arXiv:1801.10223v2 [math.CO] 29 Aug 2018

ON THE HORADAM SYMBOL ELEMENTS

arXiv:1801.10223v1 [math.CO] 30 Jan 2018

SAI GOPAL RAYAGURU, DIANA SAVIN, AND GOPAL KRISHNA PANDA Abstract. Horadam symbol elemnts are introduced. Certain properties of these elements are explored. Some well known identities such as Catalan identity, Cassini formula and d’Ocagne’s identity are obtained for these elements.

Key words: Recurrence Relations, Quaternions, Octonions, Symbol algebra. 2010 Subject classification [A.M.S.]: 11R52;11B37; 11B39; 20G20. 1. Introduction

Let N be a positive integer and K, a field such that char(K) does not divide N and it contains a primitive N th root of unity: ω. Let K ∗ = K \ {0}, a, b ∈ K ∗ , and let S be the algebra over K generated by elements x and y, with xN = a, y N = b, yx = ωxy. This algebra is
called a,b . Oba symbol algebra (also known as a power norm residue algebra) and is denoted by K,ω i j serve that S has a K-basis {x y : 0 ≤ i, j < N }. The symbol algebra of degree 2 is called the quaternion algebra and here the symbol algebras are generalization of the quaternion algebras. In 1963, Horadam[12] defined the n-th Fibonacci quaternion Qn as Qn = Fn + iFn+1 + jFn+2 + kFn+3 , where Fn denote the n-th Fibonacci number and i, j, k are the quaternion units, satisfying i2 = j 2 = k2 = −1, ij = −ji = k, jk = −kj = i, ki = −ik = j.

Subsequently many authors studied the extension and generalization of the Fibonacci quaternion and investigated their properties. Flaut and Shpakivskyi[4] studied on the properties for the generalized Fibonacci quaternions and Fibonacci-Narayana quaternions in a generalized quaternion algebra. Variants of octonion algebra involving generalized Fibonacci octonions were also studied by many authors (see [8, 9, 10, 14, 13, 16, 17, 18, 20]). Flaut and Savin[3] and after Flaut, Savin and Iorgulescu [5] considered the symbol algebra of degree 3 and defined the n-th Fibonacci symbol elements Fn and n-th Lucas symbol elements Ln as Fn = fn .1 + fn+1 .x + fn+2 .x2 + fn+3 .y + fn+4 .xy + fn+5 .x2 y + fn+6 .y 2 + fn+7 .xy 2 + fn+8 .x2 y 2 and Ln = ln .1 + ln+1 .x + ln+2 .x2 + ln+3 .y + ln+4 .xy + ln+5 .x2 y + ln+6 .y 2 + ln+7 .xy 2 + ln+8 .x2 y 2 respectively, where fn and ln are the n-th Fibonacci and and nth Lucas number. 1

Although, the notation used for defining the Fibonacci symbol elements is same as the notation for the Fibonacci numbers in the Fibonacci quaternion, we confirm their distinct behavior. We believe that the readers will not be confused with these notations. Flaut and Savin[6] studied some properties and applications of (a, b, x0 , x1 )-numbers and quaternion. Savin[19] defined and studied properties of special numbers, special quaternions and special symbol elements. In a recent paper, Flaut and Savin[7] presented some applications of special numbers obtained from a difference equation of degree three. We generalize the concept of Fibonacci and Lucas symbol elements to the Horadam symbol elements. Moreover, we study the Binet formula, generating function and some identities involving Horadam symbol elements. We will show that these elements satisfy many properties similar to that of Horadam numbers. The Horadam sequence {wk (a0 , a1 , p, q)}, k ≥ 1 is defined by

wk+1 = pwk + qwk−1 ; w0 = a0 , w1 = a1 , p+

√

(1.1) p2 +4q

and β = where a0 , a1 , p, q are integers such that ∆ = p2 + 4q > 0. Further, α = 2 √2 p− p +4q are roots of the characteristic equation. The Binet formula for this sequences are 2 given by Aαk − Bβ k , wk = α−β where A = a1 − a0 β and B = a1 − a0 α. 2. Properties Of The Horadam Sequence

In this section, we explore certain elementary properties of the Horadam sequence. Some these properties represents weighted sum formulas of this sequence. Theorem 2.1. The Horadam sequence defined in (1.1) satisfies the following identities. (a) q 2 wk + pwk+3 = (p2 + q)wk+2 , (b) q 2 wk + wk+4 = (p2 + 2q)wk+2 ,

(c) q 3 + p2 q 2 wk + pwk+5 = p4 + 3p2 q + q 2 wk+2 ,


(d) (p2 q 2 + 2q 3 )wk + wk+6 = p4 + 3q 2 + 4p2 q wk+2 ,



(e) p4 q 2 + 3p2 q 3 + q 4 wk + pwk+7 = p4 + 5p4 q + 6p2 q 2 + q 3 wk+2 , (f)

(g) (h)

k X

i=1 k X

i=1 k X i=1

pk−i qwi = wk+2 − pk w2 , pq k−i w2i−1 = w2k − q k w0 , pq k−i w2i = w2k+1 − q k w1 , 2

