(p,q)-Rogers-Szegő Matrisleri Üzerine

* Corresponding Author DOI: 10.37094/adyujsci.518782

On

(𝒑, 𝒒)-Rogers-Szegő Matrices

Adem ŞAHİN1,_*

1_{Tokat Gaziosmanpaşa University, Faculty of Education Tokat, Türkiye}

adem.sahin@gop.edu.tr, ORCID: 0000-0001-5739-4117

Received: 28.01.2019 Accepted: 29.04.2020 Published: 25.06.2020

Abstract

In the present article, we have discussed the (𝑝, 𝑞)-numbers, the Rogers-Szegő polynomial and the (𝑝, Rogers-Szegő polynomial and have defined the (𝑝, matrices and the (𝑝, 𝑞)-Rogers-Szegő matrices. We have presented some algebraic properties of these matrices and have proved them. In particular, we have obtained the factorization of these matrices, their inverse matrices, as well as the matrix representations of the (𝑝, 𝑞)-numbers, the Rogers-Szegő polynomials and the (𝑝, 𝑞)-Rogers-Szegő polynomials.

Keywords: (𝑝, 𝑞)-analogue; (𝑝, 𝑞)-Rogers-Szegő polynomials; (𝑝, 𝑞)-Rogers-Szegő matrix.

(𝒑, 𝒒)-Rogers-Szegő Matrisleri Üzerine

Öz

Bu çalışmada, (𝑝, 𝑞)-sayılarını, Rogers-Szegő polinomunu ve (𝑝, 𝑞)-Rogers-Szegő polinomunu ele aldık ve (𝑝, 𝑞)-matrislerini ve (𝑝, 𝑞)-Rogers-Szegő matrislerini tanımladık. Bu matrislere ait bazı özellikleri verdik ve bunların ispatlarını yaptık. Özellikle, bu matrislerin

çarpanlara ayrılışını, bunların ters matrislerini ve (𝑝, 𝑞)-sayılarının, Rogers-Szegő polinomlarının ve (𝑝, 𝑞)-Rogers-Szegő polinomlarının matris temsillerini elde ettik.

Anahtar Kelimeler: (𝑝, 𝑞)-analoji; (𝑝, 𝑞)-Rogers-Szegő polinomu; (𝑝, 𝑞)-Rogers-Szegő matrisi.

1. Introduction

The q-binomial is defined as;

(1)

Another way to present the q-binomial is;

where and

The q-oscillator algebra is an important part of the quantum groups [1- 3]. Accordingly, the (𝑝, 𝑞)-oscillator algebra was presented in [4] and studied in [5-6]. The (𝑝, 𝑞)-number

was introduced as a result of the studies on the (𝑝, 𝑞)-oscillator. The (𝑝, 𝑞)-number is defined as;

.

And it is obvious that; .

The (𝑝, 𝑞)-binomial coefficient is defined as;

where and 1 0 ( ; ) , and ( ; ) (1 ). ( ; ) ( ; ) m j m m j s m s q m q q a q aq s q q q q -= -é ù _{= =} -ê ú ë û

Õ

[ ]

[ ] [

]

!

!

!

q q q q

m

m

s

s

m s

é ù

=

ê ú

-ë û

[ ]

q

1

₁

m

q

m

q

-=

-

[ ] [ ] [ ] [ ]

m

q

! 1 2 ...

=

q q

m

q

.

[ ]

m

_{p q,}

[ ]

, m m p q

p

q

m

p q

-=

-[ ]

_,

[ ]

1

lim

_{p q q} p®

m

=

m

[ ]

[ ] [

]

1 , 0 , , ,

!

( , ; , )

, and ( , ; , )

(

)

( , ; , ) ( , ; , )

!

!

m p q s s m m s s m s p q p q p q

m

m

p q p q

a b p q

ap

bq

s

p q p q

p q p q

s

m s

-=

-é ù

₌

₌

₌

-ê ú

-ë û

Õ

[ ]

0

_{p q,}

! 1

=

[ ]

m

_{p q,}

! 1

=

[ ] [ ] [ ]

_{p q,}

2

_{p q,}

...

m

_{p q,}

.

After the presentation of the (𝑝, 𝑞)-number, the (𝑝, 𝑞)-calculus studied in [1, 4, 7-9]. Recently, many researchers have studied the q analogue of number sequences and polynomials, see [10-13]. The Rogers-Szegő polynomials were shown up in the studies of Rogers [14, 15] and were discussed by Szegő [16]. The single variable Rogers-Szegő polynomial is defined as;

.

