Lattice walk area combinatorics, some remarkable trigonometric sums and Ap\'ery-like numbers

TL;DR

This paper derives explicit formulas for lattice walk areas, uncovers remarkable trigonometric identities related to quantum models, and explores connections with Apéry-like numbers in number theory.

Contribution

It introduces new algebraic area enumeration formulas for generalized lattice walks and links these to trigonometric sums and Apéry-like numbers.

Findings

Derived explicit algebraic area formulas for various lattice walks.

Identified remarkable identities involving trigonometric sums.

Connected lattice walk enumeration to Apéry-like numbers in number theory.

Abstract

Explicit algebraic area enumeration formulae are derived for various lattice walks generalizing the canonical square lattice walk, and in particular for the triangular lattice chiral walk recently introduced by the authors. A key element in the enumeration is the derivation of some remarkable identities involving trigonometric sums --which are also important building blocks of non trivial quantum models such as the Hofstadter model-- and their explicit rewriting in terms of multiple binomial sums. An intriguing connection is also made with number theory and some classes of Ap\'ery-like numbers, the cousins of the Ap\'ery numbers which play a central role in irrationality considerations for {\zeta}(2) and {\zeta}(3).