2 2 = w2k w2k+6 − w2k w2k+2 − − w2k+1 (i) w2k+3


2 2 (j) w2k+3 + q 2 w2k+1 = p2 + 2q w2k w2k+4 −

q 2k ·(qa20 +pa0 a1 −a21 )·(p3 +pq+4q 3 −p2 −4q ) , p2 +4q q 2k ·(qa20 +pa0 a1 −a21 )·(p3 +p2 q 2 +pq+8q 3 ) , p2 +4q

q 2k ·(qa20 +pa0 a1 −a21 )·(p6 +4p4 q+3p2 q 2 −p3 −pq−8q 3 ) . p2 +4q


2 2 2 (k) w2k+3 − p2 + 2q w2k+2 + q 2 w2k+1 =

Proof. (a) and (b) can be proved using the recurrence relation of the sequence wk . We are proving (c). From (a) we have q 3 wk + pqwk+3 = (p2 q + q 2 )wk+2 . From (b) we have p2 q 2 wk + p2 wk+4 = (p4 + 2p2 q)wk+2 . Adding member with member the last two equalities, we obtain:

q 3 + p2 q 2 wk + p (pwk+4 + qwk+3 ) = p4 + 3p2 q + q 2 wk+2 .

This last equality is equivalent to

q 3 + p2 q 2 wk + pwk+5 = p4 + 3p2 q + q 2 wk+2 .

We are proving (d). Using (b) for k → k + 2 and then normally, we have:

q 2 wk + wk+6 = (p2 + 2q)wk+4 − q 2 wk+2 + q 2 wk =   = (p2 + 2q) · (p2 + 2q)wk+2 − q 2 wk − q 2 wk+2 + q 2 wk =   = (p2 + 2q)2 − q 2 wk+2 + q 2 (1 − p2 − 2q)wk .
This implies that (p2 q 2 + 2q 3 )wk + wk+6 = p4 + 3q 2 + 4p2 q wk+2 . The proof of (e) is similar to the proof of (c) and hence, it is omitted. The proof of (f) is based on mathematical induction. Observe that the assertion in (f) is true for k = 1. Asuume that it is true for k = t. Since, t+1 X i=1

pt+1−i qwi =

t X

pt+1−i qwi + qwt+1

i=1

=p

X t i=1

p

t−i

qwi



+ qwt+1

=p(wt+2 − pt w2 ) + qwt+1

=wt+3 − pt+1 w2 ,

the assertion is true for k = t + 1. This proves (f). The proofs of (g) and (h) are similar to that of (f) and hence, are omitted. (i) Let k, l be two positive integers, k < l. Using Binet formula for Horadam sequence, we have Aα2k − Bβ 2k Aα2l − Bβ 2l · = w2k · w2l = α−β α−β
2 Aαk+l − Bβ k+l 2ABαk+l β k+l − ABα2k β 2l − ABα2l β 2k + = = (α − β)2 (α − β)2
k+l 2k 2l−2k + β 2l−2k 2AB (−q) − AB (−q) α 2 = wk+l + . (α − β)2 So, we obtained:
2AB (−q)k+l − ABq 2k α2l−2k + β 2l−2k 2 w2k · w2l = wk+l + (2.1.) (α − β)2 3

Making l = k + 1 in the equality (2.1) and using Vi`ete’s relations for α and β, we have:
−2ABq 2k+1 − ABq 2k α2 + β 2 2 = w2k · w2k+2 = w2k+1 + (α − β)2
ABq 2k · 2q + p2 + 2q 2 = = w2k+1 − p2 + 4q
2 = w2k+1 + q 2k · qa20 + pa0 a1 − a21 . So, we obtained:

2 w2k · w2k+2 = w2k+1 + q 2k · qa20 + pa0 a1 − a21




(2.2.)

Making l = k + 2, respectively l = k + 3 in the equality (2.1), using Vi`ete’s relations for α and β and doing calculations similar to previous ones, we have:

p2 · q 2k · p2 + 2q · qa20 − a21 + pa0 a1 2 w2k · w2k+4 = w2k+2 + (2.3.) p2 + 4q and

q 2k · p3 + pq + 4q 3 · qa20 − a21 + pa0 a1 (2.4.) + w2k · w2k+6 = p2 + 4q From relations (2.2) and (2.4) we obtain:

q 2k · qa20 + pa0 a1 − a21 · p3 + pq + 4q 3 − p2 − 4q 2 2 w2k+3 − w2k+1 = w2k w2k+6 − w2k w2k+2 − . p2 + 4q 2 w2k+3

(j) From relations (2.2) and (2.4) we obtain: 2 2 +q 2 w2k+1 w2k+3



q 2k · qa20 + pa0 a1 − a21 · p3 + pq + 4q 3 + p2 q 2 + 4q 3 . = w2k ·w2k+6 +q w2k ·w2k+2 − p2 + 4q 2

Applying (b) the last equality is equivalent with:

q 2k · qa20 + pa0 a1 − a21 · p3 + p2 q 2 + pq + 8q 3 +q = p + 2q w2k · w2k+4 − p2 + 4q (2.5.) (k) From relations (2.3) and (2.4) we obtain:
2 2 2 = + q 2 w2k+1 − p2 + 2q w2k+2 w2k+3


