The perimeter generating functions of three-choice, imperfect, and one-punctured staircase polygons**Dedicated to A J Guttmann on the occasion of his 70th birthday.

Abstract

We consider the isotropic perimeter generating functions of three-choice, imperfect, and one-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 2F1 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 2F1 hypergeometric functions can be expressed with two, rational or algebraic, pullbacks.