In this paper we introduce and study polynomials, which are a generalization of Pell hybrid numbers and so called Pell hybrinomials

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Vol. 20 (2019), No. 2, pp. 1051–1062 DOI: 10.18514/MMN.2019.2971

ON PELL HYBRINOMIALS

MIROSŁAW LIANA, ANETTA SZYNAL-LIANA, AND IWONA WŁOCH Received 21 May, 2019

Abstract. Hybrid numbers generalize complex, hyperbolic and dual numbers, simultaneously.

Special kinds of hybrid numbers, related to numbers of Fibonacci type, among others Pell numbers, were introduced quite recently. In this paper we introduce and study polynomials, which are a generalization of Pell hybrid numbers and so called Pell hybrinomials.

2010Mathematics Subject Classification: 11B37; 11B39; 97F50

Keywords: Pell numbers, complex numbers, hyperbolic numbers, dual numbers, polynomials

1. INTRODUCTION

Pell numbers are well-known numbers in the number theory and they belong to the wide class of numbers of the Fibonacci type. Thenth Pell numberPnis defined recursively by the second order linear recurrence relationPnD2Pn 1CPn 2;for n2 with initial conditions P0D0; P1D1:A special version of Pell numbers is Pell-Lucas numbers Qn (also named as companion Pell numbers). Then QnD 2Qn 1CQn 2;forn2withQ0DQ1D2:

Distinct properties of Pell and Pell-Lucas numbers can be found for example in [1,2,5]. In [3] Horadam and Mahon introduced Pell and Pell-Lucas polynomials as follows.

For any variable quantity x, the Pell polynomial Pn.x/ is defined as Pn.x/D 2xPn 1.x/CPn 2.x/forn2withP0.x/D0; P1.x/D1:

The Pell-Lucas polynomialQn.x/is defined asQn.x/D2xQn 1.x/CQn 2.x/

forn2with initial termsQ0.x/D2; Q1.x/D2x:

ForxD1we obtain Pell and Pell-Lucas numbers, respectively.

For anyxlet˛.x/DxCp

x²C1andˇ.x/Dx p

x²C1:Then solving second–

order linear recurrence relations, forPn.x/andQn.x/, respectively, we have Pn.x/D˛ⁿ.x/ ˇⁿ.x/

˛.x/ ˇ.x/ (1.1)

and

Qn.x/D˛ⁿ.x/Cˇⁿ.x/: (1.2)

c 2019 Miskolc University Press

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One of the generalizations of the Pell polynomial is the Horadam polynomial, whose properties can be found in [4].

Hybrid numbers were introduced by ¨Ozdemir in [6] as a new generalization of complex, hyperbolic and dual numbers.

LetKbe the set of hybrid numbersZof the form ZDaCbiCc"Cdh;

wherea; b; c; d 2Randi; ";hare operators such that

i²D 1; "²D0; h²D1 (1.3) and

ihD hiD"Ci: (1.4)

IfZ1Da1Cb1iCc1"Cd1h;andZ2Da2Cb2iCc2"Cd2h;are any two hybrid numbers then equality, addition, substraction and multiplication by scalar are defined.

Equality:Z1DZ2only ifa1Da2; b1Db2; c1Dc2; d1Dd2;

addition:Z1CZ2D.a1Ca2/C.b1Cb2/iC.c1Cc2/"C.d1Cd2/h;

substraction:Z1 Z2D.a1 a2/C.b1 b2/iC.c1 c2/"C.d1 d2/h;

multiplication by scalars2R:sZ1Dsa1Csb1iCsc1"Csd1h:

The hybrid numbers multiplication is defined using (1.3) and (1.4). Note that using formulas (1.3) and (1.4) we can find the product of any two hybrid units. The following Table presents products of i; ";and hUsing rules given in Table 1. the

TABLE1. The hybrid number multiplication.

i " h

i 1 1 h "Ci

" hC1 0 "

h " i " 1

multiplication of hybrid numbers can be made analogously as multiplications of al- gebraic expressions. For hybrid numbers details, see [6].