2k
2 q · qa20 + pa0 a1 − a21 · p3 + p2 q 2 + +pq + 8q 3 p2 · q 2k · p2 + 2q · qa20 − a21 + pa0 a1 − . = p2 + 4q p2 + 4q This last last equality is equivalent with:


2 q 2k · qa20 + pa0 a1 − a21 · p6 + 4p4 q + 3p2 q 2 − p3 − pq − 8q 3 2 2 2 2 w2k+3 − p + 2q w2k+2 +q w2k+1 = . p2 + 4q 2 w2k+3

2

2 w2k+1

2






Theorem 2.2. For every integer n ≥ 0 and for all c ∈ R − {0},   n X i n+1 c (p − 1)wi + (c − 1)wi+1 + qwi−1 . c wn+1 = a0 + i=0

4

Proof. The proof is based on mathematical induction. For n = 0, both sides of the above equation are equal to ca1 . Assume that the assertion holds for n = m. Using the recurrence relation for the Horadam sequence, we get  m+1 X  i c (p − 1)wi + (c − 1)wi+1 + qwi−1 a0 + i=0

= a0 +

  m X ci (p − 1)wi + (c − 1)wi+1 + qwi−1 i=0

  m+1 +c (p − 1)wm+1 + (c − 1)wm+2 + qwm   m+1 m+1 =c wm+1 + c (p − 1)wm+1 + (c − 1)wm+2 + qwm = cn+2 wm+2 . Thus, the assertion is true for n = m + 1. This completes the proof.



As (p, q)−Fibonacci and (p, q)−Lucas sequence are special cases of Horadam sequence with initial values a0 = 0, a1 = 1 and a0 = 2, a1 = p respectively, they satisfy the identities in Theorem 2.1. The Binet formula for the (p, q)−Fibonacci and (p, q)−Lucas numbers are given by: Fp,q,k =

αk − β k , Lp,q,k = αk + β k . α−β

Using these Binet formulas, in the following theorem, we explore some properties of the (p, q)−Fibonacci and (p, q)−Lucas sequence. Theorem 2.3. Let Fp,q,k and Lp,q,k denote the kth term of the (p, q)−Fibonacci and (p, q)−Lucas sequence respectively. Then, we have the following identities. (a) qFp,q,k−1 + Fp,q,k+1 = Lp,q,k , (b) qLp,q,k−1 + Lp,q,k+1 = (p2 + 4q)Fp,q,k , (c) Fp,q,k+2 − q 2 Fp,q,k−2 = pLp,q,k , (d) Lp,q,k+2 − q 2 Lp,q,k−2 = p(p2 + 4q)Fp,q,k , (e) pFp,q,k + Lp,q,k = 2Fp,q,k+1 , (f) pLp,q,k + (p2 + 4q)Fp,q,k = 2Lp,q,k+1 , (g) q 2 Fp,q,k + pLp,q,k+2 = Fp,q,k+4 , (h) q 2 Lp,q,k + p(p2 + 4q)Fp,q,k+2 = Lp,q,k+4, (i) pFp,q,k+2 + qLp,q,k = (p2 + 2q)Fp,q,k+1 , 5

(j) pLp,q,k+2 + q(p2 + 4q)Fp,q,k = (p2 + 2q)Lp,q,k+1 , (k) q 3 Fp,q,k + Fp,q,k+6 = (p2 + q)Lp,q,k+3, (l) q 3 Lp,q,k + Lp,q,k+6 = (p2 + q)(p2 + 4q)Fp,q,k+3 , (m) q 4 Fp,q,k + pFp,q,k+8 = [(p2 + q)2 + pq(1 + p + q)]Fp,q,k+4 . Proof. We prove (a) only. Proofs of (b)-(m) are similar. Using the Binet formula and the fact αβ = −q, we get αk−1 − β k−1 αk+1 − β k+1 + α−β α−β   1 k q k q = α ( + α) − β ( + β) α−β α β   1 k k α (−β + α) − β (−α + β) = α−β = Lp,q,k .

qFp,q,k−1 + Fp,q,k+1 = q



The following theorem deals with the generating function and exponential generating function of a subsequence of the Horadam sequence. The proofs are similar to that of the Horadam sequence and hence are omitted. Theorem 2.4. For fixed integers k and m with k > m ≥ 0 and n ∈ N, (a) the generating function of the Horadam number wkn+m is ∞ X wm − (−q)k wm−k s , wkn+m sn = 1 − (αk + β k )s + (−q)k s2 n=0

(b) the exponential generating function of the Horadam number wkn+m is ∞ X wkn+m

n=0

n!

n

s =

ks

Aαm eα

k

− Bβ m eβ s . α−β

The following theorem provides two sum formulas of a subsequence of the Horadam sequence. Theorem 2.5. For any natural numbers m and k with k > m ≥ 0, we have (a) n X (−q)m wmn+k − wmn+m+k − (−q)m wk−m + wk , wmr+k = 1 + (−q)m − (αm + β m ) r=0

(b)

n X (−1)r wmr+k r=0

=

(−1)n+1 q m wmn+k − (−1)n+1 wmn+m+k − q m wk−m + wk . 1 + (−q)m − (−1)m (αm + β m ) 6

