Generalized Gaussian Third-Order Jacobsthal Numbers

• The paper adapts the third-order Jacobsthal recurrence by defining a Gaussian lift that combines consecutive real terms into a complex number.
• It derives a closed-form Binet-type formula that separates exponential growth from 3-periodic oscillations using characteristic roots.
• The work also presents generating functions, multiplicative identities, and q-generalized extensions, enhancing applications in complex recurrence theory.

Generalized Gaussian third-order Jacobsthal numbers are a complex-valued extension of the generalized third-order Jacobsthal sequence in which consecutive scalar terms are assembled into a single Gaussian quantity. In the formulation introduced in 2025, the underlying scalar sequence $\{\mathcal{J}_n^{(3)}\}_{n\ge 0}$ satisfies the third-order Jacobsthal recurrence

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

and the generalized Gaussian sequence $\{\mathcal{J}g_n^{(3)}\}_{n\ge 0}$ is defined so that

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

with a compatible initial term at $n=0$. The subject combines third-order Jacobsthal recurrence theory, Gaussian sequence constructions, and the broader practice of lifting scalar recurrences into richer algebras (Morales, 2 Aug 2025).

1. Definition and formal setup

The scalar base of the theory is the generalized third-order Jacobsthal sequence

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

where $a,b,c$ are arbitrary real numbers, not all zero. Its characteristic equation is

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

with roots

$\xi=2,\qquad \omega_1,\qquad \omega_2,$

where $\omega_1,\omega_2$ are the roots of $$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)},\qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c,$$0, so that

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

The paper packages the nonreal contribution through the $$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)},\qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c,$$2-periodic sequence

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

The generalized Gaussian third-order Jacobsthal numbers satisfy the same recurrence

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

with initial values

$$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)},\qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c,$$5

The defining structural relation is

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

This makes the Gaussian sequence a shifted complex lift of the scalar generalized sequence rather than an independent recurrence family (Morales, 2 Aug 2025).

The first terms are

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

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

$$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)},\qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c,$$9

The real and imaginary parts therefore each exhibit third-order Jacobsthal-type behavior, with the imaginary part shifted by one index.

2. Scalar foundation and Gaussian lift

The scalar generalized third-order Jacobsthal sequence had already been isolated in earlier work as a family with arbitrary initial values satisfying the same recurrence, unifying the ordinary third-order Jacobsthal and third-order Jacobsthal–Lucas sequences (Cerda-Morales, 2018). In the later Gaussian treatment, that scalar framework is retained exactly and then lifted to $\{\mathcal{J}g_n^{(3)}\}_{n\ge 0}$0 by the rule

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

The scalar Binet-type formula is

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

This separates the dominant exponential term $\{\mathcal{J}g_n^{(3)}\}_{n\ge 0}$3 from the oscillatory $\{\mathcal{J}g_n^{(3)}\}_{n\ge 0}$4-periodic correction encoded by $\{\mathcal{J}g_n^{(3)}\}_{n\ge 0}$5. The Gaussian lift preserves that decomposition. The main closed form is

$\{\mathcal{J}g_n^{(3)}\}_{n\ge 0}$6

The dominant mode is therefore multiplied by $\{\mathcal{J}g_n^{(3)}\}_{n\ge 0}$7, while the periodic part is governed by the same cubic-root structure as in the scalar theory (Morales, 2 Aug 2025).

The generating function is

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

with

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

for $$\mathcal{J}g_n^{(3)}=\mathcal{J}_n^{(3)}+\mathcal{J}_{n-1}^{(3)}\,i \qquad (n\ge 1),$$0. As in the scalar case, the denominator is the characteristic polynomial of the recurrence.

Several linear identities are derived directly from the closed form: $$\mathcal{J}g_n^{(3)}=\mathcal{J}_n^{(3)}+\mathcal{J}_{n-1}^{(3)}\,i \qquad (n\ge 1),$$1

$$\mathcal{J}g_n^{(3)}=\mathcal{J}_n^{(3)}+\mathcal{J}_{n-1}^{(3)}\,i \qquad (n\ge 1),$$2

and

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

These formulas compress the recurrence into explicit exponential-periodic expressions.

