Generalized Commutative Mersenne and Mersenne–Lucas Quaternion Polynomials

Dorota Bród; Anetta Szynal-Liana; Mirosław Liana

doi:10.2478/amsil-2025-0017

1. Introduction

Let p, q, n be integers. In [6], Horadam introduced a sequence {W_n(W₀, W₁; p, q)} defined by the second-order linear recurrence relation

(1)

\begin{matrix} W_{n} = p W_{n - 1} - q W_{n - 2} & for n \geq 2 \end{matrix}

with fixed real numbers W₀, W₁. For special values of W₀, W₁, p, q the equation (1) defines the well-known sequences of numbers, for example, the Fibonacci sequence F_n = W_n(0, 1; 1, −1), the Pell sequence P_n = W_n(0, 1; 2, −1) or the Jacobsthal sequence J_n = W_n(0, 1; 1, −2). Other examples of the Horadam sequence are the sequence of Mersenne numbers and the sequence of Lucas-Mersenne numbers.

The Mersenne numbers M_n are defined by the recurrence

\begin{matrix} M_{n} = 3 M_{n - 1} - 2 M_{n - 2} & for n \geq 2 \end{matrix}

with M₀ = 0, M₁ = 1 or

\begin{matrix} M_{n} = 2 M_{n - 1} + 1 & for n \geq 1 \end{matrix}

with initial condition M₀ = 0. The sequence of Mersenne–Lucas numbers {m_n} is defined by the same recurrence

\begin{matrix} m_{n} = 3 m_{n - 1} - 2 m_{n - 2} & for n \geq 2 \end{matrix}

with m₀ = 2, m₁ = 3.

The Binet formula of the Mersenne numbers and Mersenne–Lucas numbers has the form

\begin{matrix} M_{n} = 2^{n} - 1, \\ m_{n} = 2^{n} + 1, \end{matrix}

respectively. Some interesting properties of the Mersenne numbers can be found in [3, 15].

Sequences defined by the second-order linear homogeneous recurrence equation of the form

h_{n} (x) = p (x) h_{n - 1} (x) + q (x) h_{n - 2} (x),

for n ≥ 3 with h₁(x) = a and h₂(x) = bx are named as Horadam polynomials, see [7, 8]. One of them is the sequence of Mersenne polynomials {M_n(x)}, defined as follows

(2)

\begin{matrix} M_{n} (x) = 3 x M_{n - 1} (x) - 2 M_{n - 2} (x) & for n \geq 2 \end{matrix}

with M₀(x) = 0, M₁(x) = 1. Hence, we get

\begin{array}{l} M_{2} (x) = 3 x, \\ M_{3} (x) = 9 x^{2} - 2, \\ M_{4} (x) = 27 x^{3} - 12 x, \\ M_{5} (x) = 81 x^{4} - 54 x^{2} + 4 . \end{array}

The Mersenne–Lucas polynomials m_n(x) are defined by the same recurrence relation

\begin{matrix} m_{n} (x) = 3 x m_{n - 1} (x) - 2 m_{n - 2} (x) & for n \geq 2 \end{matrix}

with m₀(x) = 2, m₁(x) = 3. Hence, we obtain

\begin{array}{l} m_{2} (x) = 9 x - 4, \\ m_{3} (x) = 27 x^{2} - 12 x - 6, \\ m_{4} (x) = 81 x^{3} - 36 x^{2} - 36 x + 8, \\ m_{5} (x) = 243 x^{4} - 108 x^{3} - 162 x^{2} + 48 x + 12 . \end{array}

Binet formula for the Mersenne polynomials has the form

(3)

M_{n} (x) = \frac{λ_{1}^{n} (x) - λ_{2}^{n} (x)}{λ_{1} (x) - λ_{2} (x)},

where

(4)

\begin{matrix} λ_{1} (x) = \frac{1}{2} (3 x + \sqrt{9 x^{2} - 8}), & λ_{2} (x) = \frac{1}{2} (3 x - \sqrt{9 x^{2} - 8}), & 9 x^{2} - 8 > 0 \end{matrix}

are the roots of the characteristic equation

λ^{2} - 3 x λ + 2 = 0 .

Binet formula for the Mersenne–Lucas polynomials has the form

m_{n} (x) = A λ_{1}^{n} (x) + B λ_{2}^{n} (x),

where

(5)

\begin{matrix} A = 1 + \frac{3 - 3 x}{\sqrt{9 x^{2} - 8}}, & B = 1 + \frac{3 x - 3}{\sqrt{9 x^{2} - 8}} . \end{matrix}

2. The generalized commutative Mersenne and Mersenne–Lucas quaternions

In 1843, Hamilton ([4]) introduced the set ℍ of quaternions q of the form

q = x_{0} + x_{1} i + x_{2} j + x_{3} k,

where x₀, x₁, x₂, x₃ ∈ ℝ and

\begin{matrix} i^{2} = j^{2} = k^{2} = ijk = - 1, & ij = - ji = k, & jk = - kj = i, & ki = - ik = j . \end{matrix}

In [5], Horadam introduced the concept of Fibonacci and Lucas quaternions. Moreover, Iyer in [11] gave relations between the Fibonacci and Lucas quaternions. Iakin in [9, 10] introduced the concept of a higher-order quaternion, and established some identities for these quaternions.