The Rogers-Szegő polynomials satisfy the recursion in [17, p. 49] given as; .

Jagannathan and Sridhar [18] defined the (𝑝, Rogers-Szegő polynomial using the (𝑝, 𝑞)-number and showed that it is related to the (𝑝, 𝑞)-oscillator. The (𝑝, 𝑞)-Rogers-Szegő polynomial is defined as;

.

This polynomial is a natural generalization of the Rogers-Szegő polynomial as; .

Additionally, some matrices associated with the number sequences and the polynomials were studied. For example, in [19] the authors studied the k-Fibonacci matrix and the symmetric k-Fibonacci matrix, in [20] the authors studied the Pell matrix. Lee et al. [21] gave the factorization of the Fibonacci matrix. In [10] the authors studied the (q;x;s)-Fibonacci and Lucas matrices. Fonseca and Petronilho [22] gave explicit inverses of some tridiagonal matrices. Some of the important works on the number sequences are determinant and permanent representations [23-28].

2. (𝒑, 𝒒)-Matrices and Rogers-Szegő Polynomial

Definition 1. The lower triangular (𝑝, 𝑞)-matrices are defined by

0 ( ) m s m k _q m H x x s = é ù = _{ê ú} ë û

å

1( ) (1 ) ( ) ( 1) 1( ) m m m m H ₊ x = +x H x +x q - H _- x 0 _, ( , , ) m s m s _{p q} m H x p q x s = é ù = _{ê ú} ë û

å

1

lim

_m

( , , )

_m

( )

p®

H x p q

=

H x

n n

´

p q, _{[ ]} n = ars Ν ,

[

1] ,

0

0,

.

p q rs

r s

r s

a

otherwise

- +

- ³

ì

= í

î

Theorem 2. Let are(𝑝, 𝑞)-matrices, then

Proof. It suffices to prove that . It is obvious, for ,

and for , . For we have

which implies that .

Definition 3. The tridiagonal matrices are defined by

(3)

Theorem 4. Let be a lower triangular matrix defined as

. (4)

Then,

Proof. It is obvious by using matrix product. 3. (𝒑, 𝒒)Rogers-Szego Matrices , p q n Ν

[ ]

, 1

1,

(

),

₁

(

)

,

₂

0,

_otherwise.

p q n rs

r s

p q

_{r s}

b

pq

_{r s}

-ì

=

ï- +

ï

- =

=

_{= í}

- =

ï

ïî

Ν

, ₍ , ₎ 1 p q p q n n - =In Ν Ν

r s

<

0

0

n rk ks k

a b

=

=

å

r s

=

0

1

n rk ks rr rr k

a b

a b

=

=

=

å

r s

> >

0

2 , , , 0 , , , [ 1] [ ] ( ) [ 1] [ 1] ([ ] ( ) [ 1] ( )) 0 n s rk ks rk kr p q p q p q k k s p q p q p q a b a b r s r s p q r s pq r s r s p q r s pq + = = = = - + + - - - + -= - + - - + + - - - =

å

å

, ₍ , ₎ 1 p q p q n n - =In Ν Ν

n n

´

q [ ] n = hrs Η 1 1, ₁ 1, ( 1), ₁ 0, . rs r s r x _{r s} h x q _{r s} otherwise -ì - _{- =} ï ₊ ï = = í - _{- =} ï ïî

!

q n

Η

!

1

0

0

0

1

q n _q n

é

ù

ê

ú

ê

ú

=

ê

-

ú

ê

ú

ë

û

Η

Η

!

1

( ),

(

)

0,

otherwise.

q _{i j} n ij

H

x

i

j

h

--

₌

_{é ù}

_{= í}

ì

³

ë û î

Η

Definition 5. The lower triangular (p,q)-Rogers-Szegő matrices are defined by

The sequence is defined by , and

.

Definition 6. The lower Hessenberg matrices are defined by

Lemma 7. [28] Let be an lower Hessenberg matrix for all and define det( ) = 1. Then, and for

Lemma 8. For , .

