Gaussian Modified Third-Order Jacobsthal Numbers

• Gaussian modified third-order Jacobsthal numbers are complex-valued analogues of the real modified sequence, defined by a third-order recurrence with initial values 3, 1, 3.
• They admit a closed-form Binet-style formula and generating function that decompose into an exponential term and a periodic correction derived from nonreal cube-root components.
• The construction preserves the spectral properties of the real recurrence while enabling extensions to hypercomplex and quaternionic frameworks.

Searching arXiv for the topic and closely related papers. arXiv search query: "Gaussian modified third-order Jacobsthal numbers" Gaussian modified third-order Jacobsthal numbers are Gaussian-number-valued analogues of the modified third-order Jacobsthal sequence. In the formulation introduced by Morales, they arise as the special case $a=3$, $b=1$, $c=3$ of the generalized Gaussian third-order Jacobsthal sequence and satisfy the third-order recurrence

$$Kg_{n+3}^{(3)}=Kg_{n+2}^{(3)}+Kg_{n+1}^{(3)}+2Kg_n^{(3)},$$

with initial values

$$Kg_0^{(3)}=3-\frac12 i,\qquad Kg_1^{(3)}=1+3i,\qquad Kg_2^{(3)}=3+i.$$

The same work records the relation

$Kg_n^{(3)}=K_n^{(3)}+iK_{n-1}^{(3)},$

linking the Gaussian sequence to the real modified third-order Jacobsthal numbers $K_n^{(3)}$. Closely related literature also studies complex or unrestricted complex variants of the same underlying recurrence, so the subject sits at the intersection of linear recurrences, periodic root-of-unity corrections, and hypercomplex lifts of Jacobsthal-type sequences (Morales, 2 Aug 2025).

1. Real precursor and emergence of the Gaussian sequence

The real precursor is the modified third-order Jacobsthal sequence $\{K_n^{(3)}\}_{n\ge 0}$, introduced with recurrence

$$K_{n+3}^{(3)}=K_{n+2}^{(3)}+K_{n+1}^{(3)}+2K_n^{(3)},$$

and initial conditions

$K_0^{(3)}=3,\qquad K_1^{(3)}=1,\qquad K_2^{(3)}=3.$

Its initial segment is

$b=1$0

and it satisfies

$b=1$1

where $b=1$2 denotes the third-order Jacobsthal sequence (Cerda-Morales, 2019).

A common misconception is that the Gaussian version was already defined in this 2019 note. It was not. The note develops the real modified sequence, its generating function, Binet-style formula, periodic correction terms, and a range of identities, thereby providing the algebraic template later used for Gaussian generalization. The explicit Gaussian modified third-order Jacobsthal numbers appear later as a specialization of generalized Gaussian third-order Jacobsthal numbers, with the real initial data $b=1$3, $b=1$4, $b=1$5 selected so that the resulting Gaussian sequence is the modified Gaussian analogue of the real sequence above (Morales, 2 Aug 2025).

This historical progression is mathematically natural. The real sequence already has a characteristic polynomial with one real root and two nonreal cube-root components, so the transition from integer-valued to Gaussian-valued recurrences preserves the same spectral structure.

2. Definition, initial values, and characteristic structure

In the generalized Gaussian framework, the sequence $b=1$6 is defined by

$b=1$7

with initial values

$b=1$8

Specializing to $b=1$9, $c=3$0, $c=3$1 yields the Gaussian modified third-order Jacobsthal numbers: $c=3$2 The paper also states

$c=3$3

Accordingly, the first several terms are

$c=3$4

(Morales, 2 Aug 2025).

The characteristic equation is

$c=3$5

with roots

$c=3$6

where $c=3$7 are the roots of

$c=3$8

These satisfy

$c=3$9

The oscillatory part is encoded by the $$Kg_{n+3}^{(3)}=Kg_{n+2}^{(3)}+Kg_{n+1}^{(3)}+2Kg_n^{(3)},$$0-periodic auxiliary sequence

$$Kg_{n+3}^{(3)}=Kg_{n+2}^{(3)}+Kg_{n+1}^{(3)}+2Kg_n^{(3)},$$1

for which

$$Kg_{n+3}^{(3)}=Kg_{n+2}^{(3)}+Kg_{n+1}^{(3)}+2Kg_n^{(3)},$$2

This decomposition isolates two distinct behaviors: an exponential contribution controlled by the real root $$Kg_{n+3}^{(3)}=Kg_{n+2}^{(3)}+Kg_{n+1}^{(3)}+2Kg_n^{(3)},$$3, and a bounded $$Kg_{n+3}^{(3)}=Kg_{n+2}^{(3)}+Kg_{n+1}^{(3)}+2Kg_n^{(3)},$$4-periodic correction controlled by the nonreal roots. That separation governs essentially all closed forms and identities for the sequence.