Non-commutative quaternions and commutative quaternions were generalized and studied recently by Jafari and Yayli, see [12]. Generalized commutative quaternions were introduced by Szynal-Liana and Włoch in [17], where the authors studied generalized commutative quaternions in the special sub-family of quaternions of Fibonacci-type. Some properties of other generalized commutative quaternions can be found in [2, 13].

Let $ℍ_{α β}^{c}$ be the set of generalized commutative quaternions x of the form

x = x_{0} + x_{1} e_{1} + x_{2} e_{2} + x_{3} e_{3},

where x₀, x₁, x₂, x₃ ∈ ℝ, quaternionic units e₁, e₂, e₃ satisfy the equalities

\begin{matrix} \begin{matrix} e_{1}^{2} = α, & e_{2}^{2} = β, & e_{3}^{2} = α β, \end{matrix} \\ \begin{matrix} e_{1} e_{2} = e_{2} e_{1} = e_{3}, & e_{2} e_{3} = e_{3} e_{2} = β e_{1} & and & e_{3} e_{1} = e_{1} e_{3} = α e_{2}, \end{matrix} \end{matrix}

and α, β ∈ ℝ.

Generalized commutative quaternions generalize elliptic quaternions (α < 0, β = 1), parabolic quaternions (α = 0, β = 1), hyperbolic quaternions (α > 0, β = 1), bicomplex numbers (α = −1, β = −1), complex hyperbolic numbers (α = −1, β = 1) and hyperbolic complex numbers (α = 1, β = −1).

Let n ≥ 0 be an integer. The n-th generalized commutative Mersenne quaternion gc ℳ_n and the n-th generalized commutative Mersenne–Lucas quaternion gc ℳℒ_n are defined as follows

\begin{matrix} gc M_{n} = M_{n} + M_{n + 1} e_{1} + M_{n + 2} e_{2} + M_{n + 3} e_{3}, \\ gc M L_{n} = m_{n} + m_{n + 1} e_{1} + m_{n + 2} e_{2} + m_{n + 3} e_{3} . \end{matrix}

These quaternions are special types of generalized commutative Horadam quaternions defined in [17].

Let n ≥ 0 be an integer and x be a real variable. The n-th generalized commutative Mersenne quaternion polynomial gc ℳ_n(x) and the n-th generalized commutative Mersenne–Lucas quaternion polynomial gc ℳℒ_n(x) are defined as follows

(6)

\begin{matrix} gc M_{n} (x) = M_{n} (x) + M_{n + 1} (x) e_{1} + M_{n + 2} (x) e_{2} + M_{n + 3} (x) e_{3}, \\ gc M L_{n} (x) = m_{n} (x) + m_{n + 1} (x) e_{1} + m_{n + 2} (x) e_{2} + m_{n + 3} (x) e_{3} . \end{matrix}

For x = 1, we have gc ℳ_n(1) = gc ℳ_n and gc ℳℒ_n(1) = gc ℳℒ_n.

Generalized commutative quaternion polynomials of the Fibonacci-type are introduced in [18]. In [19], the authors considered generalized Pauli Fibonacci polynomial quaternions.

3. Main results

In this section, we give some identities for the generalized commutative Mersenne quaternion polynomials and the generalized commutative Mersenne–Lucas quaternion polynomials. We start with recurrence relations and Binet-type formulas for these quaternion polynomials.

Theorem 1

Let n ≥ 2 be an integer and x be a real variable. Then

(i) gc ℳ_n(x) = 3xgc ℳ_n−1(x) − 2gc ℳ_n−2(x),
(ii) gc ℳℒ_n(x) = 3xgc ℳℒ_n−1(x) − 2gc ℳℒ_n−2(x),

where

\begin{array}{l} gc M_{0} (x) = e_{1} + 3 x e_{2} + (9 x^{2} - 2) e_{3}, \\ gc M_{1} (x) = 1 + 3 x e_{1} + (9 x^{2} - 2) e_{2} + (27 x^{3} - 12 x) e_{3}, \\ gc M L_{0} (x) = 2 + 3 e_{1} + (9 x - 4) e_{2} + (27 x^{2} - 12 x - 6) e_{3}, \\ gc M L_{1} (x) = 3 + (9 x - 4) e_{1} + (27 x^{2} - 12 x - 6) e_{2} + (81 x^{3} - 36 x^{2} - 36 x + 8) e_{3} . \end{array}

