Applied Mathematics and Computation 215 (2009) 2317–2325

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

On statistical approximation of a general class of positive linearoperators extended in q-calculus

Cristina RaduBabes�-Bolyai University, Faculty of Mathematics and Computer Science, 1 Kogalniceanu St., 400084 Cluj-Napoca, Romania

a r t i c l e i n f o

Keywords:q-CalculusBaskakov operatorStatistical convergenceBohman–Korovkin type theoremWeighted space

0096-3003/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.amc.2009.08.023

E-mail addresses: rcristina@math.ubbcluj.ro, rad

a b s t r a c t

In this paper we present a general class of positive linear operators of discrete type basedon q-calculus and we investigate their weighted statistical approximation properties byusing a Bohman–Korovkin type theorem. We also mark out two particular cases of thisgeneral class of operators.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

In 1987 Lupas� [1] introduced a q-analogue of the Bernstein operator and investigated its approximating and shape-preserving properties. Ten years later, Phillips [2] proposed another generalization of the classical Bernstein polynomialsbased on q-integers. He obtained the rate of convergence and Voronovskaya-type asymptotic formula for these new Bern-stein operators. More results on q-Bernstein operators were obtained by Ostrovska [3,4] and Ba�rbosu [5].

Recently, some new q-type extensions of well-known positive linear operators were introduced by several authors. Forinstance q-Meyer-König and Zeller operators were studied in [6–9]. Also, have been established properties of the operatorsq-Durrmeyer [10,11] and q-Szász Mirakyan [12,13].

On the other hand, the study of the statistical convergence for sequences of positive linear operators was attempted in theyear 2002 by Gadjiev and Orhan [14].

In the present paper, inspired by Baskakov operators, we introduce a general class of positive linear operators based onq-calculus and we investigate their weighted statistical approximation properties. In Section 5 we present two particularcases, which are q-extensions of the Szász Mirakyan operator and the classical Baskakov operator, respectively.

First of all, we recall the concept of statistical convergence.A sequence ðxnÞn is said to be statistically convergent to a number L if for every e > 0,

d n 2 N : xn � Lj jP ef gð Þ ¼ 0

where

d Sð Þ :¼ limN!1

1N

XN

j¼1

vSðjÞ;

is the density of the set S # N. Here vS represents the characteristic function of S.We denote this limit by st � limnxn ¼ L (see [14,15]).

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2318 C. Radu / Applied Mathematics and Computation 215 (2009) 2317–2325

2. Notations and preliminary results

There is no general definition of a ” q-analogue”. A q-analogue, also called q-extension or q-generalization of a mathemat-ical object X, is a family of objects XðqÞ; q > 0, (in general q 2 ð0;1Þ) such that limq!1XðqÞ ¼ X.

Sometimes q-generalization is not unique. In Section 5 we present a q-extension of Szász Mirakyan operator which is notthe same with the q-analogue given in [12].

In what follows we mention some basic definitions and notations used in q-calculus. Details of q-integers can be found in[16].

For any fixed real number q > 0, we denote q-integers by ½k�q; k 2 N, where

½k�q ¼1þ qþ q2 þ � � � þ qk�1 if q – 1;k if q ¼ 1:

(

We set ½0�q ¼ 0. Note that for each k 2 N; q#½k�q; q > 0, is a continuous function with respect to q. In general, for a realnumber t 2 R, we denote the q-number t by ½t�q ¼

1�qt

1�q ; q–1.The q-factorial is defined as follows

½k�q! ¼½1�q � ½2�q � . . . � ½k�q if k ¼ 1;2; . . .

1 if k ¼ 0;

�

and the q-binomial coefficients are given by

n

k

� �q

¼½n�q!

½k�q!½n� k�q!; 0 6 k 6 n:

We use the following notations:

ðaþ bÞnq ¼Yn�1

s¼0

ðaþ qsbÞ; n 2 N; a; b 2 R; ð2:1Þ

ð1þ aÞ1q ¼Y1s¼0

ð1þ qsaÞ; a 2 R; ð2:2Þ

ð1þ aÞtq ¼ð1þ aÞ1q1þ qtað Þ1q

; a; t 2 R: ð2:3Þ

Note that the infinite product (2.2) is convergent if q 2 ð0;1Þ. Throughout the paper we consider q 2 ð0;1Þ.The q-derivative of a function f : R! R is defined by

Dqf ðxÞ ¼ f ðxÞ � f ðqxÞð1� qÞx ; x–0; Dqf ð0Þ :¼ lim

x!0Dqf ðxÞ; ð2:4Þ

and the high q-derivatives D0qf :¼ f ; Dn

qf :¼ DqðDn�1q f Þ; n ¼ 1;2; . . . For example, if hðxÞ ¼ ðx� aÞa; a; a 2 R, then

DqhðxÞ ¼ ½a�qðx� aÞa�1.From the above definition it is obvious that a continuous function on an interval, which does not include 0 is continuous

q-differentiable. We say that a function f is q-differentiable on a real interval I if for any q 2 ð0;1Þ the q-derivative of f existsand is finite in every x 2 I.