Proof. . Suppose that the result is true for all . We

prove it for . Actually, by using Lemma 7 we have

Lemma 9. Let then,

n n

´

p q,

_{[ ]}

n

=

r

ij

R

( , , ),

0

0,

.

i j ij

H

x p q

i j

r

otherwise

-

- ³

ì

= í

î

( , , )

n

h x p q

h x p q

₀

( , , ) 1

=

h x p q

₁

( , , )

= +

x

1

1 1 1 1 1

( , , ) (

1)

( , , )

n

( 1)

n s

( , , )

( , , )

n n n s s s

h x p q

x

h

_-

x p q

- -

H

_{- +}

x p q h

_-

x p q

=

= +

+

å

-n -n

´

p q, _{[ ]} n = ars S 1

( , , ),

1

0,

.

r s rs

H

x p q

r s

a

otherwise

- +

³

-ì

= í

î

n

H

n n

´

n³1 0

H

det( )

H

₁

=

a

₁₁ n³2 1 1, 1 1 , , 1 1 1

det(

)

det(

)

[( 1)

det(

)].

s s s k n s s s s k j j k k j k

H

a

H

a

a

H

-+ -+ - + -= =

=

+

å

-

Õ

1 n³ _det( p q, _{) ( , , )} n =h x p qn S , 1 1 det(_Sp q)_{=h x p q}( , , )_{= +x} 1 _m£n 1 m= +n , , 1 , 1 1, 1 1, , 1 1 1 , 1 , 2 1 1 1 , 1 2 1 1 1

det(

)

det(

)

[( 1)

det(

)]

(

1) det(

)

[( 1)

( , , ) det(

)]

(

1) det(

)

[( 1)

( , , )

( , , )]

(

n n p q p q n i p q n n n n n i j j i i j i n p q n i p q n n i i i n p q n i n n i i i n

s

s

s

x

H

x p q

x

H

x p q h

x p q

h

+ -+ + + + + -= = + -+ - -+ -= + -+ - -= +

=

+

-=

+

+

-=

+

+

-=

å

Õ

å

å

S

S

S

S

S

S

, , ).

x p q

1 n³

.

Proof. From the previous definition of we know and

.

Using this equation, we have;

Theorem 10. Let be the (p,q)-Rogers-Szegő matrices, then

Proof. The proof runs like in Theorem 2 using Lemma 9.

Corollary 11. Let be the (p,q)-Rogers-Szegő matrices and be the lower triangular matrices in Eqn. (4). Then, .

Theorem 12. Let be an lower Hessenberg matrix defined as

Then,

.

Proof. The proof runs like in Lemma 8 using Lemma 7.

The following two corollaries are easy consequences of known results in the literature.

1 1

( , , )

n

( 1)

k

( , , )

( , , )

n k n k k

H x p q

+

h x p q H

x p q

-=

=

å

-( , , )

n

h x p q

h x p q

₀

( , , ) 1

=

1 1 1 1 1

( , , ) (

1)

( , , )

n

( 1)

n k

( , , )

( , , )

n n n k k k

h x p q

x

h

_-

x p q

- -

H

_{- +}

x p q h

_-

x p q

=

= +

+

å

-1 2 2 1 1 0 1 2 2 1 1

( , , ) (

1)

( , , )

( , , )

( , , ) ...

( , , ) ( , , )

( , , ) ( , , ) 0

( , , )

( , , ) (

1)

( , , )

( , , )

( , , ) ...

( , , ) ( , , ).

n n n n n n n n n n

h x p q

x

h

x p q

H x p q h

x p q

H

x p q h x p q

H x p q h x p q

H x p q

h x p q

x

h

x p q

H x p q h

x p q

H

x p q h x p q

- - --

-- +

+

- +

-

=

Þ

=

- +

+

-+

, p q n R , 1 ( 1) ( , , ), 0 ( ) 0, otherwise. i j p q i j n ij h x p q i j t -- _{=é ù= í}ì - - - ³ ë û î R , p q n R

Η

!

qn

!

1,q

₍

q

₎

1 n n

-=

R

Η

, _{[ ]} p q n = drs n n´ D

n n

´

1

1,

₁

( 1)

( , , ),

₀

0,

_otherwise.

r s rs r s

r

s

d

-

h

x p q

_{r s}

- +

ì

-

_{+ =}

ï

= -

_í

_{- ³}

ï

î

, det p q ( , , ) n =H x p qn D

Corollary 13. Let be an tridiagonal matrix defined as

Then, , where .

Proof. . Suppose that the result is true for all the . We

prove it for .

. In fact, using Cramer’s rule we have

.

We obtain .

Therefore, holds for all positive integers n.

Corollary 14. Let be the matrix in Eqn. (3). Then, .

Proof. The proof runs like in Corollary 13.