Proof

For n = 2 we get

\begin{array}{l} gc M_{2} (x) & = 3 x gc M_{1} (x) - 2 gc (x) \\ = 3 x + 9 x^{2} e_{1} + (27 x^{3} - 6 x) e_{2} + (81 x^{4} - 36 x^{2}) e_{3} - 2 e_{1} - 6 x e_{2} - (18 x^{2} - 4) e_{3} \\ = 3 x + (9 x^{2} - 2) e_{1} + (27 x^{3} - 12 x) e_{2} + (81 x^{4} - 54 x^{2} + 4) e_{3} . \end{array}

Let n ≥ 3. By formulas (6) and (2) we get

\begin{array}{l} gc M_{n} (x) & = M_{n} (x) + M_{n + 1} (x) e_{1} + M_{n + 2} (x) e_{2} + M_{n + 3} (x) e_{3} \\ = 3 x M_{n - 1} (x) - 2 M_{n - 2} (x) + (3 x M_{n} (x) - 2 M_{n - 1} (x)) e_{1} \\ + (3 x M_{n + 1} (x) - 2 M_{n} (x)) e_{2} + (3 x M_{n + 2} (x) - 2 M_{n + 1} (x)) e_{3} \\ = 3 x (M_{n - 1} (x) + M_{n} (x) e_{1} + M_{n + 1} (x) e_{2} + M_{n + 2} (x) e_{3}) \\ - 2 (M_{n - 2} (x) + M_{n - 1} (x) e_{1} + M_{n} (x) e_{2} + M (x) e) \\ = 3 x gc M_{n - 1} (x) - 2 gc M_{n - 2} (x) \end{array}

which ends the proof of (i).

The second part can be proved similarly.

Corollary 1

Let n ≥ 2 be an integer. Then

(i) gc ℳ_n = 3gc ℳ_n−1 − 2gc ℳ_n−2,
(ii) gc ℳℒ_n = 3gc ℳℒ_n−1 − 2gc ℳℒ_n−2,

where

\begin{array}{l} gc M_{0} = e_{1} + 3 e_{2} + 7 e_{3}, \\ gc M_{1} = 1 + 3 e_{1} + 7 e_{2} + 15 e_{3}, \\ gc M L_{0} = 2 + 3 e_{1} + 5 e_{2} + 9 e_{3}, \\ gc M L_{1} = 3 + 5 e_{1} + 9 e_{2} + 17 e_{3} . \end{array}

Theorem 2

(Binet-type formula for generalized commutative Mersenne quaternion polynomials). Let n ≥ 0 be an integer, x be a real variable and 9x² − 8 > 0. Then

(7)

gc M_{n} (x) = \frac{λ_{1}^{n} (x) \hat{λ_{1} (x)} - λ_{2}^{n} (x) \hat{λ_{2} (x)}}{λ_{1} (x) - λ_{2} (x)},

where λ₁(x), λ₂(x) are given by (4) and

(8)

\begin{array}{l} \hat{λ_{1} (x)} = 1 + λ_{1} (x) e_{1} + λ_{1}^{2} (x) e_{2} + λ_{1}^{3} (x) e_{3}, \\ \hat{λ_{2} (x)} = 1 + λ_{2} (x) e_{1} + λ_{2}^{2} (x) e_{2} + λ_{2}^{3} (x) e_{3} . \end{array}

Proof

By (6) and (3) we get

\begin{array}{l} gc M_{n} (x) & = M_{n} (x) + M_{n + 1} (x) e_{1} + M_{n + 2} (x) e_{2} + M_{n + 3} (x) e_{3} \\ = \frac{λ_{1}^{n} (x) - λ_{2}^{n} (x)}{λ_{1} (x) - λ_{2} (x)} + \frac{λ_{1}^{n + 1} (x) - λ_{2}^{n + 1} (x)}{λ_{1} (x) - λ_{2} (x)} e_{1} \\ + \frac{λ_{1}^{n + 2} (x) - λ_{2}^{n + 2} (x)}{λ_{1} (x) - λ_{2} (x)} e_{2} + \frac{λ_{1}^{n + 3} (x) - λ_{2}^{n + 3} (x)}{λ_{1} (x) - λ_{2} (x)} e_{3} \\ = \frac{λ_{1}^{n} (x) (1 + λ_{1} (x) e_{1} + λ_{1}^{2} (x) e_{2} + λ_{1}^{3} (x) e_{3})}{λ_{1} (x) - λ_{2} (x)} \\ - \frac{λ_{2}^{n} (x) (1 + λ_{2} (x) e_{1} + λ_{2}^{2} (x) e_{2} + λ_{2}^{3} (x) e_{3})}{λ_{1} (x) - λ_{2} (x)} . \end{array}

Hence, we get the result.

In the same way, we can prove the following theorem.

Theorem 3

(Binet-type formula for generalized commutative Mersenne–Lucas quaternion polynomials). Let n ≥ 0 be an integer, x be a real variable and 9x² − 8 > 0. Then

gc M L_{n} (x) = A λ_{1}^{n} (x) \hat{λ_{1} (x)} + B λ_{2}^{n} (x) \hat{λ_{2} (x)},

where A, B, λ₁(x), λ₂(x),

\hat{λ_{1} (x)}

\hat{λ_{2} (x)}

are given by (5), (4), (8), respectively.