The product rule is

Dq f ðxÞgðxÞð Þ ¼ Dq f ðxÞð ÞgðxÞ þ f ðqxÞDq gðxÞð Þ:

We recall the q-Taylor theorem as it is given in [17, p. 103].

Theorem 1. If the function f ðxÞ is capable of expansion as a convergent power series and q is not a root of unity, then

f ðxÞ ¼X1n¼0

ðx� aÞnq½n�q!

Dnqf ðaÞ;

where

ðx� aÞnq ¼Yn�1

s¼0

ðx� qsaÞ ¼Xn

k¼0

n

k

� �q

qkðk�1Þ

2 xn�kð�aÞk: ð2:5Þ

C. Radu / Applied Mathematics and Computation 215 (2009) 2317–2325 2319

3. Construction of the operators

Letting Rþ :¼ ½0;1Þ and N0 :¼ f0g [N, by CBðRþÞ we denote the space of all continuous real-valued functions on Rþ andbounded on the entire positive axis. Baskakov [18] introduced the operators Ln : CBðRþÞ ! CðJÞ;n 2 N,

Lnfð ÞðxÞ ¼X1k¼0

ð�xÞk

k!uðkÞn ðxÞf

kn

� �; ð3:1Þ

which are generated by a sequence of functions ðunÞnP1;un : C! C, having the following properties:

(1) un; n 2 N, are analytic on a domain D containing the disc z 2 C : jz� Rj 6 Rf g and J :¼ ½0;R�.(2) unð0Þ ¼ 1;n 2 N.(3) un; n 2 N, are completely monotone on J, i.e., ð�1ÞkuðkÞn P 0 for x 2 J; k 2 N0;n 2 N.(4) There exists a positive integer mðnÞ such that

uðkÞn ðxÞ ¼ �nuðk�1ÞmðnÞ ðxÞ 1þ ak;nðxÞ

� �; x 2 J; ðn; kÞ 2 N�N;

where ak;nðxÞ converges to zero uniformly in k and x on J for n tending to infinity.(5) limn!1

nmðnÞ ¼ 1.

We set ei; eiðxÞ ¼ xi; i P 0.Let ð/nÞn2N be a sequence of real functions on Rþ which are continuously infinitely q-differentiable on Rþ satisfying the

following conditions.

(P1) /nð0Þ ¼ 1; n 2 N; ð3:2Þ

(P2) ð�1ÞkDkq/nðxÞP 0; n 2 N; k 2 N0; x P 0; ð3:3Þ

(P3) For all ðx; kÞ 2 N�N0 there exists a positive integer ik;0 6 ik 6 k, such that

Dkþ1q /nðxÞ ¼ ð�1Þikþ1Dk�ik

q /nðqikþ1xÞbn;k;ik ;qðxÞ; ð3:4Þ

where

st � limn

bn;k;ik ;qð0Þ

½n�ikþ1q qk

¼ 1: ð3:5Þ

Remark 1. Multiplying (3.4) by ð�1Þk�2ikþ1 we get

ð�1Þkþ1Dkþ1q /nðxÞ ¼ ð�1Þk�ik Dk�ik

q /nðqikþ1xÞbn;k;ik ;qðxÞ:

The last equality and (3.3) yield that

bn;k;ik ;qðxÞP 0; ð3:6Þ

for all x 2 Rþ; ðn; kÞ 2 N�N0; q 2 ð0;1Þ.

We set

CN Rþð Þ :¼ f 2 C Rþð Þ : 9ð Þ limx!1

f ðxÞ1þ xN

<1�

; N P 2:

Endowed with the norm k � kN this space is a Banach space, where

fk kN :¼ supxP0

f ðxÞj j1þ xN

: ð3:7Þ

Inspired by Baskakov operators (3.1) we introduce the announced q-operators as follows.

Tn f ; q; xð Þ ¼X1k¼0

ð�xÞk

½k�q!q

kðk�1Þ2 Dk

q/nðxÞf½k�q½n�qqk�1

!; ð3:8Þ

for all f 2 C2ðRþÞ; x 2 Rþ; q 2 ð0;1Þ;n 2 N, where ð/nÞn is a sequence of functions satisfying (P1)–(P3).It is obvious that Tn;n 2 N, are positive and linear operators.

2320 C. Radu / Applied Mathematics and Computation 215 (2009) 2317–2325

Lemma 1. For all n 2 N; x 2 Rþ and 0 < q < 1, we have

Tn e0; q; xð Þ ¼ 1; ð3:9Þ

Tn e1; q; xð Þ ¼ �xDq/nð0Þ½n�q

; ð3:10Þ

Tn e2; q; xð Þ ¼ x2 D2q/nð0Þq½n�2q

� xDq/nð0Þ½n�2q

: ð3:11Þ

Proof. For a fixed x 2 Rþ, by Theorem 1, we obtain

/nðtÞ ¼X1k¼0

ðt � xÞkq½k�q!

