0188-7009/021343-9
published online July 16, 2016 DOI 10.1007/s00006-016-0704-8
Advances in
Applied Clifford Algebras
On
(p, q)-Fibonacci quaternions and their
Binet formulas, generating functions and
certain binomial sums
Ahmet ˙Ipek*
Abstract. Formulas and sums involving many well-known special
quater-nion sequences (such as the Fibonacci, Pell, Jacobsthal quaterquater-nion sequences and so on) play important roles in various branches of sci-ence. Binet formulas, generating functions and certain sums of these quaternion sequences based on Fibonacci-like numbers have been inves-tigated by a number of authors in many different ways. In this paper, which is a sequel to earlier works, we first introduce a new class of (p, q) -Fibonacci quaternions which generalizes Fibonacci, Pell, and Jacob-sthal quaternions, and then we derive a new Binet formula, generating functions, and certain binomial sums for these quaternions.
Mathematics Subject Classification. Primary 11B39; Secondary 05A15;
Tertiary 11R52.
Keywords. Fibonacci quaternion, Binet formula, Generating function,
Binomial sum.
1. Introduction
The classic Fibonacci sequence{Fn}n≥0is defined byFn+2=Fn+1+Fnwith the initial conditions F0 = 0 andF1 = 1. For some recent developments on Fibonacci and Fibonacci-like sequences, we refer the reader to [1,15,17,18].
A generalization of the classic Fibonacci sequence {Fn}n≥0, which is called the (p, q)-Fibonacci sequence and denoted by {Fn(p, q)}n≥0, is defined by the following recurrence relation forn ≥ 0 and nonzero real numbers p andq such that p2+ 4q > 0:
Fn+2(p, q) = pFn+1(p, q) + qFn(p, q) (1.1)
withF0(p, q) = 0 and F1(p, q) = 1, or explicitly as:
Fn(p, q) ==γ n− δn
γ − δ ,
whereγ and δ are the roots of the characteristic equation
x2− px − q = 0 (1.2)
associated to (1.1). It is well known that forp2+ 4q > 0 solving this equation, we get two distinct characteristic roots:
γ = p + √ Δ 2 andδ = p −√Δ 2 ,
where Δ =p2+ 4q. First note that the condition p2+ 4q > 0 guarantees that the numbersγ and δ are real and that γ = δ. Also notice that
γ2+q = γ√Δ (1.3)
and
δ2+q = −δ√Δ. (1.4)
We will be using these facts frequently.
Recurrence relation (1.1) is a natural generalization of the recurrence relation
Un+2,p=pUn+1+Un (1.5) withU0= 0 andU1= 1, wherep is any nonzero integer [10].
The algebra of real quaternionsH = {a = a0+a1i + a2j + a3k | as∈ R, s = 0, 1, 2, 3} is the vector space R4and it has the standard basis ofR4 as e0= 1, e1=i, e2=j, e3=k. These elements of the basis satisfy the following
multiplication rules:
e2
0= 1, e21=e22=e23=−1, e1e2=−e2e1=e3,e2e3=−e3e2=e1,
e3e1=−e1e3=e2,
where e0 is the identity element in H. This algebra is associative and non-commutative.
Quaternions play an important role in diverse areas such as mechanics [6], kinematics [2], quantum mechanics [16] and chemistry [5].
Numerous quaternion sequences involving extensions and generaliza-tions of the Fibonacci sequence play an important role in the theory of quaternion sequences in mathematics.
Horadam [8] gave essentially first new results on the Fibonacci quater-nions by defining
Qn=Fn+iFn+1+jFn+2+kFn+3, (1.6) where Fn is the nth classic Fibonacci number. Many papers later studied some generalizations and different types of these quaternions. In [13], the authors introduced bi-periodic Fibonacci quaternions and gave a generat-ing function and Binet formula for these quaternions. Similarly, Yılmaz and S
function and Binet formula for these quaternions. Flaut and Shpakivskyi [4] gave many properties of generalized Fibonacci quaternions and Fibonacci-Narayana quaternions. Using generalized Fibonacci recurrence relation (1.5), Polatlı and Kesim [10] introduced and studied a family of the generalized Fibonacci quaternions defined by
QUn =Une0+Un+1e1+Un+2e2+Un+3e3. (1.7)
They also gave an exponential generating function for the generalized Fibonacci quaternions defined in (1.7), and some new formulas for binomial sums of these quaternions. This paper is devoted to studying the following quaternionic sequence forn ≥ 0:
QFn(p, q) = Fne0+Fn+1e1+Fn+2e2+Fn+3e3, (1.8)
where Fn is the nth (p, q)-Fibonacci number and e0, e1, e2, e3 is the basis inH.