3. Closed form, generating function, and basic linear identities

The exact Binet-type formula for the Gaussian modified third-order Jacobsthal numbers is

$$Kg_{n+3}^{(3)}=Kg_{n+2}^{(3)}+Kg_{n+1}^{(3)}+2Kg_n^{(3)},$$5

The corresponding generating function is

$$Kg_{n+3}^{(3)}=Kg_{n+2}^{(3)}+Kg_{n+1}^{(3)}+2Kg_n^{(3)},$$6

Both formulas are obtained by specializing the general Gaussian third-order Jacobsthal theory to $$Kg_{n+3}^{(3)}=Kg_{n+2}^{(3)}+Kg_{n+1}^{(3)}+2Kg_n^{(3)},$$7, $$Kg_{n+3}^{(3)}=Kg_{n+2}^{(3)}+Kg_{n+1}^{(3)}+2Kg_n^{(3)},$$8, $$Kg_{n+3}^{(3)}=Kg_{n+2}^{(3)}+Kg_{n+1}^{(3)}+2Kg_n^{(3)},$$9 (Morales, 2 Aug 2025).

Several basic linear identities follow immediately from the same specialization. The shift-by-$$Kg_0^{(3)}=3-\frac12 i,\qquad Kg_1^{(3)}=1+3i,\qquad Kg_2^{(3)}=3+i.$$0 identity is

$$Kg_0^{(3)}=3-\frac12 i,\qquad Kg_1^{(3)}=1+3i,\qquad Kg_2^{(3)}=3+i.$$1

and the three-term sum identity is

$$Kg_0^{(3)}=3-\frac12 i,\qquad Kg_1^{(3)}=1+3i,\qquad Kg_2^{(3)}=3+i.$$2

The paper also gives a first-step relation: $$Kg_0^{(3)}=3-\frac12 i,\qquad Kg_1^{(3)}=1+3i,\qquad Kg_2^{(3)}=3+i.$$3

These formulas show that the Gaussian sequence inherits the real third-order Jacobsthal dynamics without altering the denominator of the generating function or the characteristic roots. The only change is the Gaussian-valued numerator data, which modifies the coefficients of the exponential and periodic terms.

4. Summation laws, negative indices, and determinant identities

The finite-sum formula is

$$Kg_0^{(3)}=3-\frac12 i,\qquad Kg_1^{(3)}=1+3i,\qquad Kg_2^{(3)}=3+i.$$4

For negative subscripts, the recurrence is extended by

$$Kg_0^{(3)}=3-\frac12 i,\qquad Kg_1^{(3)}=1+3i,\qquad Kg_2^{(3)}=3+i.$$5

and the specialized negative-index formulas are

$$Kg_0^{(3)}=3-\frac12 i,\qquad Kg_1^{(3)}=1+3i,\qquad Kg_2^{(3)}=3+i.$$6

equivalently,

$$Kg_0^{(3)}=3-\frac12 i,\qquad Kg_1^{(3)}=1+3i,\qquad Kg_2^{(3)}=3+i.$$7

These identities provide an explicit analytic continuation of the sequence to negative indices (Morales, 2 Aug 2025).

The same paper specializes general determinant-type identities. Writing

$$Kg_0^{(3)}=3-\frac12 i,\qquad Kg_1^{(3)}=1+3i,\qquad Kg_2^{(3)}=3+i.$$8

the specialized d’Ocagne identity is

$$Kg_0^{(3)}=3-\frac12 i,\qquad Kg_1^{(3)}=1+3i,\qquad Kg_2^{(3)}=3+i.$$9

Cassini’s identity becomes

$Kg_n^{(3)}=K_n^{(3)}+iK_{n-1}^{(3)},$0

These relations are the Gaussian counterparts of classical determinant identities for second- and higher-order recurrence sequences. Their structure again splits into an exponential part and a periodic root-of-unity part, which is characteristic of third-order Jacobsthal-type families.