Proof. We prove (a) only. The proof of (b) is similar. Using the Binet formula for the Horadam sequence, we get n X

wmr+k

r=0

=

n X Aαmr+k − Bβ mr+k

α−β    mn+m  mn+m α −1 −1 1 k β Aαk − Bβ = α−β αm − 1 βm − 1  1 (−q)m (Aαmn+k − Bβ mn+k ) − (Aαmn+m+k − Bβ mn+m+k ) = α−β 1 + (−q)m − (αm + β m )  (−q)m (Aαk−m − Bβ k−m ) − (Aαk − Bβ k ) . − 1 + (−q)m − (αm + β m ) r=0



 In the following theorem we explore weighted sum formulas for a subsequence of the Horadam sequence. Theorem 2.6. For any natural numbers m and k with k > m ≥ 0, we have (a)

n X

rwmr+k =n

r=0



(1 + (−q)m )wmn+m+k − wmn+2m+k − (−q)m wmn+k 1 + (−q)m − (αm + β m )  wmn+m+k + q 2m wmn+k−m − 2(−q)m wmn+k − , (1 + (−q)m − (αm + β m ))2

(b) n X (−1)r−1 rwmr+k r=0

n+1

=(−1)



(n − 1 − 2n(−q)m )wmn+m+k + q 2m (n − 1)wmn+k−m (1 + (−q)m − (αm + β m ))2  nwmn+2m+k + (nq 2m − 2(−q)m (n − 1))wmn+k . − (1 + (−q)m − (αm + β m ))2

Proof. These identities can be proved by using the proof of Theorem 2.5 and the identity n X r=1

rxr−1 =

nxn+1 − (n + 1)xn + 1 , |x| < 1 (x − 1)2

which is obtained by differentiating the sum formula of a geometrical progression.



In the following two theorems, we obtain a binomial weighted sum formula of the Horadam sequence.   Pm m k Theorem 2.7. For the integer m ≥ 0, k=0 p wk q m−k = w2m . k 7

Proof. Using the Binet formula of wk , we get m   X m k p wk q m−k k k=0 m   X m k Aαk − Bβ k m−k q = p k α−β k=0 m   m   A X m k k m−k B X m k k m−k = p α q − p β q k k α−β α−β k=0

=

k=0

A B Aα2m − Bβ 2m (pα + q)m − (pβ + q)m = . α−β α−β α−β



Theorem 2.8. Let m be a non-negative integer. Then, ( m m   X if m even; wk+m ∆ 2 , m m−n (a) w2n+k q = k+m + Bβ k+m )∆ m−1 n 2 , if m odd. (Aα n=0 (b)

m X

(−1)n

n=0

   m m p wk+m , if m even; m−n w2n+k q = n −pm wk+m , if m odd.

Proof. The proof of this theorem is similar to that of Theorem 2.7.



3. Properties Of The Horadam Symbol Elements

In this section, we define Horadam symbol element and establish some properties for these elements. We also find the Binet formula and the generating function for these elements. Consider the symbol algebra S = So S has a K-basis

a,b
K,ω

of degree N generated by x and y over the field K.

{1, x, x2 , · · · , xN −1 , y, y 2 , · · · , y N −1 , xy, · · · , xy N −1 , x2 y, · · · , x2 y N −1 , · · · , xN −1 y, · · · , xN −1 y N −1 }. We denote the ith term of the above basis as ei−1 so that {e0 , e1 , · · · , eN 2 −1 } is an ordered basis for S. We define the k-th term of Horadam symbol element as Wk =

2 −1 NX

l=0

8

wk+l el ,

where wk is the kth Horadam number while Wk is the kth Horadam symbol element. It is easy to see that the sequence of Horadam symbol elements satisfy the same recurrence as Horadam sequence, that is, Wk+1 = pWk + qWk−1 , where the initial terms are W0 = (w0 , w1 , · · · , wn2 −1 ) and W1 = (w1 , w2 , · · · , wn2 ). The auxiliary recurrence relation is given by λ2 − pλ − q = 0 whose roots are √ equation of this √ α=

p+

p2 +4q 2

and β =

p−

p2 +4q . 2

α=

Let

2 −1 NX

l

α el and β =

l=0

2 −1 NX

β l el .

l=0

In the following theorem we obtain the Binet formula for the Horadam symbol elements. Theorem 3.1. The Binet formula of Horadam symbol elements is Aααk − Bββ k Wk = , α−β where A = a1 − a0 β, B = a1 − a0 α. Proof. Using the Binet formula for Horadam sequence, we get Wk =

2 −1 NX

wk+l el =

2 −1 NX

l=0

l=0

Aαk+l − Bβ k+l el α−β 2

2

N −1 N −1 Aαk X l βk X l = α el − β el . α−β α−β l=0

l=0

 In the following theorem, we prove two important identities for the Horadam symbol elements. Theorem 3.2. Let Wk be the kth term of the Horadam symbol elements. Then the following identities hold. (a) q 2 Wk + pWk+3 = (p2 + q)Wk+2 , (b) q 2 Wk + Wk+4 = (p2 + 2q)Wk+2 ,