Corollary 2

Let n ≥ 0 be an integer. Then

\begin{matrix} gc M_{n} = 2^{n} (1 + 2 e_{1} + 4 e_{2} + 8 e_{3}) - (1 + e_{1} + e_{2} + e_{3}), \\ gc M L_{n} = 2^{n} (1 + 2 e_{1} + 4 e_{2} + 8 e_{3}) + (1 + e_{1} + e_{2} + e_{3}) . \end{matrix}

By simple calculations we have

\begin{array}{l} λ_{1} (x) - λ_{2} (x) = \sqrt{9 x^{2} - 8}, λ_{1} (x) + λ_{2} (x) = 3 x, λ_{1} (x) \cdot λ_{2} (x) = 2, \\ λ_{1}^{2} (x) + λ_{2}^{2} (x) = 9 x^{2} - 4, λ_{1}^{3} (x) + λ_{2}^{3} (x) = 27 x^{3} - 18 x, \end{array}

\begin{array}{l} \hat{λ_{1} (x)} \hat{λ_{2} (x)} & = \hat{λ_{2} (x)} \hat{λ_{1} (x)} \\ = 1 + λ_{2} (x) e_{1} + λ_{2}^{2} (x) e_{2} + λ_{2}^{3} (x) e_{3} \\ + λ_{1} (x) e_{1} + λ_{1} (x) λ_{2} (x) α + λ_{1} (x) λ_{2}^{2} (x) e_{3} + λ_{1} (x) λ_{2}^{3} (x) α e_{2} \\ + λ_{1}^{2} (x) e_{2} + λ_{1}^{2} (x) λ_{2} (x) e_{3} + λ_{1}^{2} (x) λ_{2}^{2} (x) β + λ_{1}^{2} (x) λ_{2}^{3} (x) β e_{1} \\ + λ_{1}^{3} (x) e_{3} + λ_{1}^{3} (x) λ_{2} (x) α e_{2} + λ_{1}^{3} (x) λ_{2}^{2} (x) β e_{1} + λ_{1}^{3} (x) λ_{2}^{3} (x) α β \\ = 1 + λ_{1} (x) λ_{2} (x) α + λ_{1}^{2} (x) λ_{2}^{2} (x) β + λ_{1}^{3} (x) λ_{2}^{3} (x) α β \\ + λ_{2} (x) e_{1} + λ_{1} (x) e_{1} + λ_{1}^{2} (x) λ_{2}^{3} (x) β e_{1} + λ_{1}^{3} (x) λ_{2}^{2} (x) β e_{1} \\ + λ_{2}^{2} (x) e_{2} + λ_{1}^{2} (x) e_{2} + λ_{1} (x) λ_{2}^{3} (x) α e_{2} + λ_{1}^{3} (x) λ_{2} (x) α e_{2} \\ + λ_{2}^{3} (x) e_{3} + λ_{1}^{3} (x) e_{3} + λ_{1} (x) λ_{2}^{2} (x) e_{3} + λ_{1}^{2} (x) λ_{2} (x) e_{3} . \end{array}

Hence, we get

(9)

\hat{λ_{1} (x)} \hat{λ_{2} (x)} = 1 + 2 α + 4 β + 8 α β + (3 x + 12 x β) e_{1} + (9 x^{2} - 4) (1 + 2 α) e_{2} + (27 x^{3} - 12 x) e_{3} .

The next theorems present general bilinear index-reduction formulas for generalized commutative Mersenne quaternion polynomials and generalized commutative Mersenne–Lucas quaternion polynomials.

Theorem 4

Let a ≥ 0, b ≥ 0, c ≥ 0, d ≥ 0 be integers such that a + b = c + d. Assume that x is a real variable and 9x² − 8 > 0. Then

\begin{array}{l} gc M_{a} (x) \cdot gc M_{b} (x) - gc M_{c} (x) \cdot gc (x) \\ = \frac{[λ_{1}^{c} (x) λ_{2}^{d} (x) + λ_{2}^{c} (x) λ_{1}^{d} (x) - λ_{1}^{a} (x) λ_{2}^{b} (x) - λ_{2}^{a} (x) λ_{1}^{b} (x)] \hat{λ_{1} (x)} \hat{λ_{2} (x)}}{9 x^{2} - 8}, \end{array}

where λ₁(x), λ₂(x) are given by (4) and

\hat{λ_{1} (x)} \hat{λ_{2} (x)}

is given by (9).