A special kind of hybrid numbers, namely Pell hybrid numbers and Pell-Lucas hybrid numbers, were introduced in [7] as follows.

Thenth Pell hybrid numberPHnand thenth Pell-Lucas hybrid numberQHnare defined as

PHnDPnCiPnC1C"PnC2ChPnC3; (1.5) QHnDQnCiQnC1C"QnC2ChQnC3; (1.6) respectively.

Interesting results of Pell and Pell-Lucas hybrid numbers obtained recently can be found in [8].

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In this paper we introduce Pell and Pell-Lucas hybrinomials, i.e. polynomials, which are a generalization of Pell hybrid numbers and Pell-Lucas hybrid numbers, respectively.

Forn0Pell and Pell-Lucas hybrinomials are defined by

PHn.x/DPn.x/CiPnC1.x/C"PnC2.x/ChPnC3.x/ (1.7) and

QHn.x/DQn.x/CiQnC1.x/C"QnC2.x/ChQnC3.x/; (1.8) wherePn.x/is thenth Pell polynomial,Qn.x/is the then-th Pell-Lucas polynomial andi; ";hare hybrid units satisfy (1.3) and (1.4).

ForxD1we obtain Pell hybrid numbers and Pell-Lucas hybrid numbers, respectively.

2. PROPERTIES OF PELL ANDPELL-LUCAS HYBRINOMIALS

Theorem 1. Letn0be an integer. For any variable quantityx, we have PHn.x/D2xPHn 1.x/CPHn 2.x/forn2 (2.1) withPH0.x/DiC".2x/Ch.4x²C1/

andPH1.x/D1Ci.2x/C".4x²C1/Ch.8x³C4x/:

Proof. IfnD2we have

PH2.x/D2xPH1.x/CPH0.x/

D2x.1Ci.2x/C".4x²C1/Ch.8x³C4x//

CiC".2x/Ch.4x²C1/

D2xCi.4x²C1/C".8x³C4x/Ch.16x⁴C12x²C1/

DP2.x/CiP3.x/C"P4.x/ChP5.x/:

Ifn3then using the definition of Pell polynomials we have PH_n.x/DP_n.x/CiP_n_C₁.x/C"P_n_C₂.x/ChP_n_C₃.x/

D.2xPn 1.x/CPn 2.x//Ci.2xPn.x/CPn 1.x//

C".2xPnC1.x/CPn.x//Ch.2xPnC2.x/CPnC1.x//

D2x .P_{n 1}.x/CiP_n.x/C"P_n_C₁.x/ChP_n_C₂.x//

CPn 2.x/CiPn 1.x/C"Pn.x/ChPnC1.x/

D2xPHn 1.x/CPHn 2.x/;

which ends the proof.

In the same way one can easily prove the next theorem.

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Theorem 2. Letn0be an integer. For any variable quantityx, we have QHn.x/D2xQHn 1.x/CQHn 2.x/forn2 (2.2) withQH0.x/D2Ci.2x/C".4x²C2/Ch.8x³C6x/andQH1.x/D2xCi .4x²C2/C".8x³C6x/Ch.16x⁴C16x²C2/:

Now we give so called Binet formulas for Pell and Pell-Lucas hybrinomials.

Theorem 3. Letn0be an integer. Then PHn.x/D ˛ⁿ.x/

˛.x/ ˇ.x/ 1Ci˛.x/C"˛².x/Ch˛³.x/

ˇⁿ.x/

˛.x/ ˇ.x/ 1Ciˇ.x/C"ˇ².x/Chˇ³.x/

;

(2.3)

where˛.x/DxCp

x²C1andˇ.x/Dx p x²C1:

Proof. Using (1.1), (1.5) and (1.7) we have

PHn.x/DPn.x/CiPnC1.x/C"PnC2.x/ChPnC3.x/

D˛ⁿ.x/ ˇⁿ.x/

˛.x/ ˇ.x/ Ci˛ⁿ^C¹.x/ ˇⁿ^C¹.x/

˛.x/ ˇ.x/

C"˛ⁿ^C².x/ ˇⁿ^C².x/

˛.x/ ˇ.x/ Ch˛ⁿ^C³.x/ ˇⁿ^C³.x/

˛.x/ ˇ.x/

and after calculations the result follows.

