On t-Dimensional Gersenne Sequences and Their Symmetry Properties

Abstract

In this paper, we introduce a new class of sequences of integers called t-dimensional Gersenne sequences, with $2\le t<\infty$, which extends the classical Mersenne-type sequences into higher dimensions by incorporating complex and hyper-complex structures. One of our main goals is to establish analogous or symmetrical properties for the t-dimensional Gersenne sequences that are similar to those we know for the classical Gersenne sequence. We investigate some algebraic properties of this new sequences, and the Binet formula is given so as to allow us to study several identities, such as the Tagiuri–Vajda identity, among others. Furthermore, some results concerning the partial sums involving terms of these new sequences are presented.

1. Introduction

Let $k,a,b$ be fixed nonnegative integers, and let ${\left\{G_n^{(k,a,b)}\right\}}_{n\ge 0}={\left\{G_n^{\left(k\right)}\right\}}_{n\ge 0}$ be the Gersenne sequence defined by the recurrence

$$G_n^{\left(k\right)}=(k+1)G_{n-1}^{\left(k\right)}-k{G}_{n-2}^{\left(k\right)},\mathrm{for}n\ge 2,$$

(1)

with initial values $G_0^{\left(k\right)}=a$, $G_1^{\left(k\right)}=b$. In particular, ${\left\{G_n^{(k,0,1)}\right\}}_{n\ge 0}={\left\{v_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the classical k-Mersenne sequence. A straightforward calculation shows that the first five elements of the Gersenne sequence are

$$a,b,k(b-a)+b,(k^2+k)(b-a)+b,(k^3+k^2+k)(b-a)+b,\dots$$

(2)

Throughout the text, unless otherwise stated, k is a positive integer greater than 1, that is, $k>1$.

According [1] and the references therein, this formulation generalizes the classical Mersenne sequence, as well as others. For example, when $k=2$, $a=0$, and $b=1$ in (2) we find that the 2-Mersenne sequence is the familiar Mersenne sequence given by

$${\left\{M_n\right\}}_{n\ge 0}=\{0,1,3,7,15,31,63,127,255,\dots \},$$

which is referred to as the A000225 sequence in the OEIS [2]. In the setting of $k=10$, $a=0$, and $b=1$, we obtain the classic Repunit numbers or the 10-Mersenne numbers.

$${\left\{r_n\right\}}_{n\ge 0}=\{0,1,11,111,1111,11111,\dots \},$$

the sequence $A002275$ in OEIS [2]. However, if we consider $k={10}^2$, $a=0$, and $b=1$, we obtain the One-Zero numbers.

$${\left\{U_n\right\}}_{n\ge 0}=\{0,1,101,10101,1010101,\dots \},$$

where the id number is $A094028$ in OEIS [2].

There is a long history as well as vast literature concerning t-dimensional versions of some recurrence sequences, which have a wide range of applications in mathematics and the applied sciences. Here, we highlight some belonging to the class of Gersenne sequences. In [3], the authors present Gaussian Mersenne numbers, while in [4], the t-dimensional forms associated with the Mersenne sequence were studied. Already in [5] the t-dimensional relations of the Repunit sequence were already investigated. Recently, in [6] the authors analyzed the properties of the t-dimensional One-Zero sequence. Inspired by these and other related studies, further research has been carried out into bidimensional extensions of additional numerical sequences, building upon their unidimensional frameworks.

In this work, we investigate the structure of t-dimensional Gersenne sequences as an extension of the unidimensional Gersenne sequence, with integers $t\ge 1$ and special attention paid to the bidimensional Gersenne sequence. We conducted a detailed analysis of this sequence in order to establish a strong theoretical model that enables its generalization to both tridimensional and higher-dimensional (t-dimensional) settings. Building upon foundational results related to the classical (unidimensional) Gersenne sequence, as presented in Section 2, we derive and present a collection of identities and properties that characterize these bidimensional extensions.

The article is structured as follows: Section 2 presents the background and auxiliary results related to the unidimensional or usual Gersenne Sequence, which provide the basis for the analysis in the subsequent sections. So, we begin by revisiting the foundational properties of the unidimensional Gersenne sequence ${\left\{G_n^{\left(k\right)}\right\}}_{n\ge 0}$, defined by the recurrence (1) and with arbitrary initial values $G_0^{\left(k\right)}=a$ and $G_1^{\left(k\right)}=b$, and we recall key identities such as Binet’s formula and the Tagiuri–Vajda identity. In Section 3, we define the bidimensional version of the Gersenne sequence and exhibit some of its properties also satisfied by the bidimensional Gersenne sequence; it will in particular study some classical identities and the partial sums of its terms. Therefore, building upon this foundation, we define the bidimensional Gersenne sequence ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$, which satisfies independent second-order recurrence relations in each dimension and takes values in the field of complex numbers. We prove that the general term of these sequences can be written explicitly as $B{G}_{(m,n)}=G_m^{\left(k\right)}+G_n^{\left(k\right)}i$, and we derive a closed-form Binet-type expression for them. Several classical identities known for linear recurrences, such as those of d’Ocagne, Catalan, and Cassini, are generalized to this bidimensional context. In addition, we present partial sum identities and analyze alternating sum structures for ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$. In Section 4, we define and study the tridimensional Gersenne sequences and present some results related to them, and we again present the definition of the multidimensional Gersenne sequence and its consequence. We further extend this framework to define and explore the tridimensional Gersenne sequence ${\left\{T{G}_{(n_1,n_2,n_3)}\right\}}_{n_1,n_2,n_3\ge 0}$, introducing additional imaginary units i and j with $i^2=j^2=-1$, and we show that the general term of these sequences admits the form $T{G}_{(n_1,n_2,n_3)}=G_{n_1}^{\left(k\right)}+G_{n_2}^{\left(k\right)}i+G_{n_3}^{\left(k\right)}j$. Finally, we generalize our construction to define the multidimensional Gersenne${\left\{M{G}_{(n_1,n_2,\dots ,n_t)}\right\}}_{n_1,n_2,\dots ,n_t\ge 0}$ for all integers $t\ge 1$, providing a unified expression involving a sum of Gersenne terms weighted by independent imaginary units. Finally, we present the final remarks of the work.

This paper examines whether the bidimensional Gersenne sequence exhibits symmetric or analogous properties to those of the unidimensional Gersenne sequence. The main novelty of this work is to express some classical identities for unidimensional sequences for the bidimensional Gersenne sequence, for example, the Tagiuri–Vajda identity and its derivations.

2. Gersenne Sequence

In this section, we recap some essential results associated with the unidimensional version of the Gersenne sequence. These auxiliary results play a crucial role in establishing new properties for the t-dimensional forms of this sequence.

First we present the Binet formula for the Gersenne sequence.

Lemma 1

(Proposition 2 [1]). (Binet’s formula) For all non-negative integers n, we have

$$G_n^{\left(k\right)}={\displaystyle \frac{c{k}^n+d}{k-1}},$$

(3)

where $G_0^{\left(k\right)}=a$, $G_1^{\left(k\right)}=b$, $c=b-a$, and $d=ka-b$ with $k>1$.

Remember that the special case $a=0$ and $b=1$ is denoted as

$$G_n^{(k,0,1)}={\upsilon }_n^{\left(k\right)}={\displaystyle \frac{k^n-1}{k-1}},$$

(4)

which represents the general term of the k-Mersenne sequence.