(c) q 3 + p2 q 2 Wk + pWk+5 = p4 + 3p2 q + q 2 Wk+2 ,


(d) (p2 q 2 + 2q 3 )Wk + Wk+6 = p4 + 3q 2 + 4p2 q Wk+2 ,



(e) p4 q 2 + 3p2 q 3 + q 4 Wk + pWk+7 = p4 + 5p4 q + 6p2 q 2 + q 3 Wk+2 . 9

Proof. Using Theorem 2.1 and the definition of Horadam symbol elements, we find 2

q Wk + pWk+3 =q

2 −1 NX

2

wk+l el + p

w(k+3)+l el

l=0

l=0

=

2 −1 NX

2 −1 NX

(q 2 wk+l + pw(k+3)+l )el

l=0

=

2 −1 NX

(p2 + q)w(k+2)+l el .

l=0

Thus (a) is proved. The proofs of (b), (c), (d), (e) are similar to the proof of (a) and hence, are omitted.  In the next two theorems we establish some important sum formulas for the Horadam symbol elements. Theorem 3.3. The Horadam symbol elements satisfy (a) (b)

k X

i=1 k X i=1

pk−i qWi = Wk+2 − pk W2 ,

(c)

k X i=1

pq k−i W2i = W2k+1 − q k W1 .

pq k−i W2i−1 = W2k − q k W0 ,

Proof. We prove (a) only. The proofs of (b) and (c) are similar. Using Theorem 2.1 and the definition of Horadam symbol elements, we get 2 −1  k 2 −1  NX NX k k X X X k−i k−i k−i p qwi+l el wi+l el = p q p qWi = i=1

i=1

=

=

2 −1  NX

l=0

=

2 −1  NX

qwi − p

k

l X i=1

p

l−i



qwi el



k

w(k+2)+l − p w2+l el w(k+2)+l el

l=0

This proves (a).

p

(k+l)−i

i=1

l=0

i=1

l=0

l=0

2 −1  k+l NX X



−p

k

2 −1  NX

l=0



w2+l el . 

Theorem 3.4. For every integer n ≥ 0 and for all c ∈ R − {0},   n X i n+1 c (p − 1)Wi + (c − 1)Wi+1 + qWi−1 . c Wn+1 = W0 + i=0

Proof. The proof is similar to that of Theorem 2.2 and hence, it is omitted. 10



The two special cases of Horadam symbol elements are (p, q)−Fibonacci symbol elements Fp,q,k and (p, q)−Lucas symbol elements Lp,q,k , defined by

Fp,q,k =

2 −1 NX

l=0

Fp,q,k+l el and Lp,q,k =

2 −1 NX

l=0

Lp,q,k+lel

respectively, where Fp,q,k and Lp,q,k denote the (p, q)−Fibonacci and (p, q)−Lucas number. The Binet formula for (p, q)−Fibonacci and (p, q)−Lucas symbol elements are respectively Fp,q,k =

ααk − ββ k and Lp,q,k = ααk + ββ k . α−β

Kilic[15] studied some binomial sums involving the (p, q)−Fibonacci numbers. The following theorem provides a similar sum formula for the Horadam symbol elements. Theorem 3.5. −n Wmn = Fp,q,d

n   X n

n−j j (−q)m(n−j) Fp,q,d−m Fp,q,m Wdj .

n   X n

n−j j (−q)m(n−j) Fp,q,d−m Fp,q,m wdj+l ,

j=0

j

Proof. Using the identity wmn+l = the theorem follows directly.

−n Fp,q,d

j=0

j



The following theorem shows that the (p, q)−Fibonacci and (p, q)−Lucas symbol elements satisfy similar identities as that of (p, q)−Fibonacci and (p, q)−Lucas numbers. Theorem 3.6. The (p, q)−Fibonacci and (p, q)−Lucas symbol elements satisfy (a) qFp,q,k−1 + Fp,q,k+1 = Lp,q,k , (b) qLp,q,k−1 + Lp,q,k+1 = (p2 + 4q)Fp,q,k , (c) Fp,q,k+2 − q 2 Fp,q,k−2 = pLp,q,k , (d) Lp,q,k+2 − q 2 Lp,q,k−2 = p(p2 + 4q)Fp,q,k , (e) pFp,q,k + Lp,q,k = 2Fp,q,k+1 , (f) pLp,q,k + (p2 + 4q)Fp,q,k = 2Lp,q,k+1, (g) q 2 Fp,q,k + pLp,q,k+2 = Fp,q,k+4 , (h) q 2 Lp,q,k + p(p2 + 4q)Fp,q,k+2 = Lp,q,k+4, (i) pFp,q,k+2 + qLp,q,k = (p2 + 2q)Fp,q,k+1 , 11

(j) pLp,q,k+2 + q(p2 + 4q)Fp,q,k = (p2 + 2q)Lp,q,k+1 , (k) q 3 Fp,q,k + Fp,q,k+6 = (p2 + q)Lp,q,k+3 , (l) q 3 Lp,q,k + Lp,q,k+6 = (p2 + q)(p2 + 4q)Fp,q,k+3 , (m) q 4 Fp,q,k + Fp,q,k+8 = [(p2 + q)2 + pq(1 + p + q)]Fp,q,k+4 . Proof. We prove (a) only. The proofs of (b)-(m) are similar. Using Theorem 2.3 and the definition of (p, q)−Fibonacci and (p, q)−Lucas symbol elements, we find qFp,q,k−1 + Fp,q,k+1 =q

2 −1 NX

l=0

=

2 −1 NX

l=0

=

2 −1 NX

l=0

Fp,q,(k−1)+l el +

2 −1 NX

l=0

Fp,q,(k+1)+l el

[qFp,q,(k+l)−1 + Fp,q,(k+l)+1 ]el Lp,q,k+l el = Lp,q,k .