3. Finite sums, negative indices, and determinant-type identities

The partial sum formula is

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

This is the Gaussian counterpart of the scalar third-order Jacobsthal endpoint formula and retains the same affine dependence on the initial data (Morales, 2 Aug 2025).

The sequence is extended to all integers through

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

with

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

For $$\mathcal{J}g_n^{(3)}=\mathcal{J}_n^{(3)}+\mathcal{J}_{n-1}^{(3)}\,i \qquad (n\ge 1),$$7,

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

The negative-index formula preserves the same $$\mathcal{J}g_n^{(3)}=\mathcal{J}_n^{(3)}+\mathcal{J}_{n-1}^{(3)}\,i \qquad (n\ge 1),$$9-based periodic correction, while replacing the dominant $n=0$0 term by $n=0$1.

For multiplicative identities, the paper introduces

$n=0$2

$n=0$3

$n=0$4

$n=0$5

The d’Ocagne identity is

$n=0$6

By taking $n=0$7, one obtains the Cassini-type identity

$n=0$8

The paper also rewrites the bracket $n=0$9 as

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

giving an equivalent mod-$$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)}, \qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c,$$1 form of Cassini’s identity.

4. Specializations and explicit families

Two named specializations are obtained by fixing the scalar initials $$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)}, \qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c,$$2. These are central to the literature because they recover Gaussian lifts of the ordinary and modified third-order Jacobsthal families (Morales, 2 Aug 2025).

Sequence	Parameters $$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)}, \qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c,$$3	Initial values
$$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)}, \qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c,$$4	arbitrary	$$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)}, \qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c,$$5
$$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)}, \qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c,$$6	$$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)}, \qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c,$$7	$$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)}, \qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c,$$8
$$\mathcal{J}_{n+3}^{(3)}=\mathcal{J}_{n+2}^{(3)}+\mathcal{J}_{n+1}^{(3)}+2\mathcal{J}_{n}^{(3)}, \qquad \mathcal{J}_{0}^{(3)}=a,\ \mathcal{J}_{1}^{(3)}=b,\ \mathcal{J}_{2}^{(3)}=c,$$9	$a,b,c$0	$a,b,c$1

For the Gaussian third-order Jacobsthal numbers,

$a,b,c$2

the first terms are

$a,b,c$3

and the closed form becomes

$a,b,c$4

For the Gaussian modified third-order Jacobsthal numbers,

$a,b,c$5

the first terms are

$a,b,c$6

and the Binet formula is

$a,b,c$7

The corresponding partial sums are

$a,b,c$8

$a,b,c$9

Specialized Cassini identities are also recorded. For $\xi^{3}-\xi^{2}-\xi-2=0,$0,

$\xi^{3}-\xi^{2}-\xi-2=0,$1

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

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

The paper also introduces a further multiplicative family, called the $\xi^{3}-\xi^{2}-\xi-2=0,$4-generalized Gaussian third-order Jacobsthal numbers. If

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

then

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

The case $\xi^{3}-\xi^{2}-\xi-2=0,$7 recovers the original sequence. For $\xi^{3}-\xi^{2}-\xi-2=0,$8,

$\xi^{3}-\xi^{2}-\xi-2=0,$9

A notational caveat is explicit in the source: the abstract refers to a “$\xi=2,\qquad \omega_1,\qquad \omega_2,$0-generalized” family, but the body of the paper develops a $\xi=2,\qquad \omega_1,\qquad \omega_2,$1-generalized family (Morales, 2 Aug 2025).

5. Position within the broader Jacobsthal literature

The generalized Gaussian theory rests on a well-defined scalar base. Earlier work had already introduced a generalized third-order Jacobsthal sequence with arbitrary real initials,

$\xi=2,\qquad \omega_1,\qquad \omega_2,$2

and derived its generating function, Binet formula, Catalan identity, and Gelin–Cesàro identity (Cerda-Morales, 2018). The Gaussian theory may therefore be viewed as a complex lift of an already established generalized scalar framework.