The next auxiliary results exhibit the difference and the sum of two terms of the Gersenne sequence.

Lemma 2

(Proposition 6 [1]). Let ${\left\{G_n^{\left(k\right)}\right\}}_{n\ge 0}$ be the Gersenne sequence. For all non-negative integers m and n with $m\ge n$, the following identities are verified:

(a) 

$G_m^{\left(k\right)}+G_n^{\left(k\right)}=c{k}^n{\upsilon }_{m-n}^{\left(k\right)}+2G_n^{\left(k\right)}$,

(b) 

$G_m^{\left(k\right)}-G_n^{\left(k\right)}=c{k}^n{\upsilon }_{m-n}^{\left(k\right)}$,

where $c=b-a$, ${\left\{G_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the Gersenne sequence and ${\left\{{\upsilon }_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the k-Mersenne sequence.

Specifically, when $m=n+1$ in Lemma 2, we obtain the following:

Corollary 1.

For all non-negative integers n, we have

(a) 

$G_{n+1}^{\left(k\right)}+G_n^{\left(k\right)}=c{k}^n+2G_n^{\left(k\right)}$,

(b) 

$G_{n+1}^{\left(k\right)}-G_n^{\left(k\right)}=c{k}^n$,

where $c=b-a$ and ${\left\{G_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the Gersenne sequence.

The next result is known as the Tagiuri–Vajda identity for the Gersenne sequence.

Lemma 3

(Theorem 1 [1]). Let ${\left\{G_m^{\left(k\right)}\right\}}_{m\ge 0}$ be the Gersenne sequence, and $m,k,r,s$ non-negative integers. The following identity holds:

$$G_{m+r}^{\left(k\right)}·G_{m+s}^{\left(k\right)}-G_m^{\left(k\right)}·G_{m+r+s}^{\left(k\right)}=-cd{k}^m·{\upsilon }_r^{\left(k\right)}·{\upsilon }_s^{\left(k\right)},$$

(5)

where $G_0^{\left(k\right)}=a$, $G_1^{\left(k\right)}=b$, $c=b-a$, $d=ka-b$, and ${\left\{{\upsilon }_m^{\left(k\right)}\right\}}_{m\ge 0}$ is given by Equation (4).

Next, we present some auxiliary results that will be relevant for the demonstrations of more advanced results in this study.

Proposition 1.

Let m be any non-negative integer, then

(a) 

$G_{m+1}^{\left(k\right)}-k{G}_{m-1}^{\left(k\right)}=c{k}^m-d$;

(b) 

$G_{2m}^{\left(k\right)}=G_m^{\left(k\right)}\left((k-1){\upsilon }_m^{\left(k\right)}+2\right)-k^m{\displaystyle \frac{c+d}{k-1}}$,

where $c=b-a$, $d=ka-b$, ${\left\{G_m^{\left(k\right)}\right\}}_{m\ge 0}$ is the Gersenne sequence, and ${\left\{{\upsilon }_m^{\left(k\right)}\right\}}_{m\ge 0}$ is given by Equation (4).

The proof is obtained from a straightforward calculation and making use of (3). Specifically, when $c=1$ and $d=-1$ in Proposition 1, we obtain the following:

Corollary 2.

For all non-negative integers m, we have

(a) 

${\upsilon }_{m+1}^{\left(k\right)}-k{\upsilon }_{m-1}^{\left(k\right)}=k^m+1$,

(b) 

${\upsilon }_{2m}^{\left(k\right)}=(k-1){\left({\upsilon }_m^{\left(k\right)}\right)}^2+2{\upsilon }_m^{\left(k\right)}$,

where ${\left\{{\upsilon }_m^{\left(k\right)}\right\}}_{m\ge 0}$ is given by Equation (4).

With a routine similar to the one adopted in Proposition 1, we obtain the next auxiliary result, and the proof is also omitted.

Proposition 2.

Let k, m, n, and l be arbitrary non-negative integers. Then

$$\begin{array}{c}\hfill (G_{m+n}^{\left(k\right)}+G_{m+l}^{\left(k\right)})-(G_m^{\left(k\right)}+G_{m+n+l}^{\left(k\right)})=-c(k-1)·k^m{\upsilon }_n^{\left(k\right)}{\upsilon }_l^{\left(k\right)},\end{array}$$

where ${\left\{G_n^{\left(k\right)}\right\}}_{n\ge 0}$ and ${\left\{{\upsilon }_n^{\left(k\right)}\right\}}_{n\ge 0}$ are, respectively, the Gersenne sequence and the $k$-Mersenne sequence.

To illustrate the last identity, we present an example.

Example 1.

Considering the repunit sequence ${\left\{r_n\right\}}_{n\ge 0}$ specified when we take $k=10,a=1,a=0$, and $b=1$. So $c=1$, $d=-1$, and we have that the identity for the repunit sequence is

$$r_{m+n}+r_{m+l}-(r_m+r_{m+n+l})=-9·{10}^m{r}_n{r}_l.$$

Now, we consider the sequence of partial sums ${\displaystyle \sum _{i=0}^n{G}_i^{\left(k\right)}=G_0^{\left(k\right)}+G_1^{\left(k\right)}+\dots +G_n^{\left(k\right)}}$ for $n\ge 0$, where ${\left\{G_n^{\left(k\right)}\right\}}_{n\ge 0}$ denotes the Gersenne sequence. Regarding these sums, we have the following results:

Lemma 4

(Proposition 12. [1]). Let ${\left\{G_n^{\left(k\right)}\right\}}_{n\ge 0}$ be the Gersenne sequence; then, we have the following identities:

(a) 

${\displaystyle \sum _{i=0}^n{G}_i^{\left(k\right)}=\frac{c·{\upsilon }_{n+1}^{\left(k\right)}+d(n+1)}{k-1}}$;

(b) 

${\displaystyle \sum _{i=0}^n{G}_{2i}^{\left(k\right)}=\frac{c(k^{n+1}+1){\upsilon }_{n+1}^{\left(k\right)}}{k^2-1}+\frac{d(n+1)}{k-1}}$;

(c) 

${\displaystyle \sum _{i=0}^n{G}_{2i+1}^{\left(k\right)}=\frac{c(k^{n+2}+k){\upsilon }_{n+1}^{\left(k\right)}}{k^2-1}+\frac{d(n+1)}{k-1}}$,

where $G_0^{\left(k\right)}=a$, $G_1^{\left(k\right)}=b$, $c=b-a$, $d=ka-b$, and ${\left\{{\upsilon }_m^{\left(k\right)}\right\}}_{m\ge 0}$ is given by Equation (4).

3. Bidimensional Gersenne Sequence

In this section, we investigate the extension of the unidimensional Gersenne sequence into the set of Gaussian integers $(m,n)=m+in$, for all integers m and n, and the imaginary unit i, according to the terminology of [7,8,9,10]. In the seminal work in [11], the authors introduced the bidimensional Fibonacci, also discussed in [12,13]. As we have already said, we find several studies that explore the extension of the unidimensional of some specific Horadam-type sequences to higher dimensions with some additional conditions. Here we will do it for the whole class of Gersenne sequences.

Firstly, we analyze the recurrence relations describing the bidimensional case of this sequence, revealing a comprehensive structure and properties.

Now, we present the bidimensional Gersenne sequence and obtain some properties of this new family of Mersenne-type sequences. We begin by introducing the following definition.

Definition 1.

