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

Dorin DumitraÅcu; Liviu Suciu

combinatorics

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

arXiv:2607.25160

summary

The paper analyzes generating functions for combinatorial sequences arising from random walks on the standard Euclidean lattice, proving that the associated differential operators are irreducible Fuchsian operators and deriving minimal polynomial recurrences for the sequences.

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

Topics & keywords

#generating functions #random walks #lattice walks #differential operators #recurrence relations #irreducibilityFuchsian operatorBorel transformminimal recurrencepolynomial coefficientscombinatorial sequencesanalytic continuation