  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.
