Gaussian Third-Order Jacobsthal Numbers

• Gaussian third-order Jacobsthal numbers are Gaussian-valued analogues of the classical third-order Jacobsthal sequence, extending the recurrence to complex numbers with modulation by cube roots of unity.
• The sequence is derived via a Binet-type formula that separates a dominant geometric term from periodic correction terms, enabling closed-form expressions and generating functions.
• The framework unifies related extensions, such as quaternionic and dual-number Jacobsthal sequences, and offers new insights into recurrence identities and algebraic structures in the complex domain.

Searching arXiv for papers on Gaussian third-order Jacobsthal numbers and related third-order Jacobsthal extensions. Gaussian third-order Jacobsthal numbers are Gaussian-valued analogues of the third-order Jacobsthal sequence, obtained by extending the classical recurrence

$$J_{n+3}^{(3)}=J_{n+2}^{(3)}+J_{n+1}^{(3)}+2J_n^{(3)}$$

from real scalars to complex numbers of the form $x+yi$. In the generalized formulation, the sequence $\{\mathcal{J}g_n^{(3)}\}_{n\ge 0}$ is defined by the same third-order recurrence with complex initial data parametrized by $a,b,c\in\mathbb{R}$, not all zero, and it admits the decomposition

$$\mathcal{J}g_n^{(3)}=\mathcal{J}_n^{(3)}+\mathcal{J}_{n-1}^{(3)}\,i,\qquad n\ge 1.$$

This places Gaussian third-order Jacobsthal numbers within a broader algebraic program in which third-order Jacobsthal sequences are lifted to richer algebras while preserving the underlying recurrence and its characteristic-root structure (Morales, 2 Aug 2025).

1. Scalar framework and characteristic structure

The Gaussian theory is built on the generalized third-order Jacobsthal sequence $\{\mathcal{J}_n^{(3)}\}_{n\ge 0}$, defined by

$$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)},\qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c.$$

Its characteristic polynomial is

$\xi^{3}-\xi^{2}-\xi-2=0,$

with roots $2,\omega_1,\omega_2$, where $\omega_1,\omega_2$ are the two complex roots of $x+yi$0. These satisfy

$x+yi$1

The generalized scalar sequence has the Binet form

$x+yi$2

where

$x+yi$3

The periodic factor $x+yi$4 encodes the oscillatory contribution of the nonreal characteristic roots, while the dominant root $x+yi$5 governs asymptotic growth (Morales, 2 Aug 2025).

This same characteristic polynomial and the same cube-root periodicity already organize earlier work on third-order Jacobsthal numbers, third-order Jacobsthal-Lucas numbers, and their quaternionic and dual extensions. In those settings, the nonreal roots appear through mod-$x+yi$6 phenomena, Binet formulas, and componentwise transfer identities, which later reappear in the Gaussian case as explicit periodic correction terms (Cerda-Morales, 2017).

2. Generalized Gaussian sequence and principal special cases

The generalized Gaussian third-order Jacobsthal sequence $x+yi$7 is defined by

$x+yi$8

with initial values

$x+yi$9

A key structural remark is that, for $\{\mathcal{J}g_n^{(3)}\}_{n\ge 0}$0,

$\{\mathcal{J}g_n^{(3)}\}_{n\ge 0}$1

Thus the Gaussian sequence is not an unrelated construction; it is a direct Gaussian lifting of the real generalized third-order Jacobsthal sequence (Morales, 2 Aug 2025).

Two specializations are singled out.

Sequence	Parameter choice	Initial values
$\{\mathcal{J}g_n^{(3)}\}_{n\ge 0}$2	$\{\mathcal{J}g_n^{(3)}\}_{n\ge 0}$3	$\{\mathcal{J}g_n^{(3)}\}_{n\ge 0}$4
$\{\mathcal{J}g_n^{(3)}\}_{n\ge 0}$5	$\{\mathcal{J}g_n^{(3)}\}_{n\ge 0}$6	$\{\mathcal{J}g_n^{(3)}\}_{n\ge 0}$7

The first is the Gaussian third-order Jacobsthal sequence in the narrow sense; the second is the Gaussian modified third-order Jacobsthal sequence. The paper treats both as instances of the generalized Gaussian framework, so identities established for $\{\mathcal{J}g_n^{(3)}\}_{n\ge 0}$8 immediately specialize to these two families (Morales, 2 Aug 2025).

