Haruspicy 2: The anisotropic generating function of self-avoiding polygons is not D-finite

We prove that the anisotropic generating function of self-avoiding polygons is not a D-finite function—proving a conjecture of Guttmann [Discrete Math. 217 (2000) 167–189] and Guttman and Enting [Phys. Rev. Lett. 76 (1996) 344–347]. This result is also generalised to self-avoiding polygons on hypercubic lattices. Using the haruspicy techniques developed in an earlier paper [Rechnitzer, Adv. Appl. Math. 30 (2003) 228–257], we are also able to prove the form of the coefficients of the anisotropic generating function, which was first conjectured in Guttman and Enting [Phys. Rev. Lett. 76 (1996) 344–347].