For all integers $m\ge 0$ and $n\ge 0$, the bidimensional Gersenne sequences ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ are defined recursively by

$$\begin{array}{cc}\hfill B{G}_{(m,n)}:=& (k+1)B{G}_{(m-1,n)}-kB{G}_{(m-2,n)},\mathit{for}\mathit{all}n\mathit{and}m\ge 2,\end{array}$$

(6)

$$\begin{array}{cc}\hfill B{G}_{(m,n)}:=& (k+1)B{G}_{(m,n-1)}-kB{G}_{(m,n-2)},\mathit{for}\mathit{all}m\mathit{and}n\ge 2,\end{array}$$

(7)

with initial conditions involving complex terms

$$B{G}_{(0,0)}=a(1+i),B{G}_{(1,0)}=b+ai,B{G}_{(0,1)}=a+bi,B{G}_{(1,1)}=b(1+i),$$

and where $i^2=-1$ is the imaginary unit.

The first result shows that recurrence given in Definition 1 is well-defined.

Proposition 3.

Let $m_0\ge 2$ and $n_0\ge 2$ be integers. The term $B{G}_{(m_0,n_0)}$ is independent of the path to be taken, where ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ is the bidimensional Gersenne sequence.

Proof. 

By (6) we have $B{G}_{(m_0,n_0)}=(k+1)B{G}_{(m_0-1,n_0)}-kB{G}_{(m_0-2,n_0)}$. Now, we apply (6) and (7) and with some calculations

$$\begin{array}{ccc}\hfill (k+1)B{G}_{(m_0-1,n_0)}-kB{G}_{(m_0-2,n_0)}& \stackrel{\left(7\right)}{=}& (k+1)[(k+1)B{G}_{(m_0-1,n_0-1)}-kB{G}_{(m_0-1,n_0-2)}]\hfill \\ & & -k[(k+1)B{G}_{(m_0-2,n_0-1)}-kB{G}_{(m_0-2,n_0-2)}]\hfill \\ & =& (k+1)[(k+1)B{G}_{(m_0-1,n_0-1)}-kB{G}_{(m_0-2,n_0-1)}]\hfill \\ & & -k[(k+1)B{G}_{(m_0-1,n_0-2)}-kB{G}_{(m_0-2,n_0-2)}]\hfill \\ & \stackrel{\left(6\right)}{=}& (k+1)B{G}_{(m_0,n_0-1)}-kB{G}_{(m_0,n_0-2)},\hfill \end{array}$$

as required. □

Proposition 3 shows that Definition 1 is valid in the context of a general term $B{G}_{(m,n)}$ that is independent of the equation we consider. For instance, to find $B{G}_{(2,2)}$, we have two paths, as follows:

$$\begin{array}{c}\hfill B{G}_{(2,2)}=\left\{\begin{array}{cc}(k+1)B{G}_{(1,2)}-kB{G}_{(0,2)},\hfill & \mathrm{by}\mathrm{Equation}\left(6\right),\hfill \\ (k+1)B{G}_{(2,1)}-kB{G}_{(2,0)},\hfill & \mathrm{by}\mathrm{Equation}\left(7\right).\hfill \end{array}\right.\end{array}$$

To conclude this section, we will use the separate first pathway and second pathway and write down a few elements for two trajectories of the bidimensional Gersenne sequence as follows:

$$\begin{array}{cc}\hfill {\left\{B{G}_{(m,0)}\right\}}_{m\ge 0}=& \{a(1+i),b+ai,[k(b-a)+b]+ai,[(k^2+k)(b-a)+b]+ai,\dots \},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\left\{B{G}_{(0,n)}\right\}}_{n\ge 0}=& \{a(1+i),a+bi,a+[k(b-a)+b]i,a+[(k^2+k)(b-a)+b]i,\dots \}.\hfill \end{array}$$

3.1. Some Properties

In this section, we examine some of the general properties of bidimensional Gersenne numbers, focusing on the generalization of the unidimensional properties mentioned in previous sections, as well as deriving new results for the bidimensional context.

Proposition 4.

Let m and n denote arbitrary non-negative integers. The following properties hold:

$$\begin{array}{cc}\hfill B{G}_{(m,0)}=& G_m^{\left(k\right)}+ai,\hfill \\ \hfill B{G}_{(0,n)}=& a+G_n^{\left(k\right)}i,\hfill \\ \hfill B{G}_{(m,1)}=& G_m^{\left(k\right)}+bi,\hfill \\ \hfill B{G}_{(1,n)}=& b+G_n^{\left(k\right)}i,\hfill \end{array}$$

(8)

where ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ is the bidimensional Gersenne sequence, ${\left\{G_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the Gersenne sequence, and $i^2=-1$.

The proof can be obtained by induction and will therefore be omitted in the interest of brevity.

The following result reveals a connection involving the bidimensional Gersenne sequence and the Gaussian Gersenne sequence, as follows:

Theorem 1.

For non-negative integers m and n, the general term of the bidimensional Gersenne sequence is described as follows:

$$B{G}_{(m,n)}=G_m^{\left(k\right)}+G_n^{\left(k\right)}i,$$

where ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ is the bidimensional Gersenne sequence, ${\left\{G_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the Gersenne sequence, and $i^2=-1$.

Proof. 

We begin by applying induction to n, with m held constant. Consider $n=0$. Note that by Definition 1 we have $B{G}_{(0,0)}=a(1+i)=G_0^{\left(k\right)}+G_0^{\left(k\right)}i$ and $B{G}_{(1,0)}=b+ai$$=G_1^{\left(k\right)}+G_0^{\left(k\right)}i$. For $m\ge 2$ held constant, we have

$$\begin{array}{c}\hfill B{G}_{(m,0)}=G_m^{\left(k\right)}+ai=G_m^{\left(k\right)}+G_0^{\left(k\right)}i.\end{array}$$

This fact is validated by (8), also given that $G_0^{\left(k\right)}=a$. Assuming the validity of the theorem for all n, we will demonstrate its validity for $n+1$. From (7), the following holds:

$$\begin{array}{cc}\hfill B{G}_{(m,n+1)}=& (k+1)B{G}_{(m,n)}-kB{G}_{(m,n-1)}.\hfill \end{array}$$

Now, by the induction hypothesis, we get the following:

$$\begin{array}{cc}\hfill B{G}_{(m,n+1)}=& (k+1)(G_m^{\left(k\right)}+G_n^{\left(k\right)}i)-k(B{G}_m^{\left(k\right)}+G_{n-1}^{\left(k\right)}i)\hfill \\ \hfill =& G_m^{\left(k\right)}+\left((k+1)G_{k,n}-k{G}_{k,n-1}\right)i,\hfill \end{array}$$

and following (1),

$$\begin{array}{cc}\hfill B{G}_{(m,n+1)}=& G_m^{\left(k\right)}+G_{n+1}^{\left(k\right)}i,\hfill \end{array}$$

which verifies the result.

