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Journal of Discrete Mathematical Sciences and
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ISSN: 0972-0529 (Print) 2169-0065 (Online) Journal homepage: https://www.tandfonline.com/loi/tdmc20
On k-Pell hybrid numbers
Paula Catarino
To cite this article: Paula Catarino (2019): On k-Pell hybrid numbers, Journal of Discrete Mathematical Sciences and Cryptography, DOI: 10.1080/09720529.2019.1569822 To link to this article: https://doi.org/10.1080/09720529.2019.1569822
Published online: 13 Feb 2019.
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On k-Pell hybrid numbers Paula Catarino
Department of Mathematics
University of Trás-os-Montes e Alto Douro Quinta de Prados
Vila Real 5001-801 Portugal
Abstract
The aim of this work is to introduce a new sequence of numbers called k-Pell hybrid numbers and the presentation of some algebraic properties involving this sequence. In addition, we present the Binet formula, the generating functions and some identities related with different terms of the sequence.
Subject Classification: Primary 11B37, Secondary 11R52, 05A15, 97F50
Keywords: k-Pell numbers, Hybrid numbers, Generating functions, Recurrence relations
1. Introduction
A hybrid number can be viewed as a generalization of the complex, hyperbolic and dual number. The set of hybrid numbers, denoted by K, was introduced by Özdemir in [15] and is defined as
2 2 2
{a bi cε dh a b c d: , , , ,i 1, ε 0, h 1, ih hi ε i}.
= + + + ∈ = − = = = − = +
K
Addition of hybrid numbers is done component-wise and this operation is commutative and associative. The multiplication of two hybrid numbers z1=a b i c1+ 1 + 1ε+d h1 and z2 =a b i c2+ 2 + 2ε+d h2 is defined by distributing the terms on the right as in ordinary algebra, preserving that the multiplication order of the units 1, i, e and h and then writing the values of followings replacing each product of units by the equalities stated in the following multiplication table:
E-mail: pcatarin@utad.pt
2 P. CATARINO Table 1
The multiplication table for K
· 1 i e h
1 1 i e h
i i –1 1 – h e + i
e e h + 1 0 -e
h h – e – i e 1
The multiplication of hybrid numbers is not commutative, but it has the property of associativity. The set K of hybrid numbers forms a non-commutative ring with respect to the addition and multiplication operations stated above. The conjugate of a hybrid number z = a + bi + ce + dh is the hybrid number z a bi c= − − ε−dh and the real number
2 ( )2 2 2
z z z z a⋅ = ⋅ = + −b c − −c d is called the character of the hybrid number z. The norm of a hybrid number z is denoted by ||z|| and is given by z z⋅ . For more details related with this number system, see the work developed by Özdemir in [15], where the author has examined this new ring of numbers, which is non-commutative and has the unit element. The relation ih = – hi = e + i between the units {i, e, h} of the three number systems (complex, hyperbolic and dual numbers) was given, and the author has seen the algebraic and geometric consistency of this relation, obtaining a noncommutative algebra which unified all three number systems.
Actually in the literature several research has been dedicated to the sequences of positive integers defined by recurrence relations and many papers are dedicated to the well-known Fibonacci (and Lucas) sequence and their generalizations (see, for example, [2], [3], [5], [10], [11], [13], and [14]). Note that recurrence relations are linear difference equations and the methods of resolution of homogeneous and nonhomogeneous linear difference equations have been studied as well as the convergence of their homogeneous part. For instance, in [1] the authors have dedicated on the convergence of the homogeneous part of specific delay difference equation, having been considered the combinatorial solutions of this homogeneous part and their generating functions as playing an important role.