This proves (a).



Generating functions are useful in solving linear homogeneous recurrences with constant coefficients. In the following theorem, we establish the generating function for the Horadam symbol elements. Theorem 3.7. For fixed integers k and m with k > m ≥ 0 and n ∈ N, the generating functions of the subsequence Wkn+m of the Horadam symbol elements is given by ∞ X

Wkn+msn =

n=0

Wm − (−q)k Wm−k s . 1 − (αk + β k )s + (−q)k s2

Proof. Using Theorem 2.4 and the definition of Horadam symbol elements, we find 2 −1  ∞  NX ∞ X X n Wkn+ms = w(kn+m)+l el sn n=0

n=0

=

l=0

2 −1  ∞ NX X

=

2 −1  NX

l=0

=

"

wkn+(m+l) s

n=0

l=0

n



el

 wm+l − (−q)k w(m+l)−k s el 1 − (αk + β k )s + (−q)k s2

2 −1  NX

l=0

wm+l el



k

− (−q)

2 −1  NX

l=0

w(m−k)+l el s #

1 − (αk + β k )s + (−q)k s2 12



. 

Corollary 3.8. The generating function for the Horadam symbol elements Wn is ∞ X

Wn s n =

n=0

W0 + (W1 − pW0 )s . 1 − ps − qs2

Exponential generating functions are also used for solving both homogeneous and nonhomogeneous linear recurrences with constant coefficients. In the following theorem, we establish the exponential generating function for the subsequences Wkn+m of the Horadam symbol elements. Theorem 3.9. For m, n ∈ N, the exponential generating functions of the sub sequence Wkn+m of the Horadam symbol elements is given by ∞ X Wkn+m

n!

n=0

ks

n

s =

Aααm eα

k

− Bββ m eβ s . α−β

Proof. Using Theorem 2.4 and the definition of Horadam symbol elements, we get  n ∞  N 2 −1 ∞ X s Wkn+m n X X s = w(kn+m)+l el n! n! n=0

n=0

=

l=0

2 −1  ∞ NX X

n=0

l=0

=

2 −1  NX

wkn+(m+l) el

=

2 −1  NX

l=0

sn n!

ks

Aαm+l eα

l=0

" Aαm



l

 n k s − Bβ m+l eβ s el α−β n! 

αk s

α el e

− Bβ

α−β

m

2 −1  NX

l=0

l



k

β el eβ s #

sn . n! 

Corollary 3.10. The exponential generating function for the Horadam symbol elements Wn is ∞ X Wn n Aαeαs − Bβeβs s = . n! α−β n=0

Catalan’s identity, Cassini formula and d’Ocagne’s identity are usually obtained for the recurrent sequences where the multiplication is commutative. Since the elements of symbol algebra is associated with non-commutative multiplication, we will obtain two such identities in each case. Theorem 3.11. (Catlan Identity)Let n, r ∈ Z, then (a) AB(−q)n−r (αr − β r )(β r αβ − αr βα) Wn−r Wn+r − Wn2 = , ∆ (b) AB(−q)n−r (αr − β r )(β r βα − αr αβ) . Wn+r Wn−r − Wn2 = ∆ 13

Proof. Using the Binet formula of Horadam symbol elements and the fact αβ = −q, we get Wn−r Wn+r − Wn2

! !   Aααn−r − Bββ n−r Aααn − Bββ n 2 Aααn+r − Bββ n+r = − α−β α−β α−β     r r ABαβ(−q)n 1 − αβ r + BAβα(−q)n 1 − αβ r = (α − β)2 n−r r AB(−q) (α − β r )(β r αβ − αr βα) . = ∆ This proves (a). In a similar way, (b) can be proved.



Corollary 3.12. (Cassini Identity)Let n ∈ Z, then (a) Wn−1 Wn+1 − Wn2 = (b) Wn+1 Wn−1 − Wn2 =

AB(−q)n−1 (βαβ − αβα) , α−β AB(−q)n−1 (ββα − αβ) . α−β

Theorem 3.13. (d’Ocagne’s identity). Let m, n ∈ N with n ≥ m, then (a) (−q)m AB(αn−m αβ − β n−m βα) , Wn Wm+1 − Wn+1 Wm = α−β (b) (−q)m AB(αn−m βα − β n−m αβ) Wm+1 Wn − Wm Wn+1 = . α−β

Proof. Use of the Binet formula of Horadam symbol elements, yields

Wn Wm+1 − Wn+1 Wm = −



! Aααm+1 − Bββ m+1 α−β !  Aααm − Bββ m Aααn+1 − Bββ n+1 α−β α−β Aααn − Bββ n α−β



ABβααm β n (β − α) + ABαβαn β m (α − β) (α − β)2
(−q)m AB αn−m αβ − β n−m βα . = α−β