In the same way, using (1.2), (1.6) and (1.8), one can easily prove the next theorem.

Theorem 4. Letn0be an integer. Then

QHn.x/D˛ⁿ.x/ 1Ci˛.x/C"˛².x/Ch˛³.x/

Cˇⁿ.x/ 1Ciˇ.x/C"ˇ².x/Chˇ³.x/

; (2.4)

where˛.x/DxCp

x²C1andˇ.x/Dx p x²C1:

Now we will give some identities related to the well-known identities for classical Pell numbers

.Catalan identity/Pn rPnCr .Pn/²D. 1/^{n r}^C¹P_r²; .Cassini identity/Pn 1PnC1 .Pn/²D. 1/ⁿ;

.d⁰Ocagne identity/PmPnC1 PmC1PnD. 1/ⁿPm n:

We give their versions for Pell and Pell-Lucas hybrinomials. These identities can be proved using Binet formulas.

For simplicity of notation let .x/D˛.x/ ˇ.x/;

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O

˛.x/D1Ci˛.x/C"˛².x/Ch˛³.x/;

ˇ.x/O D1Ciˇ.x/C"ˇ².x/Chˇ³.x/:

Then we can write (2.3) and (2.4) as PHn.x/D˛ⁿ.x/

.x/˛.x/O ˇⁿ.x/

.x/ˇ.x/O and

QHn.x/D˛ⁿ.x/˛.x/O Cˇⁿ.x/ˇ.x/;O respectively.

Moreover,˛.x/ˇ.x/D 1and².x/D4x²C4:

Theorem 5(Catalan identity for Pell hybrinomials). Letn0; r0be integers such thatnr:Then

PHn r.x/PHnCr.x/ .PHn.x//² D . 1/ⁿ

4x²C4˛.x/O ˇ.x/O

1 ˇ^r.x/

˛^r.x/

C . 1/ⁿ

4x²C4ˇ.x/O ˛.x/O

1 ˛^r.x/

ˇ^r.x/

: Proof. For integersn0; r0andnrwe have

PHn r.x/PHnCr.x/ .PHn.x//² D

˛^{n r}.x/

.x/ ˛.x/O ˇ^{n r}.x/

.x/ ˇ.x/O

˛ⁿ^C^r.x/

.x/ ˛.x/O ˇⁿ^C^r.x/

.x/ ˇ.x/O ˛ⁿ.x/

.x/˛.x/O ˇⁿ.x/

.x/ˇ.x/O

˛ⁿ.x/

.x/˛.x/O ˇⁿ.x/

.x/ˇ.x/O

D ˛^{n r}.x/

.x/ ˛.x/O ˇⁿ^C^r.x/

.x/ ˇ.x/O ˇ^{n r}.x/

.x/ ˇ.x/O ˛ⁿ^C^r.x/

.x/ ˛.x/O C˛ⁿ.x/

.x/˛.x/O ˇⁿ.x/

.x/ˇ.x/O Cˇⁿ.x/

.x/ˇ.x/O ˛ⁿ.x/

.x/˛.x/O D ˛^{n r}.x/ˇⁿ^C^r.x/

².x/ ˛.x/O ˇ.x/O ˇ^{n r}.x/˛ⁿ^C^r.x/

².x/ ˇ.x/O ˛.x/O C˛ⁿ.x/ˇⁿ.x/

².x/ ˛.x/O ˇ.x/O Cˇⁿ.x/˛ⁿ.x/

².x/ ˇ.x/O ˛.x/O D˛ⁿ.x/ˇⁿ.x/

².x/ ˛.x/O ˇ.x/O

1 ˇ^r.x/

˛^r.x/

C˛ⁿ.x/ˇⁿ.x/

².x/ ˇ.x/O ˛.x/O

1 ˛^r.x/

ˇ^r.x/

D . 1/ⁿ

4x²C4˛.x/O ˇ.x/O

1 ˇ^r.x/

˛^r.x/

C . 1/ⁿ

4x²C4ˇ.x/O ˛.x/O

1 ˛^r.x/

ˇ^r.x/

;