Dkq/nðxÞ: ð3:12Þ

Choosing t :¼ 0 in the above relation and taking into account (3.2), (2.1) and ð�xÞkq ¼ ð�xÞkqkðk�1Þ

2 , we get

X1k¼0

ð�xÞk

½k�q!q

kðk�1Þ2 Dk

q/nðxÞ ¼ /nð0Þ ¼ 1;

and (3.9) is proved.Using (3.12) we can write the q-derivative of /n with respect to t as

Dq/nðtÞ ¼X1k¼1

½k�q½k�q!ðt � xÞk�1

q Dkq/nðxÞ: ð3:13Þ

For getting the above identity we used the formula Dqðt þ aÞk ¼ ½k�qðt þ aÞk�1, see (2.4).Multiplying (3.13) by ð�xÞ and choosing t :¼ 0 we obtain

�xDq/nð0Þ ¼X1k¼1

½k�q½k�q!ð�xÞkq

ðk�1Þðk�2Þ2 Dk

q/nðxÞ; ð3:14Þ

which yields (3.10).We use a similar technique to get (3.11). Differentiating (3.13) with respect to t we get

D2q/nðtÞ ¼

X1k¼2

½k�q½k� 1�q½k�q!

ðt � xÞk�2q Dk

q/nðxÞ:

Now choosing again t :¼ 0 one has

D2q/nð0Þ ¼

X1k¼2

½k�q½k� 1�q½k�q!

ð�xÞk�2qðk�2Þðk�3Þ

2 Dkq/nðxÞ: ð3:15Þ

From (3.14) and (3.15) we have

x2D2q/nð0Þ � xqDq/nð0Þ ¼

X1k¼1

½k�q½k� 1�q½k�q!

ð�xÞkqðk�2Þðk�3Þ

2 Dkq/nðxÞ þ q

X1k¼1

½k�q½k�q!ð�xÞkq

ðk�1Þðk�2Þ2 Dk

q/nðxÞ

¼X1k¼1

½k�q½k�q!ð�xÞkq

ðk�2Þðk�3Þ2 Dk

q/nðxÞ ½k� 1�q þ qk�1 �

¼X1k¼1

½k�2q½k�q!ð�xÞkq

ðk�2Þðk�3Þ2 Dk

q/nðxÞ;

where we used the fact that ½k� 1�q þ qk�1 ¼ ½k�q for all k 2 N.On the other hand

Tn e2; q; xð Þ ¼ 1

q½n�2q

X1k¼1

½k�2q½k�q!ð�xÞkq

ðk�2Þðk�3Þ2 Dk

q/nðxÞ ¼ x2 D2q/nð0Þq½n�2q

� xDq/nð0Þ½n�2q

;

and (3.11) follows. The proof is complete. h

Remark 2. Since any linear and positive operator is monotone, relation (3.11) guarantees that Tnf 2 C2ðRþÞ for eachf 2 C2ðRþÞ.

C. Radu / Applied Mathematics and Computation 215 (2009) 2317–2325 2321

4. Statistical approximation properties in weighted spaces

In this section, by using a Bohman–Korovkin type theorem proved in [15], we present the statistical approximation prop-erties of the operator Tn given by (3.8).

At this moment, we recall the concept of A-statistical convergence, weight function and weighted space considered in[15]. Let A ¼ ðajnÞj;n be a non-negative regular summability matrix. A sequence ðxnÞn is said to be A-statistically convergentto a number L if, for every e > 0; limj

Pn:jxn�LjPeajn ¼ 0. We denote this limit by stA � limnxn ¼ L. For A :¼ C1, the Cesàro matrix

of order one, A-statistical convergence reduces to statistical convergence.Let R denote the set of real numbers. A real function . is called a weight function if it is continuous on R and

limjxj!1.ðxÞ ¼ 1;.ðxÞP 1 for all x 2 R.Let denote by B.ðRÞ the weighted space of real-valued functions f defined on R with the property jf ðxÞj 6 Mf .ðxÞ for all

x 2 R, where Mf is a constant depending on the function f. We also consider the weighted subspace C.ðRÞ of B.ðRÞ given by

C.ðRÞ :¼ f 2 B.ðRÞ : f continuous on R�

:

Endowed with the norm k � k., where kfk. :¼ supx2Rjf ðxÞj.ðxÞ ;B.ðRÞ and C.ðRÞ are Banach spaces.

Using A-statistical convergence Duman and Orhan proved the following Bohman–Korovkin type theorem [15, Theorem 3].

Theorem 2. Let A ¼ ðajnÞj;n be a non-negative regular summability matrix and let ðLnÞn be a sequence of positive linear operatorsfrom C.1

ðRÞ into B.2ðRÞ, where .1 and .2 satisfy

limjxj!1

.1ðxÞ

.2ðxÞ¼ 0:

Then

stA � limn

Lnf � fk k.2¼ 0 for all f 2 C.1

ðRÞ;

if and only if

stA � limn

LnFv � Fvk k.1¼ 0; v ¼ 0;1;2;

where Fv ðxÞ ¼ xv.1ðxÞ1þx2 ;v ¼ 0;1;2.

Examining this result, clearly, replacing R by Rþ, the theorem holds true.Further on, we consider a sequence ðqnÞn; qn 2 ð0;1Þ, such that

st � limn

qn ¼ 1: ð4:1Þ

Theorem 3. Let ðqