A note on the modified q-Bernstein polynomials for functions of several variables and their reflections on q-Volkenborn integration

Applied Mathematics and Computation 218 (2011) 707–712

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

A note on the modified q-Bernstein polynomials for functions of several variables and their reflections on q-Volkenborn integration _ Mehmet Açıkgöz a, Serkan Aracı a, Ismail Naci Cangül b,⇑ a b

University of Gaziantep, Faculty of Arts and Science, Department of Mathematics, 27310 Gaziantep, Turkey Uludag University, Faculty of Science, Department of Mathematics, Görükle, 16059 Bursa, Turkey

a r t i c l e

i n f o

a b s t r a c t In this paper, we consider the modified q-Bernstein polynomials for functions of several variables on q-Volkenborn integral and investigate some new interesting properties of these polynomials related to q-Stirling numbers, Hermite polynomials and Carlitz’s type q-Bernoulli numbers. Ó 2011 Elsevier Inc. All rights reserved.

Dedicated to Professor H. M. Srivastava on the Occasion of his Seventieth Birth Anniversary Keywords: Generating function Bernstein polynomials of several variables Bernstein operator of several variables Shift difference operator Q-difference operator q-Stirling numbers q-Volkenborn integration Hermite polynomials Carlitz’s type q-Bernoulli numbers

1. Introduction, definitions and notations The Bernstein polynomials have been studied by many researchers for a long time. The history of these polynomials goes back to Bernstein in 1904. Since then, these polynomials have been studied by many mathematicians [1–6,8,11–13]. But no result is given about the generating function of these polynomials. Throughout this paper C(DW) will denote the set of continuous functions on D = [0, 1]. Then Bernstein operator for functions f 2 C(DW) of several variables is defined as

  Y  nW w  X ni kj k1 kw  x ð1  xj Þnj kj f ;; n1 nw nj j j¼1 k1 ¼0 kW ¼0   n1 nW X X k1 kw  Bk1 ;;kw ;n1 ;;nw ðx1 ;    ; xw Þ; ¼  f ;; n1 nw k ¼0 k ¼0

Bn1 ;;nw ðf ; x1 ;    ; xw Þ : ¼

n1 X

1



W

ð1Þ

  P n ¼ nðn1Þðnkþ1Þ : Here Bn1 ;;nw ðf ; x1 ;    ; xw Þ is called the Bernstein operator of several variables of order w l¼1 nl . k! k Let p be a fixed prime number. In this paper, we will use N; Zp ; Qp ; C and Cp to denote the set of natural numbers, the ring of p-adic integers, the field of p-adic rational numbers, the complex number field, and the completion of algebraic closure of Qp , respectively. Let UDðZp Þ be the space of uniformly differentiable functions on Zp . For

where

⇑ Corresponding author. _ E-mail addresses: [email protected] (M. Açıkgöz), [email protected] (S. Aracı), [email protected] (I.N. Cangül). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.01.091

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M. Açıkgöz et al. / Applied Mathematics and Computation 218 (2011) 707–712

f 2 UDðZp ; Cp Þ ¼ ff jf : Zp ! Cp is a uniformly differentiable functiong; the q-Volkenborn integral on Zp is defined by

Iq ðf Þ ¼

Z

f ðxÞdlq ðxÞ ¼ lim

N!1

Zp

pN 1 1 X f ðxÞqx ; ½pN q x¼0

qx , ½pN q

where lq ðx þ p Zp Þ ¼ (see [9,10] for details). Let q be regarded as either a complex number q 2 C or a p-adic number q 2 Cp . If q 2 C; then we assume that jqj < 1; and if 1 q 2 Cp , we assume that j1  qjp < pp1 , which yields to the relation qx = exp (xlogq) for jqj 6 1, (see [9]). Here, the symbol j.jp stands for the p-adic value on Cp with jpjp ¼ 1p. The q-analogue of n! is defined by N

(

1;

½nq ! ¼

if n ¼ 0;

½nq ½n  1q    ½2q ½1q ; if n ¼ 1; 2; . . .

In this study, we use notation for the Gaussian binomial coefficients in the form of

  n k

½nq ! ½nq :½n  1q    ½n  k þ 1q ¼ : ½kq ! ½n  kq !½kq !

¼

q

Note that limq!1

    n n ¼ ¼ nðn1Þðnkþ1Þ , (see [7] for details). k! k q k

Carlitz’s q-Bernoulli number bk,q can be defined with the relations

b0;q ¼ 1; qðqb þ 1Þk  bk;q ¼



1; if k ¼ 1; 0; if k > 1

with the usual convention of replacing bi by b i,q, (see [9]). As shown in [8] and [9], Carlitz’s q-Bernoulli numbers can be represented by q-Volkenborn integral on Zp as follows:

bn;q ¼

Z

½xnq dlq ðxÞ ¼ lim

N!1

Zp

pN 1 1 X n x ½x q ; ½pN  x¼0 q

n 2 N0 :

The k-th order factorial of a q-number [x]q defined by

½xk;q ¼ ½xq ½x  1q    ½x  k þ 1q

  ½x x ¼ ½kk;q! . q k q In this study, we consider modified q-Bernstein polynomials for functions of several variables on Zp  Zp      Zp and we investigate some interesting properties of these polynomials related to q-Stirling numbers, Hermite polynomials and Carlitz’s q -Bernoulli numbers.

is called the q-factorial of x of order k, (see [9]). We note that

2. Main results In this section, we shall study the modified q-Bernstein polynomials for functions of several variables on q-Volkenborn integral and investigate some new interesting properties of these polynomials related to q-Stirling numbers and Carlitz’s 1 type q-Bernoulli numbers. Here we assume that q 2 Cp with j1  qjp < pp1 : For

f 2 UDðZp      Zp ; Cp Þ ¼ ff jf : Zp      Zp ! Cp is a uniformly differentiable functiong; we define q-Volkenborn integral on Zp      Zp :

Dq ðf Þ ¼

Z Zp



Z

f ðx1 ;    ; xw Þdlq ðx1 Þ    dlq ðxw Þ ¼

Zp

lim

N 1 ;...;Nw !1

w Y d¼1

Nw 1 Pw p 1 1 pX 1 X x    f ðx1 ;    ; xw Þq y¼1 y ; N d ½p  x ¼0 x ¼0 N

1

w

Pw Q xy 1 y¼1 where lq ðx1 þ pN Zp Þ    lq ðxw þ pN Zp Þ ¼ w : d¼1 ½pNd  q We consider q-Bernstein type operator for functions of several variables on Zp      Zp as follows:

Bn1 ;...;nw ;q ðf ; x1 ; . . . ; xw Þ ¼

n1 X k1 ¼0



nw X kw ¼0

f

½k1 q ½kw q ;...; ½n1 q ½nw q

!

 Bk1 ;...kw ;n1 ;...;nw ðx1 ; . . . ; xw ; qÞ

for nj ; mj 2 N0 ; j 2 f1; 2; . . . ; wg, where Bk1 kw ;n1 ;;nw ðx1 ;    ; xw ; qÞ are called modified q-Bernstein polynomials of several variP ables of degree w d¼1 nd . Let (Eh)(x) = h(x + 1) be the shift operator. Consider the q-difference operator as follows:

Dnq ¼

n Y

ðE  qi1 IÞ;

i¼1

where I is the identity operator. By (2),

ð2Þ

M. Açıkgöz et al. / Applied Mathematics and Computation 218 (2011) 707–712

f ðxÞ ¼

Xx n

nP0

Dnq f ð0Þ;

709

ð3Þ

q

where

Dnq f ð0Þ ¼

  k

Xx n

nP0

ð4Þ

ð1Þk q 2 f ðn  kÞ:

q

The q-Stirling numbers of the first and second kind are respectively defined as n n X Y ð1 þ ½kq zÞ ¼ S1 ðn; k; qÞzk k¼1

ð5Þ

k¼0

and n Y k¼1

1 1 þ ½kq z

! ¼

n X

S2 ðn; k; qÞzk :

ð6Þ

k¼0

(see [9] for details). By (2), (3)–(6), we see that

  k 

S2 ðn; k; qÞ ¼

q

2

½kq !

  k   j    n X k q 2 ð1Þj q 2 ½k  jnq ¼ Dkq 0n : ½k ! j q q j¼0

ð7Þ

For 0 < q < 1, consider the q-extension as follows: w P w 1 1 w Y X X Y ðt½xj q Þkj t ½1xs  t nj F k1 ;...;kw ðt; q; x1 ; . . . ; xw Þ ¼ ¼  Bk1 ;...;kw ;n1 ;...;nw ðx1 ; . . . ; xw ; qÞ e s¼1 : kj ! nj ! j¼1 j¼1 n ¼k n ¼k 1

Qw

kj

ðt½xq Þ j¼1 kj !

Thus, we note that F k1 ;...;kw ðt; q; x1 ; . . . ; xw Þ ¼ mials of several variables. It is easy to show that w Y

½1  xd nqd kd ¼

1 X t 1 ¼0

d¼1



1 nX 1 k1 X



t w ¼0 l1 ¼0

nX w kw lw ¼0

e

t

1

Pw

s¼1

w

½1xs 

w

is the generating function of modified q-Bernstein polyno-

 w  Y ld þ t d  1 d¼1

td

 

nd  kd ld

 Pw Pw Pw l ½xd lqd þtd ð1Þ s¼1 ls þts q y¼1 y ðq  1Þ k¼1 rk :

ð8Þ

By (8), we obtain the following results: Theorem 2.1. For nj ; kj 2 N0 with nj P kj for j 2 {1, 2, . . . , w}, we have

Z



Zp

Z Zp

 

  nX 1 1 nX 1 k1 w kw Y X X ld þ t d  1 Bk1 ;...;kw ;n1 ;...;nw ðx1 ; . . . ; xw ; qÞ w     dlq ðx1 Þ    dlq ðxw Þ ¼ d¼1 Qw nd td t ¼0 t ¼0 l ¼0 l ¼0 1

d¼1 kd  Pw Pw Pw nd  kd l btd þld þkd ;q ð1Þ s¼1 ls þts q y¼1 y ðq  1Þ k¼1 rk ; ld

w

1

w

where bn,q are the nth Carlitz q-Bernoulli numbers. For x; t 2 C, the Hermite polynomials are defined by the following generating function 2

e2txt ¼

1 X

Hn ðxÞ

n¼0

tn : n!

ð9Þ

By (9), we have

e2xt ¼

1 X t 2n n! n¼0

!

1 X n¼0

Hn ðxÞ

! tn : n!

Applying Cauchy product to above relation, we obtain

710

M. Açıkgöz et al. / Applied Mathematics and Computation 218 (2011) 707–712

e2xt ¼

! 1 n   X X n t 2nl Hl ðxÞ : n! l n¼0 l¼0

ð10Þ

By substituting x = [1  y]q into (10), we have

Hn ð½1  yq Þ ¼

k!2n Bk;nþk ðy; qÞ: yk ðn þ kÞ!

ð11Þ

By (11) and using the definition of modified q-Bernstein polynomials for several variables, we obtain the following theorem: Theorem 2.2. Let n be a positive integer with kj < nj for j 2 {1, 2, . . . , w} and Hn(x) denote the Hermite polynomials. Then we have

Bk1 ;...;kw ;n1 þk1 ;...;nw þkw ðx1 ; . . . ; xw ; qÞ ¼ 2

Pw s¼1

ns

k w Y xdd ðnd þ kd Þ! Hnd ð½1  xd q Þ: kd d¼1

By [5], it is known that w Y

!i ½xj q

¼

j¼1

n1 X



k1 ¼i1

w Y

1

s¼1

ð½1  xs q þ ½xs q Þns ls

nw X

w Y

ð12Þ

;



kd



i   Bk1 ;...;kw ;n1 ;...;nw ðx1 ; . . . ; xw ; qÞ; nd

kw ¼i1 d¼1

 ð13Þ

i

for i 2 N and w Y

!i ½xj q

¼

j¼1

i X



k1 ¼0

i X



Pw y¼1

q

kd ¼0

ky 2



 w  Y xd d¼1

kd

½kd q !S2 ði; kd ; qÞ:

q

Moreover, we know that

Z Zp



Z

Pw

 w  Y xm

Zp m¼1

km

k d¼1 d

q

ð1Þ dlq ðx1 Þ    dlq ðxm Þ ¼ Qw q s¼1 ½ks þ 1q

w

Pw l¼1



kl 2

 :