The recurrence relation (1.8) is a natural extension of the recurrence relations (1.6) and(1.7) and it is closely related to the classic Fibonacci quaternion. For this reason, we refer toQFn(p, q) as a generalized Fibonacci
quaternion. Other particular cases of the quaternionQFn(p, q) were studied, e.g. in [3,7,8,10–12]
Motivated by [3,7,8,10–12], here we introduce the quaternionQFn(p, q). Our aim is to find Binet formula, generating function and certain binomial sums of the sequence{QFn(p, q)}n≥0.
The paper is organized as follows: the main results and their proofs for (p, q)-Fibonacci quaternions are stated in the next section. Conclusions are presented in the last section.
2.
(p, q)-Fibonacci quaternions
We now define new quaternions, which are called (p, q)-Fibonacci quaternions.
Definition 2.1. The (p, q)-Fibonacci quaternion sequence {QFn(p, q)}n≥0 is defined by the following recurrence relation:
QFn(p, q) = Fne0+Fn+1e1+Fn+2e2+Fn+3e3 or QFn(p, q) = 3 s=0 Fn+ses (2.1) whereFn is thenth generalized (p, q)-Fibonacci number.
Throughout this paper, we useQFn to denote thenth (p, q) -Fibonacci quaternion.
From (1.1), taking into account (2.1), we have
QFn+1=pQFn+qQFn−1 (2.2) forn ≥ 1.
Most of special sequences of mathematics, such as the Fibonacci and Fibonacci quaternion sequences, can be expressed in terms of the roots of their characteristic equation.
One immediate consequence of the recurrence relation (2.2) is the fol-lowing theorem:
Theorem 2.2. LetQFn be the nth (p, q)-Fibonacci quaternion. Then QFn =γγ
n− δδn
γ − δ , whereγ =3s=0γses andδ =3s=0δses.
Proof. From the definiton of thenth (p, q)-Fibonacci quaternion QFnin(2.1) and Binet formula for thenth (p, q)-Fibonacci number Fn, we write
QFn= 3 s=0 Fn+ses= 3 s=0 γn+s− δn+s γ − δ es.
Usingγ =3s=0γsesandδ =3s=0δsesit is easy to verify that
QFn= γγ n− δδn
γ − δ . (2.3)
Thus, the proof is completed. The formula (2.3) is a Binet formula for thenth (p, q)-Fibonacci quater-nionQFn.
We now derive the ordinary generating functionGF(x) =∞n=0QFnxn and the exponential generating function EF(x) = ∞n=0 QFnxn!n for QFn defined by (2.1).
Theorem 2.3. ForQFn defined by (2.1), the following is its ordinary
gener-ating function:
GF(x) = QF0+ (1− px − qx−pQF0+2QF1)x. (2.4)
Proof. By substituting the recurrence
QFn+1=pQFn+qQFn−1
from (2.2) into1− px − qx2GF(x), the result follows.
Theorem 2.4. ForQFn defined by (2.1), the following is its exponential
gen-erating function:
EF(x) = γe
γx− δeδx
γ − δ .
Proof. Using Binet formula for QFn in EF(x) = ∞n=0 QFnxn!n, the series may be rewritten as EF(x) = ∞ n=0 γγn− δδn γ − δ xn n!. (2.5)
We first recall that a binomial coefficientmnmay be defined form, n ∈ N0=N ∪ {0} and m ≥ n m n =m(m − 1) . . . (m − n + 1) n! .
It is well known that for every non-negative integerm (a + b)m= m n=0 m n anbm−n wherea and b are any real numbers.
Theorem 2.5. Letm be a non-negative integer. Then,
m n=0 m n QF2n+kqm−n= QFk+mΔm2, m even QLk+mΔm−12 , m odd . (2.6)
Proof. Let the left-hand side of the assertion (2.6) of Theorem2.5be denoted byS1. With the help of Binet formula (2.3), we have
S1= m n=0 m n γγ2n+k− δδ2n+k γ − δ qm−n.