If we multiply the rth row by and the sth column by of the matrices , and , then the determinant is not altered [23]. In addition, there is a connection between the determinant and permanent of the Hessenberg matrix [26, 29]. Then, it is clear that the following corollaries holds.

Corollary 15. Let be an Hessenberg matrix defined as

[ ] n nij N =

n n

´

1

1,

,

,

1

0,

.

ij

j i

p q

i

j

n

pq

i j

otherwise

- =

-ì

ï +

=

ï

= í

_-

_{- =}

ï

ïî

, detN =_n [n+1]_{p q}

i

= -

1

1 , detN =[2]p q = +p q m£ -n 1 m=n , , ,

([1] ,...,[ ] )

T

(0,...,0,[

1] )

T n p q

n

p q

n

p q

N

=

+

1 , , , , 1

det(

)[

1]

det(

)[ ]

[ ]

[

1]

det(

)

det(

)

n p q n p q p q p q n n

N

n

N

n

n

n

N

N

-+

=

Þ +

=

, detN =_n [n+1]_{p q} , detN =n [n+1]p q q n

Η

det q ( ) n =H xn Η

( 1)

_-

_i

r

_{( )}

_-

_i

s+2 n

N

p n Η

D

_np q, ,

_{[ ]}

p q n

=

e

ij

E

n n

´

3( ) 1 , 1 ( , , ), ₀ 0, _otherwise. s t st s t i t s e i - h x p q _{s t} - + ì _{- =} ï = í - ³ ï î

Then, , where .

Corollary 16. Let be (p,q)-Rogers-Szegő polynomial, and

be an Hessenberg matrices defined as

Then, , where .

Corollary 17. Let be Rogers-Szegő polynomial, then

Proof. Proof is obvious from equation .

Corollary 18. Let be a (p,q)-number and be an tridiagonal matrix

defined as

Then, , where .

Corollary 19. Let be an tridiagonal matrix defined as

Then, . , det p q ( , , ) n =H x p qn E

i

= -

1

( , , )

n

H x p q

D

np q,

=

[ ]

d

ij ,

[ ]

p q n

=

e

ij

E

n n

´

1

1

1,

( 1)

( , , ),

₀

0,

otherwise,

i j ij _{i j}

j i

d

-

h

_{- +}

x p q

_{i j}

ì

_{- =}

ïï

= -

_í

_{- ³}

ï

ïî

3( ) 1

,

1

( )

( , , ),

₀

0,

_otherwise.

s t st _{s t}

i

t s

e

i

-

h

x p q

_{s t}

- +

ì

-

_{- =}

ï

= í

- ³

ï

î

, ,

( , , )

p q p q n n _n

per

D

=

per

E

=

H x p q

i

= -

1

( )

n

H x

1, 1, 1, 1,

det

q

det

q q q

( ).

n n n

=

n

=

per

=

per

=

H x

n

D

E

D

E

1

lim

_n

( , , )

_n

( )

p®

H x p q

=

H x

[ ]

n

p q,

M

n

=

[

m

st

]

n n

´

,

₁

,

,

₁

0,

_.

st

i

_{t s}

p q

_{s t}

m

ipq

_{s t}

otherwise

-ì

_{- =}

ï +

ï

=

= í

- =

ï

ïî

, detMn =[n+1]p q

i

= -

1

[ ] q n = kst K

n n

´

1 , ₁ 1 , ( 1), ₁ 0, _. st _s i _{t s} x _{t s} k ix q _{s t} otherwise -ì - - = ï ₊ ï = = í - _{- =} ï ïî q ₌

4. Factorization of (𝒑, 𝒒)-Rogers-Szegő Matrix The lower triangular matrix , is defined by

.

Theorem 20. Let be (p,q)-Rogers-Szegő matrices and be (p,q)-matrices. Then, .

Proof. It suffices to prove that . For ,

which implies desired result.

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[ ] n ij n n K = k _´ 1( , , ) 2( , , ) 2( , , ) 3( , , ) ij i i i i k =H_- x p q -pH_- x p q -qH_- x p q + pqH_- x p q , p q n R p q, n Ν , , p q p q n =Kn n R N , ₍ , ₎ 1 p q p q n n - =Kn R N For i> j k, _ij =0.

i j

³ ³

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2 3 1 1 2 3 1 1

( , , )

( , , )(

)

( , , )

j n ik kj ik kj ik k i i i ij s s j s

r b

+

r b

r b

H

_-

x p q

H

_-

x p q p q

H

_-

x p q pq k

= = =

=

=

=

-

+ +

=

å

å

å

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