1. Introduction and Background
The Mersenne sequence is composed of non-negative integers in the form of a power of two minus one, and it is best known for some of the prime numbers that make it up, which are called Mersenne primes. These numbers are defined by the recursion relation
with initial condition
, and
, forming the sequence
which is referred to as sequence A000225 in the OEIS [
1]. Considering the initial values
and
, with the identical recurrence relation
, for all
, we have the Mersenne–Lucas numbers. The terms of this sequence are called Mersenne–Lucas numbers and are expressed in the form
, which is identified as sequence A000051 in OEIS [
1]. These two classes of numbers, which have important implications in areas such as cryptography and the identification of large prime numbers, are an indispensable concept in number theory.
Consider
k,
a and
b as fixed constants (typically integers), with
. We define the sequence
as a Mersenne-type sequence that satisfies the recurrence relation
with initial values
and
, where
a and
b are fixed constants. When
,
and
, we have the classical Mersenne numbers. If we set
,
and
, we obtain the ordinary Repunit numbers. Here, this sequence is referred to as the Gersenne sequence.
A straightforward calculation shows that the first five elements of the Gersenne sequence are
If we take
, we have a constant sequence equal to
b for every term
n greater than 1, so
when
, we have the following sequence:
Thus, by taking
and
, we have the following sequence of non-negative integers:
. Therefore, throughout the text, unless otherwise stated,
k is a positive integer greater than 1, that is,
.
The structure of the present paper is divided into four more sections, as follows. In
Section 2, we present the recurrence of the Gersenne sequence, the generalized Binet formula, and the generating function. In
Section 3, we derive several classical identities for the generalized Gersenne sequence for all integers
n, including the Tagiuri–Vajda, Catalan, Cassini, d’Ocagnes, and Gelin–Cesàro identities. In the
Section 4, we present results on partial sums of terms of the Gersenne sequence. We conclude with some final considerations and state some future work about this topic.
In the mathematical literature, there have been countless studies on the sequences of Mersenne and Mersenne–Lucas numbers. For example, in [
2], the author offers a thorough and detailed examination of these two types of special numbers; in particular, they are intimately tied to classical problems in the theory of prime numbers (see [
3,
4,
5]). It also considers practical applications and is particularly relevant in the specific context of cryptography (see [
6,
7]). In [
8], the authors defined the generalized Mersenne number as in Equation (
1), but with initial terms
and
, and present their interpretations and matrix generators for this sequence. Other generalizations or extensions of the Mersenne or Mersenne–Lucas sequences, generating function and several identities, can be found in [
9,
10,
11,
12], among others. In [
13,
14], the authors studied the generalized Gaussian Mersenne numbers with arbitrary initial values and considered two particular cases, namely, Gaussian Mersenne and Gaussian Mersenne–Lucas numbers.
2. The Gersenne Numbers
We refer to Gersenne numbers as an abbreviation of the term ’generalized Mersenne numbers’ and they are denoted by
. This name was inspired by the term Gibonacci numbers, a shortened form of generalized Fibonacci numbers, first used in [
15], as well as ‘Geonardo numbers’, which was used in [
16] to designate a generalization of the Leonardo numbers.
For a non-negative integer
n, consider the Horadam sequence
, which is defined by the second-order recurrence relation:
where
p and
q are fixed integers, and the initial conditions are
and
. This sequence was introduced by Horadam [
17,
18] and generalizes several well-known sequences that are ruled by a characteristic equation of the form
. For a more detailed discussion and further results on the Horadam sequence, see [
18,
19], as the authors of ref. [
18] introduced and studied what is now referred to as the Horadam sequence, and ref. [
19] notes that several other well-known sequences can also be expressed as Horadam sequences by selecting suitable initial terms and recurrence constants. In our study, we consider the class of Horadam-type sequences represented by the recurrence Equation (
3), where we take
and
, for all integers
. Then, for all
k,
a and
b that are fixed constants, the Gersenne sequence is given by the recurrence relation given by Equation (
1).
By looking at list (
2), we can verify the following result.
Proposition 1. Let n denote arbitrary non-negative integers. For all , the following property holds:where is the Gersenne sequence, with initial values and . Proof is carried out by mathematical induction using the recurrence (
1).
From Proposition 1, the next result follows.
Corollary 1. Let n denote arbitrary non-negative integers. For all , the following property holds:where is the Gersenne sequence, with initial values and . Equation (
5) can be rewritten in the form
2.1. The Binet Formula
In this subsection, we present the characteristic equation of the Gersenne sequence and provide Binet’s formula. Additionally, we present the exponential generating function and the ordinary generating function of the Gersenne sequence.