Let us now fix n and perform the induction on m. For $m=0$ note that $B{G}_{(0,1)}$$=a+bi=G_0^{\left(k\right)}+G_1^{\left(k\right)}i$ and given the initial terms of ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ we have

$$\begin{array}{cc}\hfill B{G}_{(0,n)}=& a+G_n^{\left(k\right)}i=0+{\upsilon }_{n}i=G_0^{\left(k\right)}+G_n^{\left(k\right)}i,\hfill \end{array}$$

which is true by (6). Suppose the equality is valid for all values of m. Let us prove that for $m+1$. By (6), we have

$$\begin{array}{c}\hfill B{G}_{(m+1,n)}=(k+1)B{G}_{(m,n)}-kB{G}_{(m-1,n)}.\end{array}$$

Now, by the induction hypothesis, we get the following:

$$\begin{array}{cc}\hfill B{G}_{(m+1,n)}=& (k+1)(G_m^{\left(k\right)}+G_n^{\left(k\right)}i)-k(G_{m-1}^{\left(k\right)}+G_n^{\left(k\right)}i)\hfill \\ \hfill =& \left((k+1)G_m^{\left(k\right)}-k{G}_{m-1}^{\left(k\right)}\right)+G_n^{\left(k\right)}i,\hfill \end{array}$$

and by (1), we have

$$\begin{array}{c}\hfill B{G}_{(m+1,n)}=G_{m+1}^{\left(k\right)}+G_n^{\left(k\right)}i,\end{array}$$

as we wanted to show. □

When we specify $c=d=1$ in (3), Theorem 1 can be rewritten as follows:

Corollary 3.

For non-negative integers m and n, the bidimensional $k$-Mersenne sequence is described as follows:

$$B{\upsilon }_{(m,n)}={\upsilon }_m^{\left(k\right)}+{\upsilon }_n^{\left(k\right)}i,$$

where ${\left\{B{\upsilon }_m^{\left(k\right)}\right\}}_{m,n\ge 0}$ is the bidimensional $k$-Mersenne sequence, ${\left\{{\upsilon }_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the $k$-Mersenne sequence, and $i^2=-1$.

By combining the preceding result with Binet’s formula, presented in Equation (3), we can derive Binet’s formula for the bidimensional Gersenne sequence, as follows.

Corollary 4

(Binet’s formula). For non-negative integers n and m, we have

$$B{G}_{(m,n)}={\displaystyle \frac{c·k^m+d}{k-1}}+{\displaystyle \frac{c·k^n+d}{k-1}}i,$$

or equivalently,

$$B{G}_{(m,n)}={\displaystyle \frac{c(k^m+k^{n}i)+d(1+i)}{k-1}},$$

where $c=b-a$, $d=ka-b$, ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ is the bidimensional Gersenne sequence, and $i^2=-1$.

From Theorem 1 the next two results follows.

Proposition 5.

Let us consider non-negative integers m, n, and r, with $m\ge n$ and $r\ge n$. Let ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ be the bidimensional Gersenne sequence, then the identities are verified as follows:

(a) 

$B{G}_{(m,r)}-B{G}_{(n,r)}=c·k^n{\upsilon }_{m-n}^{\left(k\right)}$;

(b) 

$B{G}_{(m,r)}-B{G}_{(m,n)}=c·k^n{\upsilon }_{r-n}^{\left(k\right)}i$;

(c) 

$B{G}_{(m,r)}+B{G}_{(n,r)}=c{k}^n{\upsilon }_{m-n}^{\left(k\right)}+2B{G}_{(n,r)}$;

(d) 

$B{G}_{(m,r)}+B{G}_{(m,n)}=2B{G}_{(m,n)}+c·k^n{\upsilon }_{r-n}^{\left(k\right)}i$,

where $c=b-a$, $d=ka-b$, ${\left\{{\upsilon }_n\right\}}_n$ is the k-Mersenne sequence, and $i^2=-1$.

Proof. 

(a)

According to Theorem 1 we have

$$\begin{array}{cc}\hfill B{G}_{(m,r)}-B{G}_{(n,r)}=& G_m^{\left(k\right)}+G_r^{\left(k\right)}i-\left(G_n^{\left(k\right)}+G_r^{\left(k\right)}i\right)=G_m^{\left(k\right)}-G_n^{\left(k\right)}.\hfill \end{array}$$

From item (b) of Lemma 2, we have $G_m^{\left(k\right)}-G_n^{\left(k\right)}=c·k^n{\upsilon }_{m-n}^{\left(k\right)},$ and the statement is proved.

(b)

The proof of this item is similarly to that of item (a).

(c)

According to Theorem 1 we have

$$\begin{array}{cc}\hfill B{G}_{(m,r)}+B{G}_{(n,r)}=& (G_m^{\left(k\right)}+G_r^{\left(k\right)}i)+(G_n^{\left(k\right)}+G_r^{\left(k\right)}i)=G_m^{\left(k\right)}+G_n^{\left(k\right)}+2G_r^{\left(k\right)}i.\hfill \end{array}$$

From item (a) of Lemma 2 and Theorem 1, we have

$$\begin{array}{ccc}& & G_m^{\left(k\right)}+G_n^{\left(k\right)}+2G_r^{\left(k\right)}i=c{k}^n{\upsilon }_{m-n}^{\left(k\right)}+2G_n^{\left(k\right)}+2G_r^{\left(k\right)}i\hfill \\ & =& c{k}^n{\upsilon }_{m-n}^{\left(k\right)}+2(G_n^{\left(k\right)}+G_r^{\left(k\right)}i)=c{k}^n{\upsilon }_{m-n}^{\left(k\right)}+2B{G}_{(n,r)},\hfill \end{array}$$

as required.

(d)

The proof of this item is similarly to that of item (c).

□

The following result is an immediate consequence of Proposition 5.

Corollary 5.

Let us consider non-negative integers m and n. Let ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ be the bidimensional Gersenne sequence, then the identities are verified as follows:

(a) 

$B{G}_{(m+1,n)}-B{G}_{(m,n)}=c·k^m$;

(b) 

$B{G}_{(m,n+1)}-B{G}_{(m,n)}=c·k^{m}i$;

(c) 

$B{G}_{(m+1,n)}+B{G}_{(m,n)}=c{k}^n+2B{G}_{(m,n)}$;

(d) 

$B{G}_{(m+1,n)}+B{G}_{(m,n)}i=2B{G}_{(m,n)}+c{k}^{n}i$,

where $c=b-a$.

The following result follows from the combination of Proposition 1 and Theorem 1, and the respective proof is therefore omitted.

Proposition 6.

For non-negative integers m and n and for the bidimensional Gersenne sequence ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$, the identities are verified as follows:

(a) 

$B{G}_{(m+1,n)}-B{G}_{(m-1,n)}=c{k}^m-d$;

(b) 

$B{G}_{(m,n+1)}-B{G}_{(m,n-1)}=(c{k}^m-d)i$;

(c) 

$B{G}_{(2m,n)}-2G_m^{\left(k\right)}=(k-1)G_m^{\left(k\right)}·{\upsilon }_t^{\left(k\right)}-k^m·{\displaystyle \frac{c+d}{k-1}}+G_n^{\left(k\right)}i$;

(d) 

$B{G}_{(m,2n)}-2G_n^{\left(k\right)}=G_m^{\left(k\right)}+\left((k-1)G_n^{\left(k\right)}·{\upsilon }_n^{\left(k\right)}-k^n·{\displaystyle \frac{c+d}{k-1}}\right)i$,

where $c=b-a$, $d=ka-b$, ${\left\{G_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the Gersenne sequence, ${\left\{{\upsilon }_n\right\}}_n$ is the k-Mersenne sequence, and $i^2=-1$.

3.2. Classical Identities

The classical Tagiuri–Vajda identity for the unidimensional Gersenne sequence is extended to two dimensions in the following result.

Initially, we shall consider n to be constant in the second coordinate of the indexes, whereas the first coordinate will be variable.

