On the asymptotics and the non-holonomic character of first returns in the standard Euclidean lattice

We give precise asymptotics to the number of first-time returning random walks in the standard orthogonal lattice in R d , and we prove that these numbers do not form a P -recursive sequence.In the process, the known asymptotics of the number of closed walks are obtained in an elementary way, by using a combinatorial and geometric multiplication principle together with the classical theory of Legendre polynomials.An important observation is the connection that we make between closed walks and a sequence that has been studied when analyzing both staircase polygons and abelian squares.By showing that the relevant generating functions are G-functions, we use a form of the Hadamard convolution to find their singularities in all dimensions.We use general properties to prove that the generating function of closed walks is not invertible as a G-function, which immediately implies that the generating function of the first time returning walks is not holonomic.