Note that the Horadam-type characteristic equation associated with the Gersenne sequence
is
and its real distinct roots are
and
.
The Binet formula is presented in the next result.
Proposition 2. (Binet’s formula) For all non-negative integers n, we have where , , , and with . The proof is immediate since the characteristic equation is (
6).
The special case
and
is denoted as
are called the generalized
k-Mersenne sequence.
Table 1 shows some examples of the generalized k-Mersenne sequence.
According to
Table 1, the 2-Mersenne sequence is the classic (or ordinary) Mersenne sequence, and the 10-Mersenne sequence is the repunit sequence.
See that for any integer
, the repunit or 10-Mersenne numbers are repunits in base 10, i.e.,
. Moreover, the representation of ordinary Mersenne or 2-Mersenne numbers in base 2 is of the form
. A direct calculation using Equation (
8) shows the result, as stated in the next proposition.
Proposition 3. Let be a non-negative integer. For all and , the representation of numbers in base k iswhere is the n-th k-Mersenne number given by Equation (8). 2.2. Generating Function
Now, we present the ordinary and exponential generating functions for the Gersenne sequence
. Taking into account the literature, the formal summation
is known as the ordinary generating function for the sequence
.
Our next result presents the ordinary generating function for the Gersenne sequence.
Proposition 4. The ordinary generating function for the Gersenne sequence , denoted by , iswhere and . Proof. According to Equation (
9), the generating function for the Gersenne sequence is
, and then, using the equations
and
, we obtain
When we add to both sides of these equations and use Equation (
1), we obtain the result. □
The exponential generating function, designated as
of a sequence
, is a power series of the form
In the next result, we consider the Binet Equation (
7), and obtain the exponential generating function for the Gersenne sequence
Proposition 5. For all , the exponential generating function for the Gersenne sequence iswhere , , , and with . The proof is immediate by Binet’s formula.
The Poisson generating function
for a sequence
is given by
where
is the exponential generating function of the sequence
. Consequently, the corresponding Poisson generating function is derived.
Corollary 2. For all , the Poisson generating function for the Gersenne sequence is 3. Some Identities for Gersenne Numbers
In this section, we establish some identities for the generalized Gersenne, some of which are classical identities, for example, the Tagiuri–Vajda, Catalan, Cassini, d’Ocagnes and Gelin–Cesàro identities.
The first result shows the addition and difference between two terms of the Gersenne sequence, and it is a direct consequence of Equation (
7).
Proposition 6. Let be the Gersenne sequence. For all non-negative integers m and n with , the following identities are verified:where , , , , and is the generalized k-Mersenne sequence given by Equation (8), with . The next result establishes the linear combination for a product of two terms of the Gersenne sequence.
Proposition 7. Let be the Gersenne sequence. For all non-negative integers m and n, the following identity holds:where , , , and , with . Proof. According to Binet’s Equation (
7), we have
as required. □
The following corollary follows directly from Proposition 7, considering and .
Corollary 3. Let be the generalized k-Mersenne sequence given by Equation (8). For all non-negative integers m and n, we obtain the identitywhere k is an integer (. As a direct consequence of Corollary 3, the following result is established.
Corollary 4. Let be the generalized k-Mersenne sequence given by Equation (8). For all non-negative integers r, the following identities hold: - (1)
.
- (2)
where k is an integer (.
Using the same technique employed in Proposition 7, we omit the proof of the next result in the interest of brevity.
Proposition 8. Let be the Gersenne sequence. For all non-negative integers , if , thenwhere , , , , and is the generalized k-Mersenne sequence given by Equation (8), with . The Tagiuri–Vajda identity for the Gersenne sequence is given below.
Theorem 1. Let be the Gersenne sequence, and non-negative integers. The following identity holds:where , , , , and is the generalized k-Mersenne sequence given by Equation (8), with . Proof. According to Binet’s Equation (
7), we have
and we have the validity of the result. □
The following identities will be derived as a consequence of the Tagiuri–Vajda identity, as demonstrated in Theorem 1.
Proposition 9. (d’Ocagne’s identity) Let be the Gersenne sequence and be non-negative integers with . Then, the following identity holds:where , , , , and is the generalized k-Mersenne sequence given by Equation (8), with . Proof. Take
,
and
in Theorem 1, so
Since , we obtain the result. □
Similar to Proposition 9, we have the Catalan identity.
Proposition 10. (Catalan’s identity) Let denote the Gersenene sequence, and let n and r be non-negative integers such that . Under these conditions, the following identity holds:where , , , , and is the generalized k-Mersenne sequence given by Equation (8), with . Proof. In Theorem 1, taking
and
, we obtain
Thus, the result is obtained. □
Applying the Catalan identity yields the following result.