The Fibonacci sequence {Fn}n ŒN is defined by Fn = Fn–1 + Fn–2, n ≥ 2,
beginning with the values F0 = 0 and F1 = 0 and, for any positive integer
number k, the sequence of k-Fibonacci numbers Fk, n introduced in
[10] are similarly defined by Fk n, =kFk n, 1− +Fk n, −2, 2, ,n≥ k∈ with the initial conditions Fk, 0 = 0, Fk, 1 = 1. In particular, for k = 1 and k = 2 we
obtain the classic Fibonacci numbers and the Pell numbers, respectively (see, for example, the study of Horadam, in [12] for more details). One generalization of the Pell sequence is the k-Pell sequence {Pk, n}n defined
recursively by
,0 0, 1, ,1 , 1 2 , , 1, 1. (1.1)
k k k n k n k n
P = P = P + = P +kP − n≥
The Binet-style formulae for this sequence is given by
1 2 , 1 2 ( ) ( ) , (1.2) n n k n r r P r r − = −
where r1= +1 1+k and r2 = −1 1+k are the roots of the characteristic equation r2−2r k− =0 associated with (1.1). Note that
1 2 2, 1 2
r r+ = r r = −k
and r r1− =2 2 1+k. For more details about these sequences see, for example, [4], [6], [7], [8] and [9].
Motivated essentially by the recent work of Özdemir in [15] about hybrid numbers, in this paper we introduce the k-Pell hybrid numbers sequence. The next three sections are dedicated to this new hybrid numbers and we give some algebraic properties of them, including the Binet formula, the generating functions and some identities related with different terms of the sequence.
2. The k-Pell hybrid numbers
This section aims to set out the definition of the k-Pell hybrid numbers and some elementary results. Following the defintion of hybrid number in [15], the nth k-Pell hybrid number HPk, n is defined, by
, , , 1 , 2 , 3
HPk n = Pk n+Pk n+i P+ k n+ε+Pk n+ h. (2.1)
Note that the sequence {HPk, n} of k-Pell hybrid numbers satisfies the
following second order recursive relation HPk n, 1+ =2HPk n, +kHPk n, 1− with initial conditions HPk,0 = +i 2ε+ +(4 k h) and HPk,1 = 1 + 2i + (4 + k) e +
(8 + 4k)h.
Now, using the definition of norm of a hybrid number, we easily have that the norm of k-Pell hybrid number is given by ||HPk, n||2 = P2k, n –
4P2
k, n+2 + (1 – k2)P2k, n+1 – 2 (1 + 2k)Pk, n+1 Pk, n+2 .
A matrix representation for a k-Pell hybrid number of order n is the
2 × 2 matrix , , 2 , 1 , 2 , 1 , , 2 , (1 ) , 2 3 ( 1) . k n k n k n k n k n k n k n HPk n P P Pk n k P P k P P P A + + + + + + + + + + − − = There is a bijection
4 P. CATARINO AHPk,n by the hybrid matrix corresponding to the k-Pell hybrid number
HPk,n.
The following result is easy to show by using the recurrence relation of second order (1.1):
Proposition 2.1: If AHPk,n is the hybrid matrix corresponding to the k-Pell hybrid
number HPk,n, then ||HPk,n||2 = det(AHPk,n).
3. Generating functions and Binet’s formula
Next we shall give the generating functions for {HPk,n}. We shall
write such sequence as a power series where each term of the sequence correspond to coefficients of the series. Considering this sequence, the associated generating function gHPk,n(t) is defined by , ( ) 0HP, .
n n HPk n k n g t ∞ t = = ∑
Theorem 3.1: The generating function for the k-Pell hybrid sequences is
, , , , ( ) ( ) k 0 k 1 k 0 . HPk n 2 HP HP 2HP t g t 1 2t kt + − = − − Proof. 1. We have gHPk,n(t) = HPk,0 + HPk,1t + HPk,1t 2+ + HP k,ntn + , –2tgHPk,n(t) = –2HPk,0t – 2HPk,1t 2 – 2HP k,2t3– – 2HPk,ntn+1 – and –kgHPk,n(t)t 2 = –kHPk,0t2 – kHPk,1t3 – kHPk,2t4– – kHPk,ntn+2 –. Hence (1 – 2t – kt2) gHPk,n(t) =
HPk,0 + (HPk,1 – 2HPk,0)t and the result follows.
The next result gives the Binet formula for this sequence. Theorem 3.2: (The Binet formula) For n ≥ 0 we have HP, ( ) ( ) ,
n n 1 1 2 2 1 2 k n r rr rr r − − = where 1
r and r2 are defined by r 1 r i 2r1= + 1 + 1ε+(3k r k 4 h+ 1( + )) and r 1 r i2= + 2 +
( ( )) ,
2 2
2rε + 3k r k 4 h+ + respectively.