Proof

By formula (7) we have

\begin{array}{l} gc M_{a} (x) \cdot gc M_{b} (x) - gc M_{c} (x) \cdot gc M_{d} (x) \\ = \frac{- λ_{1}^{a} (x) λ_{2}^{b} (x) \hat{λ_{1} (x)} \hat{λ_{2} (x)} - λ_{2}^{a} (x) λ_{1}^{b} (x) \hat{λ_{2} (x)} \hat{λ_{1} (x)}}{9 x^{2} - 8} \\ + \frac{- λ_{1}^{c} (x) λ_{2}^{d} (x) \hat{λ_{1} (x)} \hat{λ_{2} (x)} - λ_{2}^{c} (x) λ_{1}^{d} (x) \hat{λ_{2} (x)} \hat{λ_{1} (x)}}{9 x^{2} - 8} \\ = \frac{[λ_{1}^{c} (x) λ_{2}^{d} (x) + λ_{2}^{c} (x) λ_{1}^{d} (x) - λ_{1}^{a} (x) λ_{2}^{b} (x) - λ_{2}^{a} (x) λ_{1}^{b} (x)] \hat{λ_{1} (x)} \hat{λ_{2} (x)}}{9 x^{2} - 8}, \end{array}

which ends the proof.

Theorem 5

Let a ≥ 0, b ≥ 0, c ≥ 0, d ≥ 0 be integers such that a + b = c + d. Assume that x is a real variable and 9x² − 8 > 0. Then

\begin{array}{l} gc M L_{a} (x) \cdot gc M L_{b} (x) - gc M L_{c} (x) \cdot gc M L_{d} (x) \\ = \frac{18 x - 17}{9 x^{2} - 8} (λ_{1}^{a} (x) λ_{2}^{b} (x) + λ_{2}^{a} (x) λ_{1}^{b} (x) - λ_{1}^{c} (x) λ_{2}^{d} (x) - λ_{2}^{c} (x) λ_{1}^{d} (x)) \hat{λ_{1} (x)} \hat{λ_{2} (x)}, \end{array}

where λ₁(x), λ₂(x) are given by (4) and

\hat{λ_{1} (x)} \hat{λ_{2} (x)}

is given by (9).

Corollary 3

Let a ≥ 0, b ≥ 0, c ≥ 0, d ≥ 0 be integers such that a + b = c + d. Then

\begin{array}{r} gc M_{a} \cdot gc M_{b} - gc M_{c} \cdot gc M_{d} = (2^{c} + 2^{d} - 2^{a} - 2^{b}) \hat{1} \cdot \hat{2}, \\ gc M L_{a} \cdot gc M L_{b} - gc M L_{c} \cdot gc M L_{d} = (2^{a} + 2^{b} - 2^{c} - 2^{d}) \hat{1} \cdot \hat{2}, \end{array}

where

\begin{matrix} \hat{1} = 1 + e_{1} + e_{2} + e_{3}, & \hat{2} = 1 + 2 e_{1} + 4 e_{2} + 8 e_{3} \end{matrix}

and

(10)

\hat{1} \cdot \hat{2} = 1 + 2 α + 4 β + 8 α β + (3 + 12 β) e_{1} + (5 + 10 α) e_{2} + 15 e_{3} .

It is easily seen that for special values of a, b, c, d, by Theorem 4 and Theorem 5 we get new identities for generalized commutative Mersenne quaternion polynomials and generalized commutative Mersenne–Lucas quaternion polynomials. Assume that $\hat{λ_{1} (x)} \hat{λ_{2} (x)}$ is given by (9) and 9x² − 8 > 0.