In the same way one can easily prove the next theorem, which gives Catalan identity for Pell-Lucas hybrinomials.

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Theorem 6(Catalan identity for Pell-Lucas hybrinomials). Letn0; r0be integers such thatnr:Then

QHn r.x/QHnCr.x/ .QHn.x//² D. 1/ⁿ˛.x/O ˇ.x/O

ˇ^r.x/

˛^r.x/ 1

C. 1/ⁿˇ.x/O ˛.x/O

˛^r.x/

ˇ^r.x/ 1

:

Note that forrD1we get Cassini identities for Pell and Pell-Lucas hybrinomials.

Moreover, forrD1we have

1 ^{ˇ .x/}_˛.x/D ^{˛.x/ ˇ .x/}_˛.x/ D^.x/_˛.x/ and1 _{ˇ .x/}^˛.x/D^{ˇ .x/ ˛.x/}_{ˇ .x/} D ^.x/_{ˇ .x/}.

Corollary 1(Cassini identities for Pell and Pell-Lucas hybrinomials). Letn0 be an integer. Then

PHn 1.x/PHnC1.x/ .PHn.x//² D. 1/^{n 1}ˇ.x/

.x/ ˛.x/O ˇ.x/O . 1/^{n 1}˛.x/

.x/ ˇ.x/O ˛.x/:O QHn 1.x/QHnC1.x/ .QHn.x//²

D. 1/ⁿ˛.x/O ˇ.x/O ˇ.x/

˛.x/ 1

C. 1/ⁿˇ.x/O ˛.x/O ˛.x/

ˇ.x/ 1

:

Theorem 7 (d’Ocagne identity for Pell hybrinomials). Letm0; n0 be integers such thatmn. Then

PHm.x/PHnC1.x/ PHmC1.x/PHn.x/

D. 1/ⁿ˛^{m n}.x/

.x/ ˛.x/O ˇ.x/O . 1/ⁿˇ^{m n}.x/

.x/ ˇ.x/O ˛.x/:O Proof. Letm; nbe as in the statement of the Theorem. Then

PHm.x/PHnC1.x/ PHmC1.x/PHn.x/

D

˛^m.x/

.x/ ˛.x/O ˇ^m.x/

.x/ ˇ.x/O

˛ⁿ^C¹.x/

.x/ ˛.x/O ˇⁿ^C¹.x/

.x/ ˇ.x/O ˛^m^C¹.x/

.x/ ˛.x/O ˇ^m^C¹.x/

.x/ ˇ.x/O

˛ⁿ.x/

.x/˛.x/O ˇⁿ.x/

.x/ˇ.x/O

D ˛^m^Cⁿ^C¹.x/

².x/ ˛O².x/ ˛^m.x/ˇⁿ^C¹.x/

².x/ ˛.x/O ˇ.x/O ˛ⁿ^C¹.x/ˇ^m.x/

².x/ ˇ.x/O ˛.x/O Cˇ^m^Cⁿ^C¹.x/

².x/ ˇO².x/ ˛^m^C¹^Cⁿ.x/

².x/ ˛O².x/C˛^m^C¹.x/ˇⁿ.x/

².x/ ˛.x/O ˇ.x/O C˛ⁿ.x/ˇ^m^C¹.x/

².x/ ˇ.x/O ˛.x/O ˇ^m^C¹^Cⁿ.x/

².x/ ˇO².x/

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D˛^m^C¹.x/ˇⁿ.x/ ˛^m.x/ˇⁿ^C¹.x/