Proof. 1. Using (2.1), we have HPk,n = Pk,n + Pk,n + 1i + Pk,n + 2Π+ Pk,n + 3h = Pk,n
+ (i + kh) Pk,n + 1 + (Œ + 2h) Pk,n + 2 = Pk,n + (i + kh) Pk,n + 1 + 2Pk,n + 1 Œ+ 4Pk,n + 1 h + kPk,n Œ+ 2kPk,n h = Pk,n (1 + kh = 2kh) + Pk,n + 1 (i + kh + 2Œ + 4h) and the result
follows by the use of (1.2).
4. Some identities involving this sequence
As a consequence of the Binet formula of Theorem 3.2, we get the following interesting identities.
Proposition 4.1: (Catalan’s identity) For natural numbers n, r, with n ≥ r and a positive integer k, if HPk,n is the nth k-Pell hybrid number, then the following identity is true: , , , , ( ) ( ) HP HP (HP ) ( ) (HP ) , ( ) 2r 2r 2 n r 2 2 1 1 2 k n r k n r k n k r s r s r k 4 k 1 − − + − − = − − +
where r1 and r2 are defined as in Theorem 3.2 and s1, s2 are defined by
( ) 2
1 2 1 1
s =r r − r and ( ) , 2
2 1 2 2
s =r r − r respectively.
Proof. Using Theorem 3.2, and the fact that the multiplication of two k-Pell
hybrid numbers is not commutative, we have
2 1 1 2 2 1 1 2 2 , , , 1 2 1 2 2 1 1 2 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) n r n r n r n r k n r k n r k n n n r r r r r r r r HP HP HP r r r r r r r r r r − − + + − + − − − = − − − − −
(
)
( )
( )
2 1 1 2 2 1 1 2 2 1 1 2 2 1 2 2 2 1 2 2 2 1 1 1 2 1 2 2 1 1 2 2 1 2 2 2 2 2 2 1 2 2 1 2 1 1 2 , 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (HP ) ( ) r r n n r r r r r r r n r k r r r k r r r r r r r r r r r r k r r r r r r r r r r r r r r r r r r r r r r r r k r r − − − − − + + = − − − − + + = − − − − = − − + − 2 2 2 2 2 1 1 , ( ) ( ) ( ) (HP ) 4( 1) r r n r k r s r s r k k − − = − − + and then the result follows.
Observe that the Cassini identity is a particular case of Catalan’s identity for r = 1 and then we easily have the following result.
6 P. CATARINO Proposition 4.2: (Cassini’s identity) For a natural number n and a positive integer k, if HPk,n is the nth k-Pell hybrid number, then the following identity is true: , , , , ( ) ( ) HP HP (HP ) ( ) (HP ) , ( ) 2 2 2 n 1 2 2 1 1 2 k n 1 k n 1 k n k 1 s r s r k 4 k 1 − − + − − = − − +
where r1 and r2 are defined as in Theorem 3.2 and s1, s2 are defined as in Proposition 4.1.
The d’Ocagne identity for this sequence is stated in the following result which proof is similar to the proof of Proposition 4.1.
Proposition 4.3: (d’Ocagne’s identity) Suppose that n is a nonnegative integer number and m any natural number. If m > n and HPk,n is the nth k-Pell hybrid number, then the expression of the d’Ocagne’s identity is given by
(
)
, ( )n ( )m n ( )m n , k m n 1 1 2 2 1 k HP d r d r 2 k 1 − − − − + − + where r1 and r2 are defined as in Theorem 3.2 and d1, d2 are defined by
, ,
1 1 2 1 2 2 1 2
d =r r r d− =r r r− respectively. 5. Conclusion
In this paper, the sequence of k-Pell hybrid numbers was introduced and some properties, the Binet formula and the generating function were presented. It is important to refer that for k = 1, the results stated in this paper are the correspondent for the sequences of the Pell hybrid numbers. Acknowledgment
This research was financed by Portuguese Funds through FCT – Fundação para a Ciência e a Tecnologia, within the Projects UID/ MAT/00013/2013 and UID/CED/00194/2013. Also the author thanks the referee for all comments and suggestions made to make improve the final version.
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