5. Alternative complex formulations and normalization issues

A related but distinct line of work studies unrestricted complex modified third-order Jacobsthal numbers in the form

$Kg_n^{(3)}=K_n^{(3)}+iK_{n-1}^{(3)},$1

with the “usual” complex specialization

$Kg_n^{(3)}=K_n^{(3)}+iK_{n-1}^{(3)},$2

In that setting, the sequence is treated as a specialization of unrestricted modified third-order Jacobsthal quaternions, and the complex sequence satisfies the same third-order recurrence

$Kg_n^{(3)}=K_n^{(3)}+iK_{n-1}^{(3)},$3

where $Kg_n^{(3)}=K_n^{(3)}+iK_{n-1}^{(3)},$4 (Morales, 2024).

The corresponding Binet-like formula is

$Kg_n^{(3)}=K_n^{(3)}+iK_{n-1}^{(3)},$5

and the generating function is

$Kg_n^{(3)}=K_n^{(3)}+iK_{n-1}^{(3)},$6

This formulation is closely related to, but not identical with, Morales’s Gaussian modified sequence. In (Morales, 2 Aug 2025) the principal normalization is

$Kg_n^{(3)}=K_n^{(3)}+iK_{n-1}^{(3)},$7

for $Kg_n^{(3)}=K_n^{(3)}+iK_{n-1}^{(3)},$8, together with the generalized initialization

$Kg_n^{(3)}=K_n^{(3)}+iK_{n-1}^{(3)},$9

In (Morales, 2024) the standard complex choice is instead

$K_n^{(3)}$0

whose first terms begin

$K_n^{(3)}$1

The difference is therefore a matter of shift convention and initialization rather than recurrence structure.

A second normalization issue appears in quaternion literature. One 2025 paper uses a companion “modified third-order Jacobsthal” sequence with initial values

$K_n^{(3)}$2

while still retaining the same recurrence and characteristic polynomial (Morales, 12 Feb 2025). This shows that the term “modified third-order Jacobsthal” is not completely uniform across the literature. For Gaussian modified third-order Jacobsthal numbers in the sense of Morales, however, the underlying real sequence is the one with initial values $K_n^{(3)}$3.

6. Hypercomplex extensions and broader structural context

The Gaussian sequence belongs to a larger program of lifting third-order Jacobsthal-type recurrences into hypercomplex algebras. One such development introduces third-order Jacobsthal $K_n^{(3)}$4-parameter generalized quaternions and their modified counterparts by embedding consecutive scalar terms into quaternion coordinates: $K_n^{(3)}$5

$K_n^{(3)}$6

These quaternion sequences satisfy the same recurrence componentwise and admit Binet formulas, generating functions, sum identities, Cassini identities, and d’Ocagne/Vajda-type relations (Morales, 12 Feb 2025).

Earlier work on third-order Jacobsthal quaternions develops the same recurrence backbone using the companion matrix

$K_n^{(3)}$7

together with quaternion generating functions, Binet formulas, and matrix representations (Cerda-Morales, 2017). This suggests that the Gaussian modified sequence is best understood not as an isolated construction, but as one instance of a recurrence-preserving lift from $K_n^{(3)}$8 to a richer coefficient domain.

From this perspective, the central invariant is the scalar recurrence

$K_n^{(3)}$9

with characteristic polynomial

$\{K_n^{(3)}\}_{n\ge 0}$0

Once that recurrence is fixed, integer, Gaussian, complex, quaternionic, and generalized quaternionic versions differ primarily in their initial data and ambient algebra. The repeated appearance of the same denominator

$\{K_n^{(3)}\}_{n\ge 0}$1

in generating functions, and the same decomposition into a $\{K_n^{(3)}\}_{n\ge 0}$2 term plus a $\{K_n^{(3)}\}_{n\ge 0}$3-periodic correction, is the unifying feature across these families.

In that sense, Gaussian modified third-order Jacobsthal numbers occupy a precise intermediate position. They retain the scalar third-order Jacobsthal spectral data, but package it in Gaussian form; they are more structured than a purely formal complexification, yet remain commutative and considerably simpler than quaternionic generalizations.