².x/ ˛.x/O ˇ.x/O C˛ⁿ.x/ˇ^m^C¹.x/ ˛ⁿ^C¹.x/ˇ^m.x/

².x/ ˇ.x/O ˛.x/O D˛^m.x/ˇⁿ.x/.˛.x/ ˇ.x//

².x/ ˛.x/O ˇ.x/O C˛ⁿ.x/ˇ^m.x/.ˇ.x/ ˛.x//

².x/ ˇ.x/O ˛.x/O D˛^m.x/ˇⁿ.x/

.x/ ˛.x/O ˇ.x/O ˛ⁿ.x/ˇ^m.x/

.x/ ˇ.x/O ˛.x/O D. 1/ⁿ˛^{m n}.x/

.x/ ˛.x/O ˇ.x/O . 1/ⁿˇ^{m n}.x/

.x/ ˇ.x/O ˛.x/:O

Thus the Theorem is proved.

In the same way we can prove next theorems.

Theorem 8(d’Ocagne identity for Pell-Lucas hybrinomials). Letm0; n0be integers such thatmn. Then

QHm.x/QHnC1.x/ QHmC1.x/QHn.x/

D. 1/ⁿˇ^{m n}.x/.x/ˇ.x/O ˛.x/O . 1/ⁿ˛^{m n}.x/.x/˛.x/O ˇ.x/:O Theorem 9. Letm0; n0be integers. Then

PHm.x/QHn.x/ QHm.x/PHn.x/

D2. 1/ⁿ˛^{m n}.x/

.x/ ˛.x/O ˇ.x/O 2. 1/ⁿˇ^{m n}.x/

.x/ ˇ.x/O ˛.x/:O

Some identities for Pell and Pell-Lucas hybrinomials can be found by analogy to well–known identities for the Pell and Pell-Lucas polynomials. In the next part of this paper we indicate such identities.

Theorem 10([3]). Letn1be an integer. Then

PnC1.x/CPn 1.x/DQn.x/D2xPn.x/C2Pn 1.x/: (2.5) Theorem 11. Letn1be an integer. Then

PHnC1.x/CPHn 1.x/DQHn.x/D2xPHn.x/C2PHn 1.x/:

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Proof. Using (2.5) we have

PHnC1.x/CPHn 1.x/

DPnC1.x/CiPnC2.x/C"PnC3.x/ChPnC4.x/

CPn 1.x/CiPn.x/C"PnC1.x/ChPnC2.x/

D.PnC1.x/CPn 1.x//Ci.PnC2.x/CPn.x//

C".PnC3.x/CPnC1.x//Ch.PnC4.x/CPnC2.x//

DQn.x/CiQnC1.x/C"QnC2.x/ChQnC3.x/

DQHn.x/:

On the other hand

2xPHn.x/C2PHn 1.x/

D2x.Pn.x/CiPnC1.x/C"PnC2.x/ChPnC3.x//

C2 .Pn 1.x/CiPn.x/C"PnC1.x/ChPnC2.x//

D.2xPn.x/CPn 1.x//Ci.2xPnC1.x/CPn.x//

C".2xPnC2.x/CPnC1.x//Ch.2xPnC3.x/CPnC2.x//

DQn.x/CiQnC1.x/C"QnC2.x/ChQnC3.x/

DQHn.x/;

so the result follows.

QnC1.x/CQn 1.x/D4.x²C1/Pn.x/: (2.6) Theorem 13. Letn1be an integer. Then

QHnC1.x/CQHn 1.x/D4.x²C1/PHn.x/:

Proof. Using (2.6) and proceeding in the same way as in the Theorem11the result

follows.

n 1

X

lD1

Pl.x/DPn.x/CPn 1.x/ 1

2x : (2.7)

n 1

X

lD1

PHl.x/DPHn.x/CPHn 1.x/ PH0.x/ PH1.x/

2x :

