Basic properties of a generalized third order sequence of numbers

TL;DR

This paper investigates the fundamental properties of a generalized third order linear recurrence sequence, including sum formulas and binomial identities, with arbitrary complex initial conditions and coefficients.

Contribution

It introduces and analyzes a broad class of third order sequences, deriving new sum formulas and binomial identities for these sequences.

Findings

Derived formulas for partial sums of sequence terms in arithmetic progression.

Established double binomial summation identities for the sequence.

Extended properties to sequences with complex initial conditions and coefficients.

Abstract

We study the properties of the third order sequence $(w_n)=\left(w_n(a,b,c; r, s,t)\right)$ defined by the recurrence relation $w_n = rw_{n - 1} + sw_{n - 2} + tw_{n - 3}\, (n \ge 3)$ with $w_0 = a,\,w_1 = b,\,w_2=c$, where $a$, $b$, $c$, $r$, $s$ and $t$ are arbitrary complex numbers and $t\ne 0$. Properties examined include the partial sum of the terms of the sequence, with indices in arithmetic progression, as well as double binomial summation identities.