On the Irreducibility of the Differential Operators Associated to Random Walks in the Standard Euclidean Lattice

Dorin Dumitraşcu; Liviu Suciu

paper

On the Irreducibility of the Differential Operators Associated to Random Walks in the Standard Euclidean Lattice

arXiv:2607.25160

Abstract

For a positive integer $d \geq 1$ , we consider the sequences $(A_{n}^{(d)})_{n}$ and $(x_{n}^{(d)})_{n}$ given by $A_{n}^{(d)} = n_{1} + \dots + n_{d} = n \sum \frac{( 2 n )!}{( n _{1} ! ) ^{2} ( n _{2} ! ) ^{2} \dots ( n _{d} ! ) ^{2}} and x_{n}^{(d)} = \frac{A _{n}^{(d)}}{( n 2 n )} .$ They have rich combinatorial interpretations, but we focus on the analytical properties of their generating functions $A_{d}$ and $F_{d}$ . We use a modified Borel transform, and algebraic and combinatorial considerations to prove that $F_{d}$ is annihilated by an irreducible Fuchsian differentiable operator $L_{d - 1, F}$ of order $d - 1$ . We determine the structure of $F_{d}$ as a global analytic function (analytic continuations from the original disk of definition, branches, finite singularities, and the structure of $F_{d}$ near the finite singularities). Additionally, we show that the sequence $(x_{n}^{(d)})_{n}$ satisfies a minimal recurrence of width $r = ⌊(d + 1) /2 ⌋$ with polynomial coefficients $Q_{r} (n + r) x_{n + r} + \dots + Q_{0} (n) x_{n} = 0, n \geq 0.$ These polynomials are shown to have very specific symmetries and we compute explicitly $Q_{0}$ , $Q_{1}$ , and $Q_{r}$ . Similar results about the functions $A_{d}$ are obtained.

34 pages, 1 figure