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Proof. For an integern2we have

n 1

X

lD1

PHl.x/DPH1.x/CPH2.x/C: : :CPHn 1.x/

DP1.x/CiP2.x/C"P3.x/ChP4.x/

CP2.x/CiP3.x/C"P4.x/ChP5.x/

C

DP1.x/CP2.x/C CPn 1.x/

Ci.P2.x/CP3.x/C CPn.x/CP1.x/ P1.x//

C".P3.x/CP4.x/C CPnC1.x/CP1.x/CP2.x/

P1.x/ P2.x//

Ch.P4.x/CP5.x/C CPnC2.x/CP1.x/CP2.x/CP3.x/

P1.x/ P2.x/ P3.x//:

Using (2.7) we obtain

n 1

X

lD1

PH_l.x/DPn.x/CPn 1.x/ 1 2x

Ci

PnC1.x/CPn.x/ 1

2x P1.x/

C"

PnC2.x/CPnC1.x/ 1

2x P₁.x/ P₂.x/

Ch

PnC3.x/CPnC2.x/ 1

2x P1.x/ P2.x/ P3.x/

: Bringing to the common denominator we have

n 1

X

lD1

PH_l.x/DP_n.x/CP_{n 1}.x/ 1 2x

Ci

PnC1.x/CPn.x/ 1 2x 2x

C"

PnC2.x/CPnC1.x/ 1 2x 4x² 2x

Ch

PnC3.x/CPnC2.x/ 1 2x 4x² 2x.4x²C1/

2x

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and finally

n 1

X

lD1

PH_l.x/D Pn.x/CiPnC1.x/C"PnC2.x/ChPnC3.x/

2x

C .0C1/ i.1C2x/ ".2xC.4x²C1// h..4x²C1/C.8x³C4x//

2x DPHn.x/CPHn 1.x/ PH0.x/ PH1.x/

2x :

Thus the Theorem is proved.

n 1

X

lD1

Q_l.x/DQn.x/CQn 1.x/ 2 2x

2x : (2.8)

n 1

X

lD1

QHl.x/DQHn.x/CQHn 1.x/ QH0.x/ QH1.x/

2x :

Proof. Using (2.8) and proceeding in the same way as in the Theorem15the result

follows.

Next we shall give the generating function for Pell hybrinomials.

Theorem 18. The generating function for Pell hybrinomial sequence fPHn.x/gis

G.t /DiC".2x/Ch.4x²C1/C.1C"Ch.2x//t

1 2xt t² :

Proof. Assume that the generating function of the Pell hybrinomial sequencefPHn.x/g has the formG.t /D P¹

nD0

PHn.x/tⁿ. Then

G.t /DPH0.x/CPH1.x/tCPH2.x/t²C: : :

Multiply the above equality on both sides by 2xt and then by t²we obtain G.t /.2x/t D PH0.x/.2x/t PH1.x/.2x/t² PH2.x/.2x/t³ : : :

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G.t /t²D PH0.x/t² PH1.x/t³ PH2.x/t⁴ : : : By adding these three equalities above, we will get

G.t /.1 2xt t²/DPH0.x/C.PH1.x/ PH0.x/.2x//t

sincePHn.x/D2xPHn 1.x/CPHn 2.x/(see (2.1)) and the coefficient oftⁿ, for n2, are equal to zero. Moreover,PH0.x/DiC".2x/Ch.4x²C1/; PH1.x/

PH0.x/.2x/D1C"Ch.2x/:

In the same way we obtain the generating functiong.t /for Pell-Lucas hybrinomials.

Theorem 19. The generating function for the Pell-Lucas hybrinomial sequence fQHn.x/gis

g.t /D QH0.x/C.QH1.x/ QH0.x/.2x//t

1 2xt t² ;

whereQH0.x/D2Ci.2x/C".4x²C2/Ch.8x³C6x/;

andQH1.x/ QH0.x/.2x/D 2xC2iC".2x/Ch.4x²C2/:

We will give the matrix representation of Pell hybrinomials.