Theorem 2

(First Tagiuri–Vajda identity). For non-negative integers $m,r,s$, and n, we have

$$B{G}_{(m+r,n)}B{G}_{(m+s,n)}-B{G}_{(m,n)}B{G}_{(m+r+s,n)}=-c{k}^m{\upsilon }_r^{\left(k\right)}{\upsilon }_s^{\left(k\right)}(d+(k-1)G_n^{\left(k\right)}i),$$

where $c=b-a$, $d=ka-b$, ${\left\{G_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the Gersenne sequence, ${\left\{{\upsilon }_n\right\}}_{n\ge 0}$, and $i^2=-1$.

Proof. 

By Theorem 1 we have

$$\begin{array}{ccc}& & B{G}_{(m+r,n)}B{G}_{(m+s,n)}-B{G}_{(m,n)}B{G}_{(m+r+s,n)}\hfill \\ & =& (G_{m+r}^{\left(k\right)}+G_n^{\left(k\right)}i)(G_{m+s}^{\left(k\right)}+G_n^{\left(k\right)}i)-(G_m^{\left(k\right)}+G_n^{\left(k\right)}i)(G_{m+r+s}^{\left(k\right)}+G_n^{\left(k\right)}i)\hfill \\ & =& \left(G_{m+r}^{\left(k\right)}{G}_{m+s}^{\left(k\right)}-G_m^{\left(k\right)}{G}_{m+r+s}^{\left(k\right)}\right)+(G_{m+r}^{\left(k\right)}+G_{m+s}^{\left(k\right)}-G_m^{\left(k\right)}-G_{m+r+s}^{\left(k\right)})G_n^{\left(k\right)}i\hfill \end{array}$$

Applying Proposition 2 and Lemma 3 simultaneously, we get

$$\begin{array}{cc}\hfill & B{G}_{(m+r,n)}B{G}_{(m+s,n)}-B{G}_{(m,n)}B{G}_{(m+r+s,n)}=\hfill \\ \hfill =& -cd{k}^m·{\upsilon }_r^{\left(k\right)}·{\upsilon }_s^{\left(k\right)}-c{k}^m(k-1){\upsilon }_r^{\left(k\right)}{\upsilon }_s^{\left(k\right)}{G}_n^{\left(k\right)}i,\hfill \end{array}$$

and we have the validity of the result. □

Building upon the Tagiuri–Vajda identity, we derive the results establishing d’Ocagne’s identity and Catalan’s identity for the bidimensional Gersenne sequence ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$.

Proposition 7

(First d’Ocagne identity). For non-negative integers $m,n$, and t with $t\ge m$, the following holds:

$$B{G}_{(m,n)}B{G}_{(t+1,n)}-B{G}_{(t,n)}B{G}_{(m+1)}=-c{k}^t{\upsilon }_{m-t}^{\left(k\right)}(d+(k-1)G_n^{\left(k\right)}i)$$

where $c=b-a$, $d=ka-b$, ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ is the bidimensional Gersenne sequence, ${\left\{G_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the Gersenne sequence, ${\left\{{\upsilon }_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the $k$-Mersenne sequence, and $i^2=-1$.

Proof. 

Taking $m=t$, $r=0$, and $s=1$ in Equation (2). □

Proposition 8

(First Catalan identity). For non-negative integers t and r with $t\ge r$, the following holds:

$${\left(B{G}_{(t,n)}\right)}^2-B{G}_{(t-r,n)}B{G}_{(t+r,n)}=-c{k}^{t-r}{\left({\upsilon }_r^{\left(k\right)}\right)}^2(d+(k-1)G_n^{\left(k\right)}i).$$

where $c=b-a$, $d=ka-b$, ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ is the bidimensional Gersenne sequence, ${\left\{G_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the Gersenne sequence, ${\left\{{\upsilon }_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the $k$-Mersenne sequence, and $i^2=-1$.

Proof. 

Using $r=s$ and $m+r=t$ in Equation (2). □

At the expense of the above result, we obtain the Cassini identity.

Corollary 6

(First Cassini identity). Let $t\ge 1$ be any integers, then

$${\left(B{G}_{(t,n)}\right)}^2-B{G}_{(t-1,n)}B{G}_{(t+1,n)}=-c{k}^{t-1}(d+(k-1)G_n^{\left(k\right)}i),$$

where $c=b-a$, $d=ka-b$, ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ is the bidimensional Gersenne sequence, ${\left\{G_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the Gersenne sequence, ${\left\{{\upsilon }_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the $k$-Mersenne sequence, and $i^2=-1$.

To illustrate the above results, we present an example with the identities for the bidimensional Mersenne sequence.

Example 2.

Let $m,n,$ and r be any non-negative integers. Considering $k=2,a=0$, and $b=1$ in Definition 1, we have the bidimensional Mersenne sequence ${\left\{B{M}_{(m,n)}\right\}}_{m,n\ge 0}$. We obtain that the first Tagiuri–Vajda, D’Ocagne, Catalan, and Cassini identities are expressed as follows:

$$\begin{array}{cc}\hfill B{M}_{(m+r,n)}B{M}_{(m+s,n)}-B{M}_{(m,n)}B{M}_{(m+s+r,n)}=& 2^m{M}_n{M}_s(1-M_{n}i);\hfill \\ \hfill B{M}_{(m,n)}B{M}_{(r+1,n)}-B{M}_{(m+1,n)}B{M}_{(r,n)}=& -2^r{M}_{m-r}(-1+M_{n}i);\hfill \\ \hfill B{M}_{(m,n)}^2-B{M}_{(m+s,n)}B{M}_{(r-s,n)}=& -2^{m-s}{M}_s^2(-1+M_{n}i);\hfill \\ \hfill B{M}_{(m,n)}^2-B{M}_{(m+s,n)}B{M}_{(r-s,n)}=& -2^{m-1}(-1+M_{n}i)\hfill \end{array}$$

where ${\left\{M_n\right\}}_{n\ge 0}$ is the Mersenne sequence, $i^2=-1$, and $m,n,r$, and s are the non-negative integers.

Similarly, results may be derived by considering the variation of the indexes in the second coordinate. The demonstrations of these results are omitted in the interest of brevity.

Theorem 3

(Second Tagiuri–Vajda identity). For non-negative integers $m,r,s$, and n, we have

$$B{G}_{(m,n+r)}B{G}_{(m,n+s)}-B{G}_{(m,n)}B{G}_{(m,n+r+s)}=-c{k}^n{\upsilon }_r^{\left(k\right)}{\upsilon }_s^{\left(k\right)}((k-1)G_m^{\left(k\right)}i+d),$$

where $c=b-a$, $d=ka-b$, ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ is the bidimensional Gersenne sequence, ${\left\{G_m^{\left(k\right)}\right\}}_{m\ge 0}$ is the Gersenne sequence, ${\left\{{\upsilon }_m^{\left(k\right)}\right\}}_{m\ge 0}$ is the $k$-Mersenne sequence, and $i^2=-1$.

Proposition 9

(Second d’Ocagne identity). For non-negative integers $m,n$, and t with $t\ge n$, it holds that

$$B{G}_{(m,n)}B{G}_{(m,t+1)}-B{G}_{(m,t)}B{G}_{(m,n+1)}=-c{k}^t{\upsilon }_{n-t}^{\left(k\right)}((k-1)G_m^{\left(k\right)}i+d)$$

where $c=b-a$, $d=ka-b$, ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ is the bidimensional Gersenne sequence, ${\left\{G_m^{\left(k\right)}\right\}}_{m\ge 0}$ is the Gersenne sequence, ${\left\{{\upsilon }_m^{\left(k\right)}\right\}}_{m\ge 0}$ is the $k$-Mersenne sequence and $i^2=-1$.