Catalan-type identities for a = n + r, b = n − r, c = d = n, r ≥ 0 and n ≥ r
$\begin{matrix} gc M_{n + r} (x) \cdot gc M_{n - r} (x) - {(gc M_{n} (x))}^{2} = \frac{2^{n}}{9 x^{2} - 8} 2 - {(\frac{λ_{1} (x)}{λ_{2} (x)})}^{r} - {(\frac{λ_{2} (x)}{λ_{1} (x)})}^{r}] \hat{λ_{1} (x)} \hat{λ_{2} (x)}, \\ gc M L_{n + r} (x) \cdot gc M L_{n - r} (x) - {(gc M L_{n} (x))}^{2} = \frac{2^{n} (18 x - 17)}{9 x^{2} - 8} {(\frac{λ_{1} (x)}{λ_{2} (x)})}^{r} + {(\frac{λ_{2} (x)}{λ_{1} (x)})}^{r} - 2] \hat{λ_{1} (x)} \hat{λ_{2} (x)}, \end{matrix}$
Cassini-type identities for a = n + 1, b = n − 1, c = d = n and n ≥ 1
$\begin{matrix} gc M_{n + 1} (x) \cdot gc M_{n - 1} (x) - {(gc M_{n} (x))}^{2} = \frac{2^{n}}{9 x^{2} - 8} 2 - \frac{λ_{1} (x)}{λ_{2} (x)} - \frac{λ_{2} (x)}{λ_{1} (x)}] \hat{λ_{1} (x)} \hat{λ_{2} (x)}, \\ gc M L_{n + 1} (x) \cdot gc M L_{n - 1} (x) - {(gc M L_{n} (x))}^{2} = \frac{2^{n} (18 x - 17)}{9 x^{2} - 8} \frac{λ_{1} (x)}{λ_{2} (x)} + \frac{λ_{2} (x)}{λ_{1} (x)} - 2] \hat{λ_{1} (x)} \hat{λ_{2} (x)}, \end{matrix}$
ďOcagne-type identities for a = n, b = m + 1, c = n + 1 and d = m n ≥ m
$\begin{matrix} gc M_{n} (x) \cdot gc M_{m + 1} (x) - gc M_{n + 1} (x) \cdot gc M_{m} (x) = \frac{[λ_{1}^{n} (x) λ_{2}^{m} (x) - λ_{2}^{n} (x) λ_{1}^{m} (x)] \hat{λ_{1} (x)} \hat{λ_{2} (x)}}{\sqrt{9 x^{2} - 8}}, \\ gc M L_{n} (x) \cdot gc M L_{m + 1} (x) - gc M L_{n + 1} (x) \cdot gc M L_{m} (x) = \frac{18 x - 17}{\sqrt{9 x^{2} - 8}} (λ_{2}^{n} (x) λ_{1}^{m} (x) - λ_{1}^{n} (x) λ_{2}^{m} (x)) \hat{λ_{1} (x)} \hat{λ_{2} (x)}, \end{matrix}$
Vajda-type identities for a = m + p, b = n − p, c = m, d = n and m ≥ 0, p ≥ 0, n ≥ p
$\begin{array}{l} gc M_{m + p} (x) \cdot gc M_{n - p} (x) - gc M_{m} (x) \cdot gc M_{n} (x) \\ = \frac{λ_{1}^{m} (x) λ_{2}^{n} (x) (1 - {(\frac{λ_{1} (x)}{λ_{2} (x)})}^{p}) + λ_{2}^{m} (x) λ_{1}^{n} (x) (1 - {(\frac{λ_{1} (x)}{λ_{2} (x)})}^{p})]}{9 x^{2} - 8} \hat{λ_{1} (x)} \hat{λ_{2} (x)}, \\ gc M L_{m + p} (x) \cdot gc M L_{n - p} (x) - gc M L_{m} (x) \cdot gc M L_{n} (x) \\ = \frac{(18 x - 17) λ_{1}^{m} (x) λ_{2}^{n} (x) ({(\frac{λ_{1} (x)}{λ_{2} (x)})}^{p} - 1) + λ_{2}^{m} (x) λ_{1}^{n} (x) ({(\frac{λ_{2} (x)}{λ_{1} (x)})}^{p} - 1)] \hat{λ_{1} (x)} \hat{λ_{2} (x)}}{9 x^{2} - 8} . \end{array}$

Now, we give such identities for generalized commutative Mersenne quaternions and generalized commutative Mersenne–Lucas quaternions. Assume that $\hat{1} \cdot \hat{2}$ is given by (10).

Catalan-type identities for r ≥ 0 and n ≥ r
$\begin{array}{r} gc M_{n + r} \cdot gc M_{n - r} - {(gc M_{n})}^{2} = (2^{n + 1} - 2^{n + r} - 2^{n - r}) \hat{1} \cdot \hat{2}, \\ gc M L_{n + r} \cdot gc M L_{n - r} - {(gc M L_{n})}^{2} = (2^{n + r} + 2^{n - r} - 2^{n + 1}) \hat{1} \cdot \hat{2}, \end{array}$
Cassini-type identities for n ≥ 1
$\begin{matrix} gc M_{n + 1} \cdot gc M_{n - 1} - {(gc M_{n})}^{2} & = - 2^{n - 1} \cdot \hat{1} \cdot \hat{2}, \\ gc M L_{n + 1} \cdot gc M L_{n - 1} - {(gc M L_{n})}^{2} & = 2^{n - 1} \cdot \hat{1} \cdot \hat{2}, \end{matrix}$
ďOcagne-type identities for n ≥ m
$\begin{array}{r} gc M_{n} \cdot gc M_{m + 1} - gc M_{n + 1} \cdot gc M_{m} = (2^{n} - 2^{m}) \hat{1} \cdot \hat{2}, \\ gc M L_{n} \cdot gc M L_{m + 1} - gc M L_{n + 1} \cdot gc M L_{m} = (2^{m} - 2^{n}) \hat{1} \cdot \hat{2}, \end{array}$
Vajda-type identities for m ≥ 0, p ≥ 0, n ≥ p
$\begin{matrix} gc M_{m + p} \cdot gc M_{n - p} - gc M_{m} \cdot gc M_{n} = (2^{m} (1 - 2^{p}) + 2^{n} (1 - 2^{- p})) \hat{1} \cdot \hat{2}, \\ gc M L_{m + p} \cdot gc M L_{n - p} - gc M L_{m} \cdot gc M L_{n} = (2^{m} (2^{p} - 1) + 2^{n} (2^{- p} - 1)) \hat{1} \cdot \hat{2} . \end{matrix}$

4. Matrix generators and generating functions

Now, we give the matrix representations of gcℳ_n(x). By Theorem 1 we get the following result.

