On generalized Mersenne hybrid numbers

The hybrid numbers are generalization of complex, hyperbolic and dual numbers. In this paper we consider a special kind of hybrid numbers, namely the Mersenne hybrid numbers and we give some of their properties.


Introduction and preliminary results.
Let p, q, n be integers. For n ≥ 0, Horadam (see [2]) defined the numbers W n = W n (W 0 , W 1 ; p, q) by the recursive equation (1) W n = p · W n−1 − q · W n−2 , for n ≥ 2 with fixed real numbers W 0 , W 1 . For the historical reasons these numbers were later called the Horadam numbers. For special W 0 , W 1 , p, q, the equation (1) defines the well-known numbers named as the numbers of the Fibonacci type, e.g. the Fibonacci numbers F n = W n (0, 1; 1, −1), the Jacobsthal numbers J n = W n (0, 1; 1, −2), the Pell numbers P n = W n (0, 1; 2, −1) and the Mersenne numbers M n = W n (0, 1; 3, 2). Fibonacci numbers, Jacobsthal numbers, Pell numbers and others have many different generalizations, see the list of them in [1].
In [3], Ochalik and Włoch introduced the generalized Mersenne numbers as follows. Let k ≥ 3 be a fixed integer. For any integer n ≥ 0, let M (k, n) be the nth generalized Mersenne number defined by the second order linear recurrence relation of the form for n ≥ 2 with initial conditions M (k, 0) = 0 and M (k, 1) = 1.
Some identities and properties of M (k, n), also combinatorial interpretations and matrix generators were given in [3]. In the next part of the paper we use the following results. .
For other identities, see [3]. Moreover, for M (k, n) we have the following identity.

Remark 4.
For k = 3 we obtain the well-known identity for the classical Mersenne numbers M n+1 = 2 · M n + 1.
In this paper, we introduce and study the generalized Mersenne hybrid numbers. It is worth to mention that the Horadam hybrid numbers were introduced in [5] and consequently, the Fibonacci hybrid numbers and the like were studied in [6,7,8]. This paper is a sequel of them. Firstly we give the necessary definitions.
Let us consider the set K of hybrid numbers Z of the form where a, b, c, d ∈ R and i, ε, h are operators such that Let Z 1 = a 1 +b 1 i+c 1 ε+d 1 h and Z 2 = a 2 +b 2 i+c 2 ε+d 2 h be any two hybrid numbers. We define equality, addition, subtraction and multiplication by scalar in the following way: Using equalities (6) and (7), we define multiplication of hybrid numbers. Moreover, by formulas (6) and (7) the product of any two hybrid operators can be calculated, see Table 1.
From the above rules the multiplication of hybrid numbers can be made analogously as multiplications of algebraic expressions. Note that the multiplication operation in the hybrid numbers is associative, but not commutative.
The conjugate of a hybrid number Z is defined by The real number is called the character of the hybrid number Z. The hybrid numbers were introduced byÖzdemir in [4] as a generalization of complex, hyperbolic and dual numbers.
For n ≥ 0, the nth Horadam hybrid number H n is defined as Some interesting results for the Horadam hybrid numbers were obtained in [5]. Now we will define the generalized Mersenne hybrid numbers, which form a subclass of the Horadam hybrid numbers (defined by (8)).
Let n ≥ 0 be an integer. We define the generalized Mersenne hybrid sequence {M H k n } by the following recurrence: (9) M H k n = M (k, n) + iM (k, n + 1) + εM (k, n + 2) + hM (k, n + 3), where M (k, n) denotes the nth generalized Mersenne number, defined by (2). For k = 3 we have the Mersenne hybrid numbers denoted by M H n .

Theorem 5. Let n ≥ 0, k ≥ 3 be integers. Then
In [5], the character of a Horadam hybrid number was calculated.

Corollary 7.
Let n ≥ 0 be an integer. Then

Remark 8.
Using (5), we get another form of character of the generalized Mersenne hybrid number: Remark 9. For k = 3, we get the character of the Mersenne hybrid numbers: C(M H n ) = −75 · M 2 n − 128 · M n − 54. Next theorem gives the Binet formula for the generalized Mersenne hybrid numbers.
Using (3), we obtain the next results.
Theorem 14. Let n ≥ 0, k ≥ 3 be integers. Then Theorem 15. Let n ≥ 0 be an integer. Then Next we shall give the ordinary generating function for the generalized Mersenne hybrid numbers.
Proof. Assume that the generating function of the generalized Mersenne hybrid number sequence {M H k n } has the form G(t) = ∞ n=0 M H k n t n . Then 1−kt + (k − 1)t 2 G(t)