On the nature of the generating series of walks in the quarter plane

Abstract

In the present paper, we use difference Galois theory to study the nature of the generating series counting walks in the quarter plane. These series are trivariate formal power series Q(x, y, t) that count the number of discrete paths confined in the first quadrant of the plane with a fixed directions set. While the variables x and y are associated to the ending point of the path, the variable t encodes its length. In this paper, we prove that if Q(x, y, t) does not satisfy any algebraic differential relations with respect to x or y, it does not satisfy any algebraic differential relations with respect to the parameter t. Combined with [BBMR16, DHRS18, DHRS17], we are able to characterize the t-differential transcendence of the generating series for any unweighted walk.

Complexity Problems in Enumerative Combinatorics

“…We refer to [25,26] for a comprehensive overview of the background and early stages of this far-reaching project, and to [15,22] for some recent developments which are gateways to references. Finally, let us mention a remarkable recent development [49], which proves D-transcendence for many families of lattice walks. Let us single out just one of the many results in that paper:…”

On Walks Avoiding a Quadrant

“…7 (algebraic?) non-D-finite [34] non-D-finite [27,7,14] (non-D-finite?) Table 2: Algebraic nature of the conformal mapping w, the quadrant generating function Q(x, y) and the three-quarter plane counting function C(x, y)…”

“…5]. Starting from (14) we can prove that c(X 0 (y))L −0 (X 0 (y)) + X 0 (y) d(y)D(y) − X 0 (y)y = 0 for y ∈ {y ∈ C : |X 0 (y)| < 1} ∩ D, and then c(X 0 (y)) n 0,j 0 c 0,−j−1 (n)X 0 (y) j t n + d(y)D(y) − y = 0 for y ∈ {y ∈ C : |X 0 (y)| < 1 and X 0 (y) = 0} ∩ D which can be continued in G L ∪ D. Being a power series, D(y) is analytic on D and on (G L ∪ D) \ D, D(y) may have the same singularities as X 0 and d(y), namely the branch cuts [y 1 , y 2 ] and [y 3 , y 4 ]. But none of these segments belong to (G L ∪ D) \ D (see Lemma 3).…”

Transcendental holomorphic maps between real algebraic manifolds in a complex space

“…For these walks, by symmetry, the solutions (l, h, g) of (3.2) satisfy the relation h(x, t) = g(x, t) and have positive integer coefficients. In fact, it has been recently shown that the generating series corresponding to these 7 types of walks are among those that are x-hypertranscendental and yhypertranscendental [DHRS18]. That is, Q(x, 0, t) (resp.…”