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Haruspicy 3: The directed bond-animal generating function is not D-finite

Andrew Rechnitzer
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Abstract

While directed site-animals have been solved on several lattices, directed bond-animals remain unsolved on any non-trivial lattice. In this paper we demonstrate that the anisotropic generating function of directed bond-animals on the square lattice is fundamentally different from that of directed site-animals in that it is not differentiably finite. We also extend this result to directed bond-animals on hypercubic lattices. This indicates that directed bond-animals are unlikely to be solved by similar methods to those used in the solution of directed site-animals. It also implies that a solution cannot by conjectured using computer packages such as GFUN or differential approximants.

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Andrew Rechnitzer. 2004-08-04. Haruspicy 3: The directed bond-animal generating function is not D-finite. https://arxiv.org/abs/math/0408054

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