Proposition 10

(Second Catalan identity). For non-negative integers t and r with $t\ge r$, the following holds:

$${\left(B{G}_{(m,t)}\right)}^2-B{G}_{(m,t-r)}B{G}_{(m,t+r)}=-c{k}^{t-r}{\left({\upsilon }_r^{\left(k\right)}\right)}^2\left((k-1)G_m^{\left(k\right)}i+d\right).$$

where $c=b-a$, $d=ka-b$, ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ is the bidimensional Gersenne sequence, ${\left\{G_m^{\left(k\right)}\right\}}_{m\ge 0}$ is the Gersenne sequence, ${\left\{{\upsilon }_m^{\left(k\right)}\right\}}_{m\ge 0}$ is the $k$-Mersenne sequence, and $i^2=-1$.

Corollary 7

(Second Cassini identity). Let $t\ge 1$ be any integers, then

$${\left(B{G}_{(m,t)}\right)}^2-B{G}_{(m,t-1)}B{G}_{(m,t+1)}=-c{k}^{t-1}\left((k-1)G_m^{\left(k\right)}i+d\right),$$

where $c=b-a$, $d=ka-b$, ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ is the bidimensional Gersenne sequence, ${\left\{G_m^{\left(k\right)}\right\}}_{m\ge 0}$ is the Gersenne sequence, ${\left\{{\upsilon }_m^{\left(k\right)}\right\}}_{m\ge 0}$ is the $k$-Mersenne sequence, and $i^2=-1$.

To illustrate these results, we provide an example involving the identities of the bidimensional One-Zero sequence.

Example 3.

Let $m,n,$ and r be any non-negative integers. Considering $k={10}^2,a=0$ and $b=1$ in Definition 1, we have the bidimensional One-Zero sequence ${\left\{B{U}_{(m,n)}\right\}}_{m,n\ge 0}$, we the second Tagiuri–Vajda, D’Ocagne, Catalan, and Cassini identities are as follows:

$$\begin{array}{cc}\hfill B{U}_{(m+r,n)}B{U}_{(m+s,n)}-B{U}_{(m,n)}B{U}_{(m+s+r,n)}=& {10}^{2n}{U}_r{U}_s(1-99U_{m}i);\hfill \\ \hfill B{U}_{(m,n)}B{U}_{(r+1,n)}-B{U}_{(m+1,n)}B{U}_{(r,n)}=& {10}^{2n}{U}_{m-r}(1-99U_{m}i);\hfill \\ \hfill B{U}_{(m,n)}^2-B{U}_{(m+s,n)}B{U}_{(r-s,n)}=& {10}^{2(n-s)}{U}_s^2(1-99U_{m}i);\hfill \\ \hfill B{U}_{(m,n)}^2-B{U}_{(m+s,n)}B{U}_{(r-s,n)}=& {10}^{2(n-1)}(1-99U_{m}i);\hfill \end{array}$$

where ${\left\{U_n\right\}}_{n\ge 0}$ is the One-Zero sequence, $i^2=-1$, and $m,n,r$, and s are the non-negative integers.

It should be noted that the results presented in Example 3, for the bidimensional One-Zero sequence, are in accordance with Propositions 3.13, 3.14, 3.15, and 3.16 from [6].

3.3. Some Partial Sums

In this section, we present some results concerning the partial sums of the first $t+1$ terms of the bidimensional Gersenne sequence ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$.

We begin by considering the sequence of partial sums in the first coordinate of the indexes, given by

$$\sum _{m=0}^{t}B{G}_{(m,n)}=B{G}_{(0,n)}+B{G}_{(1,n)}+B{G}_{(2,n)}+\dots +B{G}_{(t,n)},$$

for all integers n, where ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ represents the bidimensional Gersenne sequence.

Proposition 11.

For non-negative integers m and n, let ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ be the bidimensional Gersenne sequence and k a non-negative integer. Then,

(a) 

${\displaystyle \sum _{m=0}^{t}B{G}_{(m,n)}=\frac{c·{\upsilon }_{t+1}^{\left(k\right)}+d(t+1)}{k-1}+(t+1)G_n^{\left(k\right)}i;}$

(b) 

${\displaystyle \sum _{m=0}^{t}B{G}_{(2m,n)}=\frac{c(k^{t+1}+1){\upsilon }_{t+1}^{\left(k\right)}}{k^2-1}+\frac{d(t+1)}{k-1}+(t+1)G_n^{\left(k\right)}i;}$

(c) 

${\displaystyle \sum _{m=0}^{t}B{G}_{(2m+1,n)}=\frac{c(k^{t+2}+k){\upsilon }_{t+1}^{\left(k\right)}}{k^2-1}+\frac{d(t+1)}{k-1}+(t+1)G_n^{\left(k\right)}i,}$

where $c=b-a$, $d=ka-b$, and ${\left\{G_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the Gersenne sequence.

Proof. 

(a)

By Lemma 4, item (a), and Theorem 1, we have

$$\begin{array}{cc}\hfill \sum _{m=0}^{t}B{G}_{(m,n)}=& \sum _{m=0}^t(G_m^{\left(k\right)}+G_n^{\left(k\right)}i)\hfill \\ \hfill =& \sum _{m=0}^t{G}_m^{\left(k\right)}+\sum _{m=0}^t{G}_n^{\left(k\right)}i\hfill \\ \hfill =& {\displaystyle \frac{c·{\upsilon }_{m+1}^{\left(k\right)}+d(t+1)}{k-1}}+(t+1)G_n^{\left(k\right)}i,\hfill \end{array}$$

which is the end of the proof.

(b)

Based on Lemma 4, item (b), and Theorem 1, we obtain the required result after some calculations.

(c)

Similarly, by Lemma 4, item (c), and Theorem 1 we get the result.

□

Consider now the sequence of alternating partial sums given by

$$\sum _{m=0}^t{(-1)}^{m}B{G}_{(m,n)}=B{G}_{(0,n)}-B{G}_{(1,n)}+B{G}_{(2,n)}-B{G}_{(3,n)}+\dots +{(-1)}^{t}B{G}_{(t,n)}$$

for all integers n, where ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ denotes the bidimensional Gersenne sequence.

Proposition 12.

For non-negative integers $m,n,t$, and k, let ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ be the bidimensional Gersenne sequence. Then,

(a) 

${\displaystyle \sum _{t=0}^{2m+1}{(-1)}^{t}B{G}_{(t,n)}=-\frac{c(k^{m+1}+1){\upsilon }_{m+1}^{\left(k\right)}}{k+1};}$

(b) 

${\displaystyle \sum _{t=0}^{2m}{(-1)}^{t}B{G}_{(t,n)}=\frac{c·k^{2m}{\upsilon }_{m+1}^{\left(k\right)}}{k+1}-\frac{m+2}{k^2-1}+G_n^{\left(k\right)}i,}$

where $c=b-a$, $d=ka-b$, ${\left\{G_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the Gersenne sequence, and ${\left\{{\upsilon }_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the $k$-Mersenne sequence.

Proof. 