ð14Þ

By (13) and (14), we have

w bw i;q ¼ q

i X



k1 ¼0

i X

Pw q



d¼1

kw ¼0



  kn kn w 1 Y ð1Þ 2 q 2 ½k  !S ði; k ; qÞ: q n q 2 n ½kn þ 1q n¼1

kd

We know that

S2 ðn; k; qÞ ¼

    k X jþn kj k þ n ð1Þ kj q j ð1  qÞk j¼0 q 1

ð15Þ

and that

  n   X n n ¼ ðq  1Þjk S2 ðk; j  k; qÞ: k q j j¼0 By simple calculation, we obtain

qnx

  k ( )     n n n X X X n n 2 k ½xk;q ¼ S1 ðk; m; qÞ ½xm ¼ ðq  1Þ qq ðq  1Þkq q k k m¼0 k¼0 k¼m

ð16Þ

and

Z Zp

qnx dlq ðxÞ ¼

 1  X n ðq  1Þm bm;q : m m¼0

By (16) and (17), we can write

ð17Þ

M. Açıkgöz et al. / Applied Mathematics and Computation 218 (2011) 707–712





n

  n X n ðq  1Þmþk S1 ðk; m; qÞ: q k q k¼m

¼

m

711

Theorem 2.3. For nj ; kj 2 N0 with nj P kj where j 2 {1, 2, . . . , w}, we have n1 X

Bk1 ;...;kw ;n1 ;n2 ;...;nw ðx1 ; . . . ; xw ; qÞ ¼

l1 ¼k1

 w  Y ny ly

y¼1



Pw ðq  1Þ d¼1 kd þld ;

nw X lw ¼kw

 S1 ðly ; ky ; qÞ½xy kqy ½1  xy nqy ky :

q

From the definition of the q-Stirling numbers of the first kind,

    n   n n X x 2 q ½nq ! ¼ ½xn;q q 2 ¼ S1 ðn; k; qÞ½xkq : n q k¼0

ð18Þ

Theorem 2.4. For nj ; mj 2 N0 ; j 2 f1; 2; . . . ; wg and i 2 N; we have w Y s¼1

1 ns ls

ð½1  xs q þ ½xs q Þ 

w Y



d¼1

kd

n1 X



k1 ¼i1



nw X

;

kw ¼i1



k1 kw Y i i w X X X X i  Bk1 ;...;kw ;n1 ;...;nw ðx1 ; . . . ; xw ; qÞ ¼  S1 ðnd ; ld ; qÞS2 ði; kd ; qÞ½xd lqd : nd k ¼0 l ¼0 k ¼0 l ¼0 d¼1 1

i

1

w

w

Corollary 2.5. For i 2 N, we have

bi;q ¼

k1 i X X



k1 ¼0 l1 ¼0

kw Y i w X X

!w1 S1 ðnd ; ld ; qÞS2 ði; kd ; qÞbld ;q

;

kw ¼0 lw ¼0 d¼1

where bi,q are the i-th Carlitz q-Bernoulli numbers. In [9], the q-Bernoulli polynomials of order k 2 N0 are defined by ðkÞ bn;q ðxÞ ¼

  Z Z n Pk X n ix 1 ð1Þi  q l¼1 ðklþiÞxl dlq ðx1 Þ    dlq ðxk Þ: q n ð1  qÞ i¼0 i Zp Zp |fflfflfflfflfflffl{zfflfflfflfflfflffl}

ð19Þ

ktimes

By (19), we note that ðkÞ bn;q ðxÞ ¼

  n X n ði þ kÞ    ði þ 1Þ ix 1 ð1Þi q : n ð1  qÞ i¼0 i ½i þ kq    ½i þ 1q

The inverse q-Bernoulli polynomials of order k are also defined by

ðkÞ bn;q ðxÞ ¼

1 ð1  qÞn

n X i¼0

ð1Þi Z

  n xi q i

Z ; Pk  q l¼1 ðklþiÞxl dlq ðx1 Þ    dlq ðxk Þ Zp Zp |fflfflfflfflfflffl{zfflfflfflfflfflffl}