Corollary 5. Let represent the Gersenene sequence. For any non-negative integers m, the following identity is satisfied:where , , , , and is the generalized k-Mersenne sequence given by Equation (8), with . Other consequences from the Catalan identity are as follows:
Corollary 6. (Cassini’s identity) Let represent the Gersenene sequence. The following identity is satisfied for all integers where , , , and , with . The next result, which is one of the main results of this paper, gives a formula for calculating the multiplication of four consecutive elements of the Gersenne sequence, which generalizes the Gelin–Cesàro identity.
Theorem 2. Let denote the Gersenne sequence. The following identity holds for all non-negative integers :where , , , , and is the generalized k-Mersenne sequence given by Equation (8), with . Proof. By substituting
into Equation (
10), we obtain the following:
By multiplying Equations (
10) and (
11), we obtain
which completes the proof. □
By virtue of Theorem 2, and taking , we have the following corollary.
Corollary 7. (Gelin-Cesàro’s identity) Let m be any natural number and denote the Gersenne sequence. Then, the identity holdswhere , , , , and is the generalized k-Mersenne sequence given by Equation (8), with . In the first example, we specify the last identity for the repunit sequence.
Example 1. Considering the repunit sequence specified when we take and (see Table 1), and , and we have that the Gelin–Cesàro identity for the repunit sequenceaccording to Proposition 3.6 in [20]. In the following example, we also illustrate the Gelin–Cesàro’s identity for the One-Zero sequence.
Example 2. Considering the One-Zero sequence specified when we take and , and , and we have thataccording to Proposition 3.1 in [21]. In concluding this section, we present a restricted result to the generalized k-Mersenne sequence. This result shows the difference between an even order and any n-th term.
Proposition 11. Let n be any non-negative integer. We havewhere is a k-Mersenne sequence given by Equation (8), with . Proof. A straightforward calculation, and making use of Equation (
8), shows that
which proves the result. □
The above result generalizes Proposition 1 given in [
22].
4. Sum of Terms Involving the Gersenne Sequence
In this section, we present results on partial sums of terms of the Gersenne numbers with
n integers. In general, the partial sums
, for
, where
is the Horadam sequence, can be obtained by applying Equations (3.5), (3.12) and (3.13) in [
18], or the fundamental summation rule (Equation (33)) in [
23], when
, which we have not applied here.
So, we consider the sequence of partial sums
for
, where
is the Gersenne sequence.
Proposition 12. Let be the Gersenne sequence; then, we have the following formulas:
- (a)
;
- (b)
;
- (c)
;
- (d)
.
In the above, , , , and is the generalized k-Mersenne sequence given by Equation (8), with .
Proof. - (a)
By Equation (
7), we have
as required.
Using the same approach as in (a), we obtain the other points. □
In order to better visualize Proposition 12, consider the following example.
Example 3. Considering the repunit sequence specified when we take and (see Table 1), , , the following formulas hold: - (a)
;
- (b)
;
- (c)
;
- (d)
.
In the above example, the items
(a),
(b) and
(c) are direct consequences of Proposition 6.1, as presented in [
20], while item
(d) follows from Proposition 12.
To illustrate the previous result, we present in the next example the particular case for .
Example 4. Considering the Gersenne sequence, specified when we take and (see Table 1), and , and ; therefore, the following formulas hold: - (a)
;
- (b)
;
- (c)
;
- (d)
.
Consider now the sequence of alternating partial sums given by
for
, where
denotes the Gersenne sequence.
Proposition 13. Let be the Gersenne sequence, and n be a non-negative integer. Hence, the following hold:
- (a)
, if n is odd;
- (b)
, if n is even.
In the above, , , , , and is the generalized k-Mersenne sequence given by Equation (8), with .
Proof. - (a)
First, suppose that
n is an odd natural number, or equivalently, the last term is negative; thus,
According to items
(b) and
(c) in Proposition 12, it follows that
- (b)
The proof follows a similar approach to the one used in item (a).
□
5. Final Considerations
This study presents an innovative approach to the generalized Mersenne number, so that this generalization is essentially analogous to the generalized Horadam sequence for a second-order recurrence with . Motivated by some previous work on Mersenne numbers, we generalize the Mersenne numbers by allowing for arbitrary initial values and a second-order linear recurrence that depends on a positive parameter k, thereby introducing a new family of Gersenne numbers. We explore their algebraic properties and derive several well-known identities, using explicit formulas.