Combining items (b) and (c) of Proposition 11, the required result is obtained following some calculations. □

Now, in the second coordinate, we examine the sequence of partial sums of the $n+1$ first terms, given by

$$\sum _{n=0}^{t}B{G}_{(m,n)}=B{G}_{(m,0)}+B{G}_{(m,1)}+\dots +B{G}_{(m,t)}$$

for $m\ge 0$, where ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ represents the bidimensional Gersenne sequence.

Similarly to Proposition 11, we have the following result. In the interest of brevity, we will omit the proof.

Proposition 13.

For non-negative integers m and n, let ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ be the bidimensional Gersenne sequence and k a non-negative integer. Then,

(a) 

${\displaystyle \sum _{n=0}^{t}B{G}_{(m,n)}=(t+1)G_m^{\left(k\right)}+\frac{c·{\upsilon }_{t+1}^{\left(k\right)}+d(t+1)}{k-1}i;}$

(b) 

${\displaystyle \sum _{n=0}^{t}B{G}_{(m,2n)}=(t+1)G_m^{\left(k\right)}+\frac{c(k^{t+1}+1){\upsilon }_{t+1}^{\left(k\right)}}{k^2-1}i+\frac{d(t+1)}{k-1}i;}$

(c) 

${\displaystyle \sum _{n=0}^{t}B{G}_{(m,2n+1)}=(t+1)G_m^{\left(k\right)}+\frac{c(k^{t+2}+k){\upsilon }_{t+1}^{\left(k\right)}}{k^2-1}i+\frac{d(t+1)}{k-1}i,}$

where $c=b-a$, $d=ka-b$, and ${\left\{G_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the Gersenne sequence.

Proposition 14.

For non-negative integers $m,n,t$, and k, let ${\left\{B{G}_{(m,n)}\right\}}_{m,n\ge 0}$ be the bidimensional Gersenne sequence. Then,

(a) 

${\displaystyle \sum _{t=0}^{2n+1}{(-1)}^{t}B{G}_{(m,t)}=-\frac{c(k^{n+1}+1){\upsilon }_{n+1}^{\left(k\right)}}{k+1}i;}$

(b) 

${\displaystyle \sum _{t=0}^{2n}{(-1)}^{t}B{G}_{(m,t)}=G_m^{\left(k\right)}+\frac{c·k^{2n}{\upsilon }_{m+1}^{\left(k\right)}}{k+1}i-\frac{n+2}{k^2-1}i,}$

where $c=b-a$, $d=ka-b$, ${\left\{G_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the Gersenne sequence, and ${\left\{{\upsilon }_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the $k$-Mersenne sequence.

4. Multidimensional Gersenne Sequence

Initially, in this section, we define and analyze a tridimensional Gersenne sequence, presenting some of its properties. We employ the bidimensional characterization of Gersenne to extend results to tridimensional and to multidimensional, or $t$-dimensional, Gersenne sequences, for all $2\le t<\infty$ integers.

The following definition is proposed as a foundational starting point:

Definition 2.

For integers $n_1\ge 0$, $n_2\ge 0$, and $n_3\ge 0$, let ${\left\{T{G}_{(n_1,n_2,n_3)}\right\}}_{n_1,n_2,n_3\ge 0}$ be the tridimensional Gersenne sequence, and the general term $T{G}_{(n_1,n_2,n_3)}$ is given by one of the following forms:

$$\begin{array}{cc}\hfill (k+1)T{G}_{(n_1-1,n_2,n_3)}-kT{G}_{(n_1-2,n_2,n_3)}& \mathit{for}\mathit{all}{n}_2,n_3,\mathit{and}{n}_1\ge 2;\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill (k+1)T{G}_{(n_1,n_2-1,n_3)}-kT{G}_{(n_1,n_2-2,n_3)}& \mathit{for}\mathit{all}{n}_1,n_3,\mathit{and}{n}_2\ge 2;\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill (k+1)T{G}_{(n_1,n_2,n_3-1)}-kT{G}_{(n_1,n_2,n_3-2)}& \mathit{for}\mathit{all}{n}_1,n_2,\mathit{and}{n}_3\ge 2,\hfill \end{array}$$

(11)

subject to the initial conditions involving complex numbers

$$\begin{array}{ccc}T{G}_{(0,0,0)}=a(1+i+j),& T{G}_{(1,0,0)}=b+a(i+j),& T{G}_{(0,1,0)}=a(1+j)+bi,\\ T{G}_{(0,0,1)}=a(1+i)+bj,& & T{G}_{(1,1,0)}=b(1+i)+aj,\\ T{G}_{(1,0,1)}=b(1+j)+ai,& T{G}_{(0,1,1)}=a+b(i+j),& T{G}_{(1,1,1)}=b(1+i+j),\end{array}$$

where $i^2=j^2=-1$ are imaginary units with $ij=ji=0$.

In a manner analogous to the approach taken in Section 3, we will present corresponding results in this section; however, here we will only present the statement, without the respective proofs.

Proposition 15.

Let $n_1$, $n_2$, and $n_3$ denote arbitrary non-negative integers. The following properties hold:

$$\begin{array}{cc}\hfill T{G}_{(n_1,0,0)}& =G_{n_1}^{\left(k\right)}+a(i+j);\hfill \\ \hfill T{G}_{(0,n_2,0)}& =a(1+j)+G_{n_2}^{\left(k\right)}i;\hfill \\ \hfill T{G}_{(0,0,n_3)}& =a(1+i)+G_{n_3}^{\left(k\right)}j;\hfill \\ \hfill T{G}_{(n_1,1,0)}& =G_{n_1}^{\left(k\right)}+bi+aj;\hfill \\ \hfill T{G}_{(n_1,0,1)}& =G_{n_1}^{\left(k\right)}+ai+bj;\hfill \\ \hfill T{G}_{(n_1,1,1)}& =G_{n_1}^{\left(k\right)}+a(i+j);\hfill \\ \hfill T{G}_{(1,n_2,0)}& =b+G_{n_2}^{\left(k\right)}i+aj;\hfill \\ \hfill T{G}_{(1,n_2,1)}& =b(1+j)+G_{n_2}^{\left(k\right)}i;\hfill \\ \hfill T{G}_{(1,0,n_3)}& =b+ai+G_{n_3}^{\left(k\right)}j;\hfill \\ \hfill T{G}_{(1,1,n_3)}& =b(1+i)+G_{n_3}^{\left(k\right)}j;\hfill \end{array}$$

where ${\left\{T{G}_{(n_1,n_2,n_3)}\right\}}_{n_1,n_2,n_3\ge 0}$ is the tridimensional Gersenne sequence, ${\left\{G_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the Gersenne sequence, and $i^2=j^2=-1$.

The subsequent result explores the connection between the tridimensional Gersenne sequence and the unidimensional version of the Gersenne sequence.

Theorem 4.

For non-negative integers $n_1$, $n_2$, and $n_3$, the tridimensional Gersenne numbers are described as follows:

$$T{G}_{(n_1,n_2,n_3)}=G_{n_1}^{\left(k\right)}+G_{n_2}^{\left(k\right)}i+G_{n_3}^{\left(k\right)}j,$$

where ${\left\{T{G}_{(n_1,n_2,n_3)}\right\}}_{n_1,n_2,n_3\ge 0}$ is the tridimensional Gersenne sequence, ${\left\{G_n^{\left(k\right)}\right\}}_{n\ge 0}$ is the Gersenne sequence, and $i^2=j^2=-1$.

