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Using a simple transfer matrix approach we have derived very long series expansions for the perimeter generating function of punctured staircase polygons (staircase polygons with a single internal staircase hole). We find that all the terms in the generating function can be reproduced from a linear Fuchsian differential equation of order 8. We perform an analysis of the properties of the differential equation.doi:10.1088/0305-4470/39/15/002 fatcat:pdtzfjrf5vbzxcbfl3czfxymi4