=



Bolat and K¨ ose[2] investigated certain properties of the k−Fibonacci sequence. The following theorem provide similar identities for the Horadam symbol elements. 14

Theorem 3.14. For arbitrary natural numbers m and n, we get Wm Wn+1 + qWm−1 Wn =

A2 α2 αm+n − B 2 β 2 β m+n . α−β

Proof. Using the Binet formula for Horadam octonions, we have    Aααm − Bββ m Aααn+1 − Bββ n+1 Wm Wn+1 + qWm−1 Wn = α−β α−β    m−1 m−1 − Bββ Aααn − Bββ n Aαα +q α−β α−β =

A2 α2 αm+n (α + αq ) − B 2 β 2 β m+n (−β − βq )

(α − β)2
AB αβαm−1 β n (αβ + q) + βααn β m−1 (αβ + q) − (α − β)2 A2 α2 αm+n − B 2 β 2 β m+n = . α−β



Observe that, A2 α2 αm+n − B 2 β 2 β m+n . α−β In the following theorem, we establish two more such identities. Wn+1 Wm + qWn Wm−1 =

Theorem 3.15. For any natural numbers a, b, c, k and a > k, b > k, c > k, we have (a)

(b)

A2 α2 αa+b−k − B 2 β 2 β a+b−k Wa Wb − (−q)k Wa−k Wb−k = , Fp,q,k α−β Wa Wb Wc − Lp,q,k (−q)k Wa−k Wb−k Wc−k + (−q)3k Wa−2k Wb−2k Wc−2k 2 Lp,q,k Fp,q,k =

A3 α3 αa+b+c−3k − B 3 β 3 β a+b+c−3k . α−β

where Fp,q,k and Lp,q,k denote the kth (p, q)−Fibonacci and (p, q)−Lucas numbers respectively. Proof. The proof of (a) and (b) are similar to the proof of Theorem 3.14 followed by the use of the identities α2k − Lp,q,k αk + (−q)k = β 2k − Lp,q,k β k + (−q)k = 0 and (−q)k (−q)k 2 α3k − Lp,q,k (−q)k + 3k = β 3k − Lp,q,k (−q)k + 3k = Lp,q,k Fp,q,k . α β  In the following theorem, we provide some sum and weighted sum formulas for a subsequence of the Horadam symbol elements. 15

Theorem 3.16. For any natural numbers m and k with k > m ≥ 0, we have (a)

n X

Wmr+k =

r=0

(−q)m Wmn+k − Wmn+m+k − (−q)m Wk−m + Wk , 1 + (−q)m − (αm + β m )

(b)

n X

(−1)r Wmr+k

r=0

=

(−1)n+1 q m Wmn+k − (−1)n+1 Wmn+m+k − q m Wk−m + Wk , 1 + (−q)m − (−1)m (αm + β m )

(c)

n X

rWmr+k

r=0

=n



(1 + (−q)m )Wmn+m+k − Wmn+2m+k − (−q)m Wmn+k 1 + (−q)m − (αm + β m )  Wmn+m+k + q 2m Wmn+k−m − 2(−q)m Wmn+k − , (1 + (−q)m − (αm + β m ))2

(d)

n X

(−1)r−1 rWmr+k

r=0

= (−1)n+1



(n − 1 − 2n(−q)m )Wmn+m+k + q 2m (n − 1)Wmn+k−m (1 + (−q)m − (αm + β m ))2  nWmn+2m+k + (nq 2m − 2(−q)m (n − 1))Wmn+k . − (1 + (−q)m − (αm + β m ))2 16

Proof. Since all the proofs are similar, we prove (a) only. Using the Theorem 2.5, Theorem 2.6 and the definition of Horadam symbol elements, we get n X

Wmr+k

r=0

=

2

n  NX −1 X r=0

=

2 −1  n NX X

r=0

l=0

=

2 −1  NX

l=0

=

w(mr+k)+l el

l=0



 wmr+(k+l) el

 (−q)m wmn+(k+l) − wmn+m+(k+l) − (−q)m w(k+l)−m + w(k+l) el 1 + (−q)m − (αm + β m )

" (−q)m

2 −1  NX

w(mn+k)+l el

l=0



−

2 −1  NX

w(mn+m+k)+l el

l=0



1 + (−q)m − (αm + β m ) (−q)m −

2 −1  NX

w(k−m)+l el

l=0



−

2 −1  NX

l=0

1 + (−q)m − (αm + β m )



wk+l el #

. 