Theorem 20. Letn0be an integer. Then PHnC2.x/ PHnC1.x/

PH_n_C₁.x/ PH_n.x/

D

PH2.x/ PH1.x/

PH₁.x/ PH₀.x/

2x 1

1 0

n

: Proof. (by induction onn)

IfnD0then assuming that the matrix to the power 0 is the identity matrix the result is obvious. Now suppose that for anyn0holds

PHnC2.x/ PHnC1.x/

PHnC1.x/ PHn.x/

D

PH2.x/ PH1.x/

PH1.x/ PH0.x/

2x 1

1 0

n

: We shall show that

PHnC3.x/ PHnC2.x/

PHnC2.x/ PHnC1.x/

D

PH2.x/ PH1.x/

PH1.x/ PH0.x/

2x 1

1 0

nC1

: By simple calculation using induction’s hypothesis we have

PH2.x/ PH1.x/

PH1.x/ PH0.x/

2x 1

1 0

n

2x 1

1 0

D

PHnC2.x/ PHnC1.x/

PHnC1.x/ PHn.x/

2x 1

1 0

D

2xPHnC2.x/CPHnC1.x/ PHnC2.x/

2xPHnC1.x/CPHn.x/ PHnC1.x/

D

PHnC3.x/ PHnC2.x/

PHnC2.x/ PHnC1.x/

;

(12)

In the same way we obtain the matrix representation for Pell-Lucas hybrinomials.

Theorem 21. Letn0be an integer. Then QHnC2.x/ QHnC1.x/

QHnC1.x/ QHn.x/

D

QH2.x/ QH1.x/

QH1.x/ QH0.x/

2x 1

1 0

n

: COMPLIANCE WITH ETHICAL STANDARDS

Conflict of Interest: The authors declare that they have no conflict of interest.

REFERENCES

[1] A. F. Horadam, “Pell identities.”Fibonacci Quart., vol. 9, no. 3, pp. 245–263, 1971.

[2] A. F. Horadam, “Minmax Sequences for Pell Numbers.”Applications of Fibonacci Numbers, vol. 6, pp. 231–249, 1996.

[3] A. F. Horadam and B. J. M. Mahon, “Pell and Pell-Lucas Polynomials.”Fibonacci Quart., vol. 23, no. 1, pp. 7–20, 1985.

[4] T. Horzum and E. G. Kocer, “On Some Properties of Horadam Polynomials.”Int. Math. Forum, vol. 25, no. 4, pp. 1243–1252, 2009.

[5] T. Koshy,Pell and Pell-Lucas Numbers with Applications. New York: Springer, 2014.

[6] M. ¨Ozdemir, “Introduction to Hybrid Numbers.” Adv. Appl. Clifford Algebr., vol. 28, 2018, doi:

10.1007/s00006-018-0833-3.

[7] A. Szynal-Liana, “The Horadam hybrid numbers.”Discuss. Math. Gen. Algebra Appl., vol. 38, no. 1, pp. 91–98, 2018, doi:10.7151/dmgaa.1287.

[8] A. Szynal-Liana and I. Włoch, “On Pell and Pell-Lucas Hybrid Numbers.”Comment. Math. Prace Mat., vol. 58, no. 1–2, pp. 11–17, 2018, doi:10.14708/cm.v58i1-2.6364.

Authors’ addresses

Mirosław Liana

Rzeszow University of Technology, The Faculty of Management, al. Powstańców Warszawy 10, 35-959 Rzeszów, Poland

E-mail address:mliana@prz.edu.pl

Anetta Szynal-Liana

Rzeszow University of Technology, The Faculty of Mathematics and Applied Physics, al. Powstańców Warszawy 12, 35-959 Rzeszów, Poland

E-mail address:aszynal@prz.edu.pl

Iwona Włoch

Rzeszow University of Technology, The Faculty of Mathematics and Applied Physics, al. Powstańców Warszawy 12, 35-959 Rzeszów, Poland

E-mail address:iwloch@prz.edu.pl