Thus, using the previous result and Binet’s formula from (3), we derive the Binet formula for a tridimensional Gersenne sequence, as follows.

Corollary 8

(Binet’s formula). For non-negative integers $n_1$, $n_2$, and $n_3$, we have

$$T{G}_{(n_1,n_2,n_3)}={\displaystyle \frac{k^{n_1}-1}{k-1}}+{\displaystyle \frac{k^{n_2}-1}{k-1}}i+{\displaystyle \frac{k^{n_3}-1}{k-1}}j,$$

or equivalently,

$$T{G}_{(n_1,n_2,n_3)}={\displaystyle \frac{k^{n_1}+k^{n_2}i+k^{n_3}j-(1+i+j)}{k-1}},$$

where ${\left\{T{G}_{(n_1,n_2,n_3)}\right\}}_{n_1,n_2,n_3\ge 0}$ is the tridimensional Gersenne sequence, and $i^2=j^2=-1$.

Proposition 16.

For all non-negative integers $n_1$, $n_2$, $n_3$, and r, with $n_3\ge r$, let ${\left\{T{G}_{(n_1,n_2,n_3)}\right\}}_{n_1,n_2,n_3\ge 0}$ be the tridimensional Gersenne sequence; then, the following identities are verified:

(a) 

$T{G}_{(n_1,n_2,n_3)}-T{G}_{(n_1,n_2,n)}=c·k^r{G}_{n_3-r}^{\left(k\right)}j;$

(b) 

$T{G}_{(n_1,n_2,n_3+1)}-kT{G}_{(n_1,n_2,n_3-1)}=-(k-1)G_{n_1}^{\left(k\right)}-(k-1)G_{n_2}^{\left(k\right)}i+(c{k}^{n_3}-d)j;$

(c) 

$T{G}_{(n_1,n_2,2n_3)}-2G_{n_3}^{\left(k\right)}j=G_{n_1}^{\left(k\right)}+G_{n_2}^{\left(k\right)}i+\left((k-1)G_{n_3}^{\left(k\right)}·{\upsilon }_{n_3}^{\left(k\right)}-k^{n_3}·{\displaystyle \frac{c+d}{k-1}}\right)$,

where $c=b-a$, $d=ka-b$, ${\left\{G_m\right\}}_{m\ge 0}$ is the unidimensional Gersenne sequence, and $i^2=j^2=-1$.

Proof. 

The proof is performed similarly to Propositions 5 and 6. □

We can now present the generalized case. Let us now extend Definition 1 and introduce the multidimensional, or t-dimensional, Gersenne sequence for all $2\le t<\infty$ integers.

Definition 3.

For integers $n_1\ge 0$, $n_2\ge 0$, …, $n_t\ge 0$, and $t\ge 2$, let ${\left\{M{G}_{(n_1,n_2,\dots ,n_t)}\right\}}_{n_1,n_2,\dots ,n_t\ge 0}$ be the multidimensional Gersenne sequence. The term $M{G}_{(n_1,n_2,\dots ,n_t)}$ is given by one of the following forms:

$$\begin{array}{ccc}M{G}_{(n_1+1,n_2,\dots ,n_t)}\hfill & =& (k+1)M{G}_{(n_1,n_2,\dots ,n_t)}-kM{G}_{(n_1-1,n_2,\dots ,n_t)},\hfill \\ M{G}_{(n_1,n_2+1,\dots ,n_t)}\hfill & =& (k+1)M{G}_{(n_1,n_2,\dots ,n_t)}-kM{G}_{(n_1,n_2-1,\dots ,n_t)},\hfill \\ & ⋮& \\ M{G}_{(n_1,n_2,\dots ,n_t+1)}\hfill & =& (k+1)M{G}_{(n_1,n_2,\dots ,n_t)}-kM{G}_{(n_1,n_2,\dots ,n_t-1)},\hfill \end{array}$$

with initial conditions that include complex value terms

$$\begin{array}{c}M{G}_{(0,0,\dots ,0)}=a(1+i_1+i_2+\dots +i_{t-1}),\\ M{G}_{(1,0,\dots ,0)}=b+a(i_1+i_2+\dots +i_{t-1}),\\ M{G}_{(0,1,0,\dots ,0)}=b{i}_1+a(1+i_2+\dots +i_{t-1}),\\ M{G}_{(0,0,1,0,\dots ,0)}=b{i}_2+a(1+i_1+\dots +i_{t-1}),\\ M{G}_{(1,1,0,\dots ,0)}=b(1+i_1)+a(i_2+i_3+\dots +i_{t-1}),\\ M{G}_{(1,0,1,0,\dots ,0)}=b(1+i_2)+a(i_1+i_3+\dots +i_{t-1}),\\ M{G}_{(1,1,1,0,\dots ,0)}=b(1+i_1+i_2)+a(i_4+i_3+\dots +i_{t-1},\\ ⋮\\ M{G}_{(1,1,\dots ,1)}=b(1+i_2+i_3+\dots +i_{t-1}),\end{array}$$

where $i_1^2=i_2^2=\dots =i_{t-1}^2=-1$ are imaginary units.

By combining Definition 3 with the results established in Theorem 4 and Corollary 8, we arrive at the following key result:

Proposition 17.

For any t-tuple $(n_1,n_2,\dots ,n_t)$ of non-negative integers, the general term of the set ${\left\{M{G}_{(n_1,n_2,\dots ,n_t)}\right\}}_{n_1,n_2,\dots ,n_t\ge 0}$ is given by

$$M{G}_{(n_1,n_2,\dots ,n_t)}=G_{n_1}^{\left(k\right)}+G_{n_2}^{\left(k\right)}{i}_1+\dots +G_{n_t}^{\left(k\right)}{i}_{t-1},$$

or equivalently,

$$M{G}_{(n_1,n_2,\dots ,n_t)}={\displaystyle \frac{k^{n_1}+k^{n_2}{i}_1+\dots +k^{n_t}{i}_t-(1+i_2+\dots +i_t)}{k-1}},$$

where ${\left\{G_{n_l}^{\left(k\right)}\right\}}_{n_l\ge 0}$ is the Gersenne sequence for each $l\in \{1,2,\dots ,t\}$, and $i_2^2=i_3^2=\dots =i_{t-1}^2=-1$.

5. Final Considerations

Recurrent sequences represent a fundamental domain within discrete mathematics. The classical one-dimensional structure of the Gersenne sequence has served as a foundation for integrating imaginary units into the broader class of Mersenne-type sequences. This extension has enabled the development of two- and three-dimensional recurrence relations, ultimately paving the way for generalizations to the t-dimensional setting. The complexification of the unidimensional Gersenne sequence was accomplished by expanding both its recursive formulation and its initial conditions, thereby facilitating its extension into higher dimensions. This study presents an in-depth analysis of the complexification process and explores its connection with Horadam-type sequences. The inclusion of imaginary units and the algebraic extension of dimensional representation allowed for the incorporation of bidimensional recurrences and the derivation of multiple identities in their complex forms. Notably, emphasis is placed on the Tagiuri–Vajda identity and its derived results. These contributions provide significant analytical tools for understanding the structural intricacies of such sequences. The rigorous investigation into their recurrence relations has led to the discovery of new properties and, in some instances, the generalizations of results found in the prior literature. Altogether, this work highlights the remarkable flexibility of Mersenne-type sequences and sets the stage for the continued development of their theoretical and applied applications.