The following corollary is an easy consequence of the above theorem. Corollary 3.17. For the Horadam symbol elements Wn , (a) n X r=0

(b) n X n X r=0

(d)

(1 − p)W0 + W1 + (−1)(qWn + Wn+1 ) , 1 − (p + q)

(−1)r Wr =

r=0

(c)

Wr =

(1 + p)W0 − W1 + (−1)n+1 (qWn − Wn+1 ) , 1 + (p − q)

 (Wn+1 − Wn+2 ) + q(Wn − Wn+1 ) q 2 Wn−1 + 2qWn + Wn+1 rWr = n − , 1 − (p + q) (1 − (p2 + q))2 

 n 2 X r−1 n+1 (n − 1 + 2nq)Wn+1 + q (n − 1)Wn−1 (−1) rWr =(−1) (1 − (p + q))2 r=0

 nWn+2 + (nq 2 + 2(q(n − 1)))Wn . − (1 − (p2 + q))2 17

In the following theorem we provide some more identities involving Horadam symbol elements. Theorem 3.18. Let m be a non-negative integer. Then, ( m m   X Wk+m ∆ 2 , if m even; m m−n (a) W2n+k q = k+m + Bββ k+m )∆ m−1 n 2 , if m odd. (Aαα n=0 (b)

m X

   m m p Wk+m , if m even; m−n (−1) W2n+k q = n −pm Wk+m, if m odd. n

n=0

(c)

m   X m

n=0

(d)

n

m   X m

n=0

n

Wn Wn+k q

m−n

=

(

m−2

(A2 α2 αm+k + B 2 β 2 β m+k )∆ 2 , if m even; m−2 (A2 α2 αm+k − B 2 β 2 β m+k )∆ 2 , if m odd.

pn Wn q m−n = W2m .

Proof. We prove (a) only. Rest of the proofs are similar. Using Binet formula of Wk , we find m   X m W2n+k q m−n n n=0 ! m   X Aαα2n+k − Bββ 2n+k m q m−n = n α − β n=0 m   m  

n Bββ k X Aααk X m m 2 n m−n = α q − β 2 q m−n n n α−β α−β =

n=0 k Aαα (α2 +

n=0

q)m

Bββ k (β 2

− α−β

+

q)m

√ √ Aααk (α ∆)m − Bββ k (−β ∆)m = α−β   k+m Aαα + (−1)m+1 Bββ k+m m = ∆2. α−β



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[3] C. Flaut and D. Savin, Some properties of symbol algebras of degree three, Mathematical Reports 16 (2014),(66),3, 443–463. [4] C. Flaut and V. Shpakivskyi, On generalized Fibonacci quaternions and Fibonacci-Narayana quaternions, Adv. Appl. Clifford Algebras 23 (2013),673–688. doi: 10.1186/s13662-015-0627-z [5] C. Flaut, D. Savin and G. Iorgulescu, Some properties of Fibonacci and Lucas symbol elements, Journal of Mathematical Sciences: Advances and Applications 20 (2013), 37-43. [6] C. Flaut and D. Savin, Some remarks regarding (a, b, x0 , x1 )numbers and (a, b, x0 , x1 )- quaternions https://arxiv.org/pdf/1705.00361.pdf [7] C. Flaut and D. Savin, Some special number sequences obtained from a difference equation of degree three, Chaos, Solitons and Fractals 106 (2018), 67-71. [8] S. Halici, On Fibonacci quaternions, Adv. Appl. Clifford Algebr. 22 (2012),(2), 321–327. [9] S. Halici, On Complex Fibonacci Quaternions, Adv. Appl. Clifford Algebr. 23 (2013), 105–112. [10] S. Halici and A. Karata¸s, On a generalization for Fibonacci quaternions, Chaos, Solitons and Fractals 98 (2017), 178-182. [11] M. Hazewinkel, Handbook of Algebra, Vol. 2, Amsterdam, North Holland, 39 (2000). [12] A. F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly 70 (1963), 289–291. ˙ On (p, q)-Fibonacci quaternions and their Binet formulas, generating functions and certain bino[13] A. Ipek, mial sums, Adv. Appl. Clifford Algebr. (2013). doi: 10.1007/s00006-016-0704-8. [14] A. Karata¸s and S. Halici, Horadam Octonions, An. S ¸ t. Univ. Ovidius Constant¸a, Mat. Ser., 25 (3) (2017), 97-106. [15] E. Kilic, On binomial sums for the general second order linear recurrence, Integers 10 (2010), 801–806. [16] H. H. K¨ osal and M. Tosun, Some Equivalence Relations and Results over the Commutative Quaternions and their matrices, An. S ¸ t. Univ. Ovidius Constant¸a, Mat. Ser., 25 (3) (2017), 125142. [17] D. Savin, Some properties of Fibonacci numbers, Fibonacci octonions and generalized Fibonacci-Lucas octonions, Advances in Difference Equations (2015), doi: 10.1186/s13662-015-0627-z [18] D. Savin, About Special Elements in Quaternion Algebras Over Finite Fields, Advances in Applied Clifford Algebras, June 2017, Vol. 27, Issue 2, p. 1801-1813. [19] D. Savin, Special numbers, special quaternions and special symbol elements https://arxiv.org/pdf/1712.01941.pdf [20] N. Yilmaz, Y. Yazlik, N, Taskara, On the Bi-Periodic Lucas Octonions, Advances in Applied Clifford Algebras, June 2017, Vol. 27, Issue 2, p. 1927-1937. National Institute of Technology, Rourkela, India E-mail address: [email protected] Ovidius University,Bd. Mamaia 124, 900527,, Constanta, Romania E-mail address: [email protected]; [email protected] National Institute of Technology, Rourkela, India E-mail address: gkpanda [email protected]

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