The perimeter generating functions of three-choice, imperfect, and 1-punctured staircase polygons

TL;DR

This paper analyzes the generating functions of three types of staircase polygons, revealing their relationships, expressing solutions in hypergeometric functions, and linking them to modular forms, thus advancing understanding of their mathematical structure.

Contribution

It derives a common 12th order differential equation for the three generating functions and expresses solutions in terms of hypergeometric functions and modular forms.

Findings

All three generating functions satisfy a shared 12th order Fuchsian ODE.

Each 8th order differential operator decomposes into smaller operators of up to 3rd order.

Solutions can be expressed using $_2F_1$ hypergeometric functions and modular forms.

Abstract

We consider the isotropic perimeter generating functions of three-choice, imperfect, and 1-punctured staircase polygons, whose 8th order linear Fuchsian ODEs are previously known. We derive simple relationships between the three generating functions, and show that all three generating functions are joint solutions of a common 12th order Fuchsian linear ODE. We find that the 8th order differential operators can each be rewritten as a direct sum of a direct product, with operators no larger than 3rd order. We give closed-form expressions for all the solutions of these operators in terms of $_2F_1$ hypergeometric functions with rational and algebraic arguments. The solutions of these linear differential operators can in fact be expressed in terms of two modular forms, since these $_2F_1$ hypergeometric functions can be expressed with two, rational or algebraic, pullbacks.