ð20Þ

ktimes

ðkÞ ðkÞ ðkÞ ðkÞ (see [9]). In the special case of x = 0, bn;q ¼ bn;q ð0Þ are called the n-th q-Bernoulli numbers of order k and bn;q ¼ bn;q ð0Þ are called the n-th inverse q-Bernoulli numbers of order k. By using (20), we obtain

ðnÞ bk;q

(  )      k k X X ½j þ nq    ½j þ 1q ½nq ! jþn 1 j k j kþn  : ¼ ð1Þ ð1Þ ¼ kþn j ðj þ nÞ    ðj þ 1Þ kj q n q ð1  qÞk j¼0 ð1  qÞk j¼0 n! n 1

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M. Açıkgöz et al. / Applied Mathematics and Computation 218 (2011) 707–712

By (19), we have

S2 ðn; k; qÞ ¼



kþn



n

n! ðnÞ b : ½nq ! k;q

ð21Þ

By (21), we obtain the following theorem: Theorem 2.6. Let nj ; mj 2 N0 for j 2 {1, 2, . . . , w} and i 2 N. Then we have

bi;q ¼ q

  d i w X Y ½iq ! kd þ i ð1Þkd ðiÞ  ½kd q ! bkd ;q ½k þ 1 i! k d d q ¼0 k ¼0 d¼1

i i X X k1 ¼0 k2

!w1 :

w

Acknowledgement The third author is supported by the Commission of Scientific Research Projects of Uludag University, Project numbers 2006/40, 2008/31 and 2008/54. References [1] M. Acıkgoz, S. Araci, On the generating function of Bernstein polynomials, in Proceedings of the 8th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2010), AIP, Rhodes, Greece, 2010. [2] M. Acıkgoz, S. Araci, New generating function of Bernstein type polynomials for two variables, in Proceedings of the 8th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2010), AIP, Rhodes, Greece, 2010. [3] M. Açıkgöz, S. Aracı, A study on the integral of the product of several type Bernstein polynomials, IST Transaction of Applied Mathematics Modelling and Simulation, 2010, vol. 1, no. 1(2), ISSN: 1913-8342, pp. 10–14. [4] M. Açıkgöz, S. Aracı, The relations between Bernoulli, Bernstein and Euler polynomials, in Proceedings of the 8th International conference of Numerical Analysis and Applied Mathematics (ICNAAM 2010), AIP, Rhodes, Greece, 2010. [5] M. Açıkgöz, S. Aracı, A new approach to modified q-Bernstein polynomials for functions of two variables with their generating function and interpolation function, in: International Congress in Honour of Prof. H.M. Srivastava on his 70th Birth Anniversary, 18–21 August 2010, Uludag˘ University, Bursa Turkey, submitted for publication. _ Buyukyazici, E. Ibikli, _ [6] I. The approximation properties of generalized Bernstein polynomials of two variables, Appl. Math. Comput. 156 (2004) 367–380. [7] V. Kac, P. Cheung, Quantum Calculus, Springer-Verlag, New York, 2002. ISBN 0-387-95341-8. [8] T. Kim, L.-C. Jang, H. Yi, Note on the modified q-Bernstein polynomials, Discrete. Dyn. Nat. Soc. (2010) 12 (Article ID 706483). [9] T. Kim, J. Choi, Y.-H. Kim, Some identities on the q-Bernstein polynomials, q-Stirling numbers and q-Bernoulli numbers, Adv. Stud. Contemp. Math. 20 (2010) 335–341. no. 3. [10] T. Kim, q-Volkenborn integration, Russ. J. Math. Phys. 9 (2002) 288–299. no. 3. [11] H. Oruc, G.M. Phillips, A generalization of the Bernstein polynomials, Proc. Edinburgh Math. Soc. 42 (1999) 403–413. [12] G.M. Phillips, A survey of results on the q-Bernstein polynomials, IMA Journal of Numerical Analysis, 2009, pp. 1–12 (published online on June 23). [13] Y. Simsek, M. Açıkgöz, A new generating function of q-Bernstein-type polynomials and their interpolation function, Abstr. Appl. Anal. vol. 2010, Article ID 769095, p. 12. doi:10.1155/2010/769095.01-313.