At a still broader level, the recurrence belongs to the general third-order class

$\xi=2,\qquad \omega_1,\qquad \omega_2,$3

with arbitrary complex parameters. The Jacobsthal specialization is obtained by setting

$\xi=2,\qquad \omega_1,\qquad \omega_2,$4

and this general framework supplies recurrence identities, generating functions, negative-index formulas, arithmetic-progression sums, and double-binomial summations for complex-valued sequences (Adegoke, 2019). Generalized Gaussian third-order Jacobsthal numbers therefore fit naturally into a pre-existing complex third-order recurrence theory.

The literature also contains several higher-algebra lifts of the same scalar recurrence. Third-order Jacobsthal 3-parameter generalized quaternions package four consecutive scalar terms,

$\xi=2,\qquad \omega_1,\qquad \omega_2,$5

and inherit the same recurrence, generating function, Binet formula, and determinant-type identities (Morales, 12 Feb 2025). Related constructions exist for dual numbers, quaternions, octonions, bihyperbolic numbers, and modified-sequence complex specializations (Cerda-Morales, 2017, Cerda-Morales, 2018, Cerda-Morales, 2017, Cerda-Morales, 2024, Morales, 2024). This suggests that the Gaussian case is the commutative two-dimensional member of a wider family of hypercomplex Jacobsthal lifts.

A particularly close neighboring construction occurs in the modified-sequence setting, where unrestricted complex modified third-order Jacobsthal numbers are defined as

$\xi=2,\qquad \omega_1,\qquad \omega_2,$6

That arbitrary-shift formulation is explicitly identified as a complex, Gaussian-type specialization of an unrestricted quaternion theory (Morales, 2024). It parallels the role of the $\xi=2,\qquad \omega_1,\qquad \omega_2,$7-generalized family in the ordinary generalized Gaussian third-order Jacobsthal setting.

6. Scope, distinctions, and common confusions

A recurrent confusion is to identify generalized Gaussian third-order Jacobsthal numbers with the one-parameter second-order generalization of Jacobsthal and Jacobsthal–Lucas numbers. That construction is

$\xi=2,\qquad \omega_1,\qquad \omega_2,$8

with a corresponding Lucas companion and characteristic factorization $\xi=2,\qquad \omega_1,\qquad \omega_2,$9, but it remains strictly second order and does not define Gaussian or third-order sequences (Al-Kateeb, 2019). Generalized Gaussian third-order Jacobsthal numbers require a change in recurrence order, not merely a deformation of second-order coefficients.

A second distinction concerns ambient algebra. Quaternionic, octonionic, and generalized-quaternion Jacobsthal objects inherit the same scalar recurrence but carry noncommutative multiplication laws or higher-dimensional basis structure. The Gaussian sequence is simpler algebraically: it is built in a commutative complex setting, so identities such as the recurrence, Binet formula, generating function, and endpoint sum formula transfer directly, whereas product identities simplify relative to the quaternionic case (Morales, 12 Feb 2025).

A third distinction concerns exact historical priority. Earlier papers supplied the scalar generalized third-order Jacobsthal sequence, complex third-order recurrence machinery, and numerous hypercomplex analogues, but they did not explicitly define “generalized Gaussian third-order Jacobsthal numbers.” That exact notion, with arbitrary initial values, explicit Gaussian initial conditions, a Binet formula, negative-index theory, d’Ocagne and Cassini identities, and the $\omega_1,\omega_2$0-generalized family, appears as a named object only in the dedicated 2025 treatment (Morales, 2 Aug 2025).

The central structural idea across the literature is consistent: the recurrence

$\omega_1,\omega_2$1

has one dominant real root $\omega_1,\omega_2$2 and two cubic-root-of-unity corrections $\omega_1,\omega_2$3, and every lift—scalar, Gaussian, quaternionic, or otherwise—is organized by that same spectral decomposition. In the generalized Gaussian case, this produces a sequence whose real and imaginary parts are shifted generalized third-order Jacobsthal terms and whose algebraic properties are governed simultaneously by exponential growth and mod-$\omega_1,\omega_2$4 periodicity (Morales, 2 Aug 2025).