to subnormal solutions.

Abstract

New necessary and sufficient conditions are given for the quantization of a class of periodic second order non-homogeneous ordinary differential equations in the complex plane in this paper. The problem is studied from the viewpoint of complex oscillation theory first developed by Bank and Laine (1982, 1983) and Gundersen and Steinbart (1994). We show that when a solution is complex non-oscillatory (finite exponent of convergence of zeros) then the solution, which can be written as special functions, must degenerate. This gives a necessary and sufficient condition when the Lommel function has finitely many zeros in every branch and this is a type of quantization for the non-homogeneous differential equation. The degenerate solutions are of polynomial/rational-type functions which are of independent interest. In particular, this shows that complex non-oscillatory solutions of this class of differential equations are equivalent to the subnormal solutions considered in a previous paper of the authors (to appear). In addition to the asymptotics of special functions, the other main idea we apply in our proof is a classical result of E. M. Wright which gives precise asymptotic locations of large zeros of a functional equation.