Arithmetic of Some Sequences Via $2$-determinants

TL;DR

This paper explores the properties of $2$-determinants related to second-order linear recurrence sequences, deriving generalized identities and connecting them to classical results and combinatorial interpretations.

Contribution

It introduces a generalized identity of d'Ocagne for $2$-determinants and links various classical identities to these generalized forms.

Findings

Derived a generalized identity of d'Ocagne for $2$-determinants

Connected classical identities like Cassini, Catalan, Vajda to the $2$-determinant framework

Provided combinatorial interpretations for the sequences involved

Abstract

We extend our investigation of $2$-determinants, which we defined in a previous paper. For a linear homogenous recurrence of the second order, we consider relations between different sequences satisfying the same linear homogeneous recurrence of the second order. After we prove a generalized identity of d'Ocagne, we derive, from a single identity, a number of classical identities (and their generalizations) such as d'Ocagne's, Cassini's, Catalan's and Vajda's. Along the way, the corresponding combinatorial interpretations in terms of restricted words over a finite alphabet are stated for the sequences we investigate.