On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis

1 Introduction

In [1], we introduced new sequences called Quaternaccis. The idea of Quaternaccis arose during research on two-parametric quasi-Fibonacci numbers of 7th and 9th order [2] (see also [3], where some basic ideas were presented). For instance, two-parametric quasi-Fibonacci numbers of 7th order are members of sequences ( A n , 7 ( δ , γ ) ) , ( B n , 7 ( δ , γ ) ) and ( C n , 7 ( δ , γ ) ) defined by the following relations:

where k ∈ N ⧹ 7 N , n ∈ N 0 , δ , γ ∈ C and ζ ∈ C is a primitive 7th root of unity. From one side n th roots of unity form a cyclic group under multiplication, and from the other some sums of these roots are linearly independent (like 1 , ζ k + ζ 6 k and ζ 2 k + ζ 5 k used in the above definition). That made us extend the idea of quasi-Fibonacci numbers to real quaternions (more about quaternion algebra can be found e.g. in [4]).

Thus, from Definition 1 we obtain an explicit formula for powers of quaternions. During our investigations we obtained a lot of interesting results, also on quaternion structure itself [1]. For example, each of the four families of Quaternacci sequences satisfies the same recurrent relation:

but with different initial conditions

which led us to explicit formulae for Quaternaccis for b c d ≠ 0 :

where μ = b 2 + c 2 + d 2 .

We decided to introduce the aforementioned notions in the context of (real) split quaternions.

The aim of the first part of this paper is to present the results of our investigations. In the second part, we generate some new identities, called “bridges,” connecting sequences in question with the other known sequences (we give analogous identities for Quaternaccis as well).

2 Preliminaries

The split quaternion algebra (see [5]) was introduced by James Cockle in 1849 and it can be defined as follows.

If N ( q ) is equal to 1, then q is called a unit split quaternion. Split quaternion algebra is an associative, noncommutative, nondivision ring. Unlike quaternion algebra, split quaternion algebra contains zero divisors, nilpotent elements and nontrivial idempotents.

The split quaternion q is called spacelike, timelike or lightlike, if q q ¯ < 0 , q q ¯ > 0 or q q ¯ = 0 , respectively. Polar form of the split quaternion q = a 0 + a 1 i + a 2 j + a 3 k is known (see [5]). Let us denote scalar and vector part of the split quaternion q by S q = a 0 and V q = a 1 i + a 2 j + a 3 k , respectively, then:

1. Every spacelike quaternion can be written in the form:

   q = N ( q ) ( sinh θ + ω → cosh θ ) ,

   where sinh θ = a 0 N ( q ) , cosh θ = − a 1 2 + a 2 2 + a 3 2 N ( q ) and ω → = a 1 i + a 2 j + a 3 k − a 1 2 + a 2 2 + a 3 2 .

2. Every timelike quaternion with spacelike vector part (that is V q V q ¯ < 0 ) can be written in the form:

   q = N ( q ) ( cosh θ + ω → sinh θ ) ,

   where cosh θ = a 0 N ( q ) , sinh θ = − a 1 2 + a 2 2 + a 3 2 N ( q ) and ω → = a 1 i + a 2 j + a 3 k − a 1 2 + a 2 2 + a 3 2 .

3. Every timelike quaternion with timelike vector part (that is V q V q ¯ > 0 ) can be written in the form:

   q = N ( q ) ( cos θ + ω → sin θ ) ,

   where cos θ = a 0 N ( q ) , sin θ = − a 1 2 + a 2 2 + a 3 2 N ( q ) and ω → = a 1 i + a 2 j + a 3 k a 1 2 − a 2 2 − a 3 2 .

3 Split Quaternaccis and their basic properties

In this section, we define Split Quaternacci sequences and we present their basic properties. It is worth pointing out that our ideas can be transferred to other similar algebras, such as e.g. dual quaternions, bicomplex numbers, octonions, Clifford algebras (see Section 5).

Notation. From now on, we will use ( β , γ , δ ) only to denote arguments of Split Quaternaccis and i , j , k to denote the respective basis vectors of split quaternion algebra. Moreover, while describing properties valid for any argument ( β , γ , δ ) we shall omit it in formulation of results, in particular we will use the following abbreviations:

Also, we set

Some results in this section are pretty straightforward generalizations of those from [1], so we shall just give sketches of proofs.

From (3) by (4) we obtain two more recurrence relations for Split Quaternaccis:

In the next theorem, we collect the basic properties of transition matrices in (3) and (6). Note that they differ only by signs of some entries.

In particular that means that each entry in the first column (and row) of every power of these matrices contains only members of one of the sequences. Thus, we can separate the sequences using minimal polynomials of these matrices (which are of degree 2 by (c)). More precisely, thanks to minimal polynomials and the above remark we obtain a linear relation between members of each sequence separately (the same for each of them), which proves the following theorem.

We can obtain the aforementioned formulae (and analogous formulae for Quaternaccis) also using generating functions.

To prove the analogue of Theorem 2 for matrix in (7) we need the following fact.

Although there are explicit forms of Split Quaternaccis, some reduction formulae can also be useful.

From Binet formulae for Split Quaternaccis given in Corollary 1 we can derive some reduction formulae also for parameters.

Moreover, by Binet formulae for Split Quaternaccis we obtained the following interesting identities:

where n , m , r ∈ N , n > m and X ∈ { B , C , D } . As a special case of (29) and (30) we derive analogues of known identities

1. (Catalan identity) for n = m :

   ( A n ) 2 − A n + r A n − r = 1 2 ( 1 − λ 2 ) n − r ( ( 1 − λ 2 ) r − A 2 r ) , ( X n ) 2 − X n + r X n − r = ( 1 − λ 2 ) n − r ( X r ) 2 ;

2. (Cassini identity) for n = m and r = 1 :

   ( A n ) 2 − A n + 1 A n − 1 = − λ 2 ( 1 − λ 2 ) n − 1 , ( X n ) 2 − X n + 1 X n − 1 = − ( 1 − λ 2 ) n − 1 ;

3. (d’Ocagne identity) for n = n + 1 and r = 1 :

   A n + 1 A m − A m + 1 A n = 1 2 ( 1 − λ 2 ) m ( A n − m + 1 − A n − m ) , X n + 1 X m − X m + 1 X n = − x ( 1 − λ 2 ) m X n − m .