Note that mn=0mn(γ2)nqm−n = (γ2 +q)m and mn=0mn(δ2)nqm−n = (δ2+q)m. Combining this withγ2+q = γ√Δ andδ2+q = −δ√Δ we obtain
S1= γγ k γ − δ γ√Δ m − δδk γ − δ −δ√Δ m . Ifm is even, then S1= γγk+m− δδk+m γ − δ Δm2 and hence S1=QFk+mΔm2. Ifm is odd, then S1=γγk+m+δδk+mΔm−12 (2.7) sinceγ − δ =√Δ. Finally, we apply Binet formula [9]
QLn=γγn+δδn
for thenth (p, q)-Lucas quaternion in (2.7) and we arrive at the desired result
for any oddm.
Theorem 2.6. Letm be a non-negative integer. Then,
m n=0 m n (−1)nQF2n+kqm−n= pmQF k+m, m even −pmQFk+m, m odd . (2.8)
Proof. For convenience, let the left-hand side of the assertion (2.8)of Theo-rem2.6be denoted byS2. Applying Binet formula (2.3), we obtain
S2= m n=0 m n (−1)n γγ2n+k− δδ2n+k γ − δ qm−n. (2.9) Ifmn=0mn(−γ2)nqm−n = (−γ2+q)mandmn=0mn(−δ2)nqm−n = (−δ2+
q)mare substituted into (2.9), we find in this case
S2= γγ k γ − δ −γ2+qm− δδk γ − δ −δ2+qm. (2.10)
We know from the characteristic Eq. (1.2) that the roots of this equation can be written as−pγ = −γ2+q and −pδ = −δ2+q. Inserting these into (2.10) gives S2= (−p)m γγk+m− δδk+m γ − δ = (−p)mQFk+m.
Thus, this completes the proof.
Theorem 2.7. Letm be a non-negative integer. Then,
m n=0 m n pnQF nqm−n =QF2m. (2.11)
Proof. Let S3 = mn=0mnpnQFnqm−n. Applying Binet formula (2.3), we transform the left-hand side of (2.11) into:
S3= m n=0 m n pnγγn− δδn γ − δ qm−n. Elementary calculations imply:
S3= γ γ − δ m n=0 m n (pγ)nqm−n− δ γ − δ m n=0 m n (pδ)nqm−n. If we use mn=0mn(pγ)nqm−n = (pγ + q)m and mn=0mn(pδ)nqm−n = (pδ + q)m, then we obtain S3=γγ 2m− δδ2m γ − δ ,
which completes the proof of Theorem2.7.
Theorem 2.8. Letm be a non-negative integer. Then,
m n=0 m n (QFn)2qm−n= γ2γm+δ2δmΔm−2 2 , m even γ2γm− δ2δmΔm−2 2 , m odd .
Proof. LetS4 =mn=0mn(QFn)2qm−n. It follows from (2.3) that the sum
S4can be written in a concise form in terms of the roots of (1.2): S4= m n=0 m n γγn− δδn γ − δ 2 qm−n
or S4= γ 2 (γ − δ)2 m n=0 m n γ2nqm−n+ δ2 (γ − δ)2 m n=0 m n δ2nqm−n − γδ + δγ (γ − δ)2 m n=0 m n (γδ)nqm−n (2.12)
Notice that the sums mn=0mn γ2nqm−n and mn=0mn δ2nqm−n respectively equal m n=0 m n γ2nqm−n =γ2+qm (2.13) and m n=0 m n δ2nqm−n=δ2+qm. (2.14)
We use (2.13) and (2.14) withγ2+q = γ√Δ,δ2+q = −δ√Δ andγδ = −q to obtain, by (2.12), that
S4=
γ2γ√Δm+δ2−δ√Δm
(γ − δ)2 . (2.15)
Ifm is even, the equality (2.15) becomes the following:
S4=γ2γm+δ2δmΔm−22 . Similarly, ifm is odd, the equality (2.15) becomes
S4=γ2γm− δ2δmΔm−22 .
3. Conclusions
In this work, we introduced the (p, q)-Fibonacci quaternion sequence and constructed Binet formula for thenth (p, q)-Fibonacci quaternion QFn. By using the Binet formula, we derived some generating functions involving the quaternionQFn. Also, we gave some binomial sum formulas related to these quaternions.
Acknowledgments
The author would like to express his gratitude to Prof. Rafal Ablamowicz for careful reading, valuable comments and helpful suggestions which led to the improvement of the paper. The author are also grateful to the anonymous reviewers for their helpful comments and suggestions which have helped us in improving the quality of this paper.
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Ahmet ˙Ipek
Department of Mathematics Kamil ¨Ozda˘g Science Faculty Karamano˘glu Mehmetbey University 70100 Karaman, Turkey
e-mail: ahmetipek@kmu.edu.tr Received: March 17, 2016. Accepted: June 30, 2016.