On non-oscillatory solutions of the generalized Hill equation

Abstract

Let ϱ≠ −1 be a complex number with modulus |ϱ|=1 and K(x,y)=P₁(x)+P₂(y) for two polynomials P₁ and P₂. We consider the non-oscillatory solutions such that λ(f)<∞ of the generalized Hill equation f''−K(e^z,e^{−ϱ z})f=0 (♯). When ϱ=1, the Hill equation f''−K(e^z,e^−z)f=0 (†) is also written as x²u''−[K(x,x⁻¹)−1/4]u=0 (‡). We point out that there is a full correspondence between the non-oscillatory solutions of equation (†) and the Liouvillian solutions of equation (‡). Then this paper has two purposes. First, we show that if equation (♯) has a nonzero non-oscillatory solution, then ϱ=1. To this end, we solve entire solutions of a general Tumura--Clunie type differential equation. Second, for the particular Hill equation f''−(e^kz+b_se^sz+b₀)f=0, where k>s≥ 1 are integers and b_snot=0, we use Kovacic's algorithms to determine the non-oscillatory solutions with relatively few zeros.

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BibTeX RIS

BibTeX

@misc{zhang2026nonoscillatory,
  title = {{On non-oscillatory solutions of the generalized Hill equation}},
  author = {Yueyang Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24008},
  eprint = {2609.24008},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24008},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - On non-oscillatory solutions of the generalized Hill equation
AU  - Yueyang Zhang
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.24008
UR  - https://arxiv.org/abs/2609.24008
AB  - Let ϱ≠ −1 be a complex number with modulus |ϱ|=1 and K(x,y)=P_1(x)+P_2(y) for two polynomials P_1 and P_2. We consider the non-oscillatory solutions such that λ(f)<∞ of the generalized Hill equation f''−K(e^z,e^−ϱ z)f=0 (♯). When ϱ=1, the Hill equation f''−K(e^z,e^−z)f=0 (†) is also written as x^2u''−[K(x,x^−1)−1/4]u=0 (‡). We point out that there is a full correspondence between the non-oscillatory solutions of equation (†) and the Liouvillian solutions of equation (‡). Then this paper has two purposes. First, we show that if equation (♯) has a nonzero non-oscillatory solution, then ϱ=1. To this end, we solve entire solutions of a general Tumura--Clunie type differential equation. Second, for the particular Hill equation f''−(e^kz+b_se^sz+b_0)f=0, where k>s≥ 1 are integers and b_snot=0, we use Kovacic's algorithms to determine the non-oscillatory solutions with relatively few zeros.
ER  -

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