This construction suggests that the Gaussian extension is algebraically conservative: the recurrence is unchanged, while the initial conditions are chosen so that the imaginary part tracks a shifted copy of the underlying real sequence.

3. Closed forms, periodic corrections, and asymptotic profile

A central result is the Binet-type formula

$\{\mathcal{J}g_n^{(3)}\}_{n\ge 0}$9

Its derivation uses the ansatz

$a,b,c\in\mathbb{R}$0

with $a,b,c\in\mathbb{R}$1 determined from the initial data. The formula separates the sequence into a dominant geometric term proportional to $a,b,c\in\mathbb{R}$2 and two periodic correction terms carried by $a,b,c\in\mathbb{R}$3 and $a,b,c\in\mathbb{R}$4 (Morales, 2 Aug 2025).

For the Gaussian third-order Jacobsthal numbers themselves,

$a,b,c\in\mathbb{R}$5

and for the Gaussian modified sequence,

$a,b,c\in\mathbb{R}$6

Several recurrence-derived identities make the same structure explicit: $a,b,c\in\mathbb{R}$7

$a,b,c\in\mathbb{R}$8

and

$a,b,c\in\mathbb{R}$9

These formulas show that Gaussian third-order Jacobsthal numbers are not governed by simple doubling. Rather, the dominant $$\mathcal{J}g_n^{(3)}=\mathcal{J}_n^{(3)}+\mathcal{J}_{n-1}^{(3)}\,i,\qquad n\ge 1.$$0 growth is continuously modulated by period-$$\mathcal{J}g_n^{(3)}=\mathcal{J}_n^{(3)}+\mathcal{J}_{n-1}^{(3)}\,i,\qquad n\ge 1.$$1 oscillations inherited from the cube roots of unity. This is the same mod-$$\mathcal{J}g_n^{(3)}=\mathcal{J}_n^{(3)}+\mathcal{J}_{n-1}^{(3)}\,i,\qquad n\ge 1.$$2 mechanism that appears in scalar and quaternionic third-order Jacobsthal identities, but in the Gaussian case it is encoded directly in the sequence values rather than only in auxiliary closed forms (Cerda-Morales, 2017).

4. Negative indices, generating functions, and summation formulas

The generalized Gaussian sequence is extended to negative indices by the backward recurrence

$$\mathcal{J}g_n^{(3)}=\mathcal{J}_n^{(3)}+\mathcal{J}_{n-1}^{(3)}\,i,\qquad n\ge 1.$$3

with

$$\mathcal{J}g_n^{(3)}=\mathcal{J}_n^{(3)}+\mathcal{J}_{n-1}^{(3)}\,i,\qquad n\ge 1.$$4

$$\mathcal{J}g_n^{(3)}=\mathcal{J}_n^{(3)}+\mathcal{J}_{n-1}^{(3)}\,i,\qquad n\ge 1.$$5

$$\mathcal{J}g_n^{(3)}=\mathcal{J}_n^{(3)}+\mathcal{J}_{n-1}^{(3)}\,i,\qquad n\ge 1.$$6

Using $$\mathcal{J}g_n^{(3)}=\mathcal{J}_n^{(3)}+\mathcal{J}_{n-1}^{(3)}\,i,\qquad n\ge 1.$$7, the negative-index Binet form is

$$\mathcal{J}g_n^{(3)}=\mathcal{J}_n^{(3)}+\mathcal{J}_{n-1}^{(3)}\,i,\qquad n\ge 1.$$8

with an equivalent rewritten form also given in the paper (Morales, 2 Aug 2025).