Theorem 6

Let n ≥ 1 be an integer and x be a real variable. Then

\begin{array}{l} gc M_{n + 1} (x) \\ gc M_{n} (x) \end{array}] = \begin{matrix} 3 x & - 2 \\ 1 & 0 \end{matrix}] \cdot \begin{matrix} gc M_{n} (x) \\ gc M_{n - 1} (x) \end{matrix}] .

Theorem 7

Let n ≥ 0 be an integer and x be a real variable. Then

(11)

\begin{matrix} gc M_{n + 2} (x) & gc M_{n + 1} (x) \\ gc M_{n + 1} (x) & gc M_{n} (x) \end{matrix}] = \begin{matrix} gc M_{2} (x) & gc M_{1} (x) \\ gc M_{1} (x) & gc M_{0} (x) \end{matrix}] \cdot {\begin{array}{l} 3 x & 1 \\ - 2 & 0 \end{array}]}^{n} .

Proof

We use induction on n. If n = 0 then the result is obvious. Assuming the formula (11) holds for n ≥ 0, we shall prove it for n + 1.

Using induction’s hypothesis and Theorem 1, we have

\begin{array}{l} \begin{matrix} gc M_{2} (x) & gc M_{1} (x) \\ gc M_{1} (x) & gc M_{0} (x) \end{matrix}] \cdot {\begin{array}{l} 3 x & 1 \\ - 2 & 0 \end{array}]}^{n} \cdot \begin{array}{l} 3 x & 1 \\ - 2 & 0 \end{array}] \\ = \begin{array}{l} gc M_{n + 2} (x) & gc M_{n + 1} (x) \\ gc M_{n + 1} (x) & gc M_{n} (x) \end{array}] \cdot \begin{array}{l} 3 x & 1 \\ - 2 & 0 \end{array}] \\ = \begin{array}{l} 3 x gc M_{n + 2} (x) - 2 gc M_{n + 1} (x) & gc M_{n + 2} (x) \\ 3 x gc M_{n + 1} (x) - 2 gc M_{n} (x) & gc M_{n + 1} (x) \end{array}] \\ = \begin{matrix} gc M_{n + 3} (x) & gc M_{n + 2} (x) \\ gc M_{n + 2} (x) & gc M_{n + 1} (x) \end{matrix}], \end{array}

which ends the proof.

Corollary 4

Let n ≥ 0 be an integer. Then

\begin{array}{l} gc M_{n + 2} & gc M_{n + 1} \\ gc M_{n + 1} & gc M_{n} \end{array}] = \begin{array}{l} gc M_{2} & gc M_{1} \\ gc M_{1} & gc M_{0} \end{array}] \cdot {\begin{array}{l} 3 & 1 \\ - 2 & 0 \end{array}]}^{n} .

In the same way, we can prove the following results.

Theorem 8

Let n ≥ 0 be an integer and x be a real variable. Then

\begin{array}{l} gc M L_{n + 2} (x) & gc M L_{n + 1} (x) \\ gc M L_{n + 1} (x) & gc M L_{n} (x) \end{array}] = \begin{matrix} gc M L_{2} (x) & gc M L_{1} (x) \\ gc M L_{1} (x) & gc M L_{0} (x) \end{matrix}] \cdot \begin{array}{l} 3 x & 1 \\ - 2 & 0 \end{array}]

Corollary 5

Let n ≥ 0 be an integer. Then

\begin{array}{l} gc M L_{n + 2} & gc M L_{n + 1} \\ gc M L_{n + 1} & gc M L_{n} \end{array}] = \begin{array}{l} gc M L_{2} & gc M L_{1} \\ gc M L_{1} & gc M L_{0} \end{array}] \cdot {\begin{array}{l} 3 & 1 \\ - 2 & 0 \end{array}]}^{n} .

Theorem 9

The generating function of the generalized commutative Mersenne quaternion polynomials has the following form

f (t) = \frac{e_{1} + 3 x e_{2} + (9 x^{2} - 2) e_{3} + (1 - 2 e_{2} - 6 x e_{3}) t}{1 - 3 xt + 2 t^{2}} .

Proof

Let

f (t) = gc M_{0} (x) + t gc M_{1} (x) + t^{2} gc M_{2} (x) + \dots + t^{n} gc M_{n} (x) + \dots

be the generating function of the generalized commutative Mersenne quaternion polynomials. Then

\begin{array}{l} 3 xtf (t) = 3 txgc M_{0} (x) + 3 t^{2} x gc M_{1} (x) + 3 t^{3} xgc (x) + \dots + 3 t^{n} x gc M_{n} (x) + \dots \\ 2 t^{2} f (t) = 2 t^{2} gc M_{0} (x) + 2 t^{3} gc M_{1} (x) + 2 t^{4} gc M_{2} (x) + \dots + 2 t^{n} gc M_{n - 2} (x) + \dots . \end{array}

Hence, by the recurrence gc ℳ_n(x) = 3xgc ℳ_n−1(x) − 2gc ℳ_n−2(x), we get

\begin{array}{l} f (t) - 3 xtf (t) + 2 t^{2} f (t) & = gc M_{0} (x) + (gc M_{1} (x) - 3 xgc (x)) t \\ + (2 gc M_{0} (x) + gc M_{2} (x) - 3 x gc M_{1} (x)) t^{2} + \dots \\ = gc M_{0} (x) + (gc M_{1} (x) - 3 x gc M_{0} (x)) t . \end{array}

Thus

f (t) = \frac{gc M_{0} (x) + (gc M_{1} (x) - 3 x gc M_{0} (x)) t}{1 - 3 xt + 2 t^{2}} .

