Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hypergeometric Expressions for Generating Functions of Walks with Small
  Steps in the Quarter Plane   We study nearest-neighbors walks on the two-dimensional square lattice, that
is, models of walks on $\mathbb{Z}^2$ defined by a fixed step set that is a
subset of the non-zero vectors with coordinates 0, 1 or $-1$. We concern
ourselves with the enumeration of such walks starting at the origin and
constrained to remain in the quarter plane $\mathbb{N}^2$, counted by their
length and by the position of their ending point. Bousquet-M\'elou and Mishna
[Contemp. Math., pp. 1--39, Amer. Math. Soc., 2010] identified 19 models of
walks that possess a D-finite generating function; linear differential
equations have then been guessed in these cases by Bostan and Kauers [FPSAC
2009, Discrete Math. Theor. Comput. Sci. Proc., pp. 201--215, 2009]. We give
here the first proof that these equations are indeed satisfied by the
corresponding generating functions. As a first corollary, we prove that all
these 19 generating functions can be expressed in terms of Gauss'
hypergeometric functions that are intimately related to elliptic integrals. As
a second corollary, we show that all the 19 generating functions are
transcendental, and that among their $19 \times 4$ combinatorially meaningful
specializations only four are algebraic functions.

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The categories of this paper are: funct-an
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The categories of this paper are: plasm-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: chao-dyn
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The categories of this paper are: stat
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-alg
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The categories of this paper are: econ
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The categories of this paper are: solv-int
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The categories of this paper are: adap-org
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The categories of this paper are: hep-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: q-fin