The generating function is

$$\mathcal{J}g_n^{(3)}=\mathcal{J}_n^{(3)}+\mathcal{J}_{n-1}^{(3)}\,i,\qquad n\ge 1.$$9

Its specializations are

$\{\mathcal{J}_n^{(3)}\}_{n\ge 0}$0

and

$\{\mathcal{J}_n^{(3)}\}_{n\ge 0}$1

The partial sums satisfy

$\{\mathcal{J}_n^{(3)}\}_{n\ge 0}$2

with the corollaries

$\{\mathcal{J}_n^{(3)}\}_{n\ge 0}$3

and

$\{\mathcal{J}_n^{(3)}\}_{n\ge 0}$4

Together, these formulas provide the standard analytic toolkit for the sequence: bilateral extension, rational generating series, and closed summation identities (Morales, 2 Aug 2025).

5. d’Ocagne and Cassini identities

For positive integers $\{\mathcal{J}_n^{(3)}\}_{n\ge 0}$5 with $\{\mathcal{J}_n^{(3)}\}_{n\ge 0}$6, the paper proves the d’Ocagne-type identity

$\{\mathcal{J}_n^{(3)}\}_{n\ge 0}$7

where

$\{\mathcal{J}_n^{(3)}\}_{n\ge 0}$8

and

$\{\mathcal{J}_n^{(3)}\}_{n\ge 0}$9

The proof writes

$$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)},\qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c.$$0

with $$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)},\qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c.$$1, and simplifies the determinant-like expression (Morales, 2 Aug 2025).

Setting $$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)},\qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c.$$2 yields Cassini’s identity: $$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)},\qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c.$$3 and the paper further expands

$$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)},\qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c.$$4

These identities show that classical determinant identities persist in the Gaussian setting, but acquire explicitly complex coefficients and periodic correction terms. A plausible implication is that the Gaussian extension preserves the algebraic architecture of Jacobsthal theory while replacing scalar invariants by complex-valued ones controlled by the same period-$$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)},\qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c.$$5 skeleton.

6. Relation to other third-order Jacobsthal extensions and scope

Gaussian third-order Jacobsthal numbers belong to a family of third-order Jacobsthal extensions that also includes quaternionic and dual-number versions, but these should not be conflated. In the quaternionic setting, third-order Jacobsthal quaternions are defined by

$$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)},\qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c.$$6

and many identities are obtained by applying scalar relations componentwise. This includes lifts such as

$$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)},\qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c.$$7

together with generating functions, Binet formulas, norms, products, and matrix representations (Cerda-Morales, 2017).

In the dual-number setting, the extension is instead

$$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)},\qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c.$$8

with parallel recurrence, Binet, Cassini-type, and vector identities. That paper explicitly states that it is not a Gaussian extension; the complex roots appear only in the Binet formulas of the underlying real sequence, not as Gaussian-valued sequence terms (Cerda-Morales, 2017).

This distinction corrects a common misconception. The presence of $$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)},\qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c.$$9 and $\xi^{3}-\xi^{2}-\xi-2=0,$0 in scalar third-order Jacobsthal formulas does not by itself make those sequences Gaussian. In the Gaussian case, the sequence values themselves lie in the Gaussian algebra, and the paper of 2025 develops this setting directly through complex initial data, Gaussian-valued Binet formulas, partial sums, negative-index elements, d’Ocagne’s identity, Cassini’s identity, and the further $\xi^{3}-\xi^{2}-\xi-2=0,$1-generalized family (Morales, 2 Aug 2025).

The $\xi^{3}-\xi^{2}-\xi-2=0,$2-generalized Gaussian third-order Jacobsthal numbers extend the theory one step further. If $\xi^{3}-\xi^{2}-\xi-2=0,$3 with $\xi^{3}-\xi^{2}-\xi-2=0,$4, then the paper imposes the relation

$\xi^{3}-\xi^{2}-\xi-2=0,$5

In particular,

$\xi^{3}-\xi^{2}-\xi-2=0,$6

and the paper proves additional recurrence-like and summation identities involving

$\xi^{3}-\xi^{2}-\xi-2=0,$7

Within the current literature represented here, the Gaussian theory is therefore best understood as the complex-valued branch of a broader third-order Jacobsthal extension program. Its distinctive features are the explicit Gaussian lifting

$\xi^{3}-\xi^{2}-\xi-2=0,$8

the direct incorporation of complex initial values, and the systematic role of period-$\xi^{3}-\xi^{2}-\xi-2=0,$9 corrections generated by the nonreal roots of $2,\omega_1,\omega_2$0.