After simple calculations we obtain

f (t) = \frac{e_{1} + 3 x e_{2} + (9 x^{2} - 2) e_{3} + (1 - 2 e_{2} - 6 x e_{3}) t}{1 - 3 xt + 2 t^{2}} .

Theorem 10

The generating function of the generalized commutative Mersenne–Lucas quaternion polynomials has the following form

g (t) = \frac{gc M L_{0} (x) + (gc M L_{1} (x) - 3 x gc M L_{0} (x)) t}{1 - 3 xt + 2 t^{2}},

where

\begin{matrix} gc M L_{0} (x) = 2 + 3 e_{1} + (9 x - 4) e_{2} + (27 x^{2} - 12 x - 6) e_{3}, \\ gc M L_{1} (x) - 3 x gc M L_{0} (x) = 3 + (3 x - 4) e_{1} - 6 e_{2} + (- 18 x + 8) e_{3} . \end{matrix}

Corollary 6

The generating function of the generalized commutative Mersenne quaternions has the following form

f_{M} (t) = \frac{e_{1} + 3 e_{2} + 7 e_{3} + (1 - 2 e_{2} - 6 e_{3}) t}{1 - 3 t + 2 t^{2}} .

Corollary 7

The generating function of the generalized commutative Mersenne–Lucas quaternions has the following form

g_{L} (t) = \frac{2 + 3 e_{1} + 5 e_{2} + 9 e_{3} + (3 - e_{1} - 6 e_{2} - 10 e_{3}) t}{1 - 3 t + 2 t^{2}} .

Concluding remarks

For any positive integer n, the n-th bivariate Horadam polynomial h_n(x, y) was defined in [16] as h_n(x, y) = pxh_n−1(x, y) + qyh_n−2(x, y) for n ≥ 3 with the initial values h₁(x, y) = a and h₂(x, y) = bx. It is easy to see that h_n(x, 1) = h_n(x). Bivariate Mersenne polynomials M_n(x, y) and bivariate Mersenne Lucas polynomials m_n(x, y) were defined in [1] and [14], respectively, as follows

\begin{matrix} M_{n} (x, y) = 3 y M_{n - 1} (x, y) - 2 x M_{n - 2} (x, y) & for n \geq 2 \end{matrix}

with M₀(x, y) = 0, M₁(x, y) = 1 and

\begin{matrix} m_{n} (x, y) = 3 y m_{n - 1} (x, y) - 2 x m_{n - 2} (x, y) & for n \geq 2 \end{matrix}

with m₀(x, y) = 2, m₁(x, y) = 3y.

It is worth noting that, unlike before, M_n(1, x) = M_n(x) and m_n(1, x) = m_n(x). Using the above definitions, we can define, for any variables x, y and any nonnegative integer n, the n-th bivariate generalized commutative Mersenne quaternion polynomial gc ℳ_n(x, y) and the n-th bivariate generalized commutative Mersenne–Lucas quaternion polynomial gc ℳℒ_n(x, y) as follows

\begin{array}{l} gc M_{n} (x, y) = M_{n} (x, y) + M_{n + 1} (x, y) e_{1} + M_{n + 2} (x, y) e_{2} + M_{n + 3} (x, y) e_{3}, \\ gc M L_{n} (x, y) = m_{n} (x, y) + m_{n + 1} (x, y) e_{1} + m_{n + 2} (x, y) e_{2} + m_{n + 3} (x, y) e_{3} . \end{array}

Further work may involve research on these polynomials.

Acknowledgements

The authors would like to thank the referees for helpful valuable suggestions which resulted in improvements to this paper.

Generalized Commutative Mersenne and Mersenne–Lucas Quaternion Polynomials

Full Article

1. Introduction

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2. The generalized commutative Mersenne and Mersenne–Lucas quaternions

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3. Main results

Theorem 1

Proof

Corollary 1

Theorem 2

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Proof

Theorem 3

Corollary 2

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Theorem 4

Proof

Theorem 5

Corollary 3

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4. Matrix generators and generating functions

Theorem 6

Theorem 7

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Proof

Corollary 4

Theorem 8

Corollary 5

Theorem 9

Proof

Theorem 10

Corollary 6

Corollary 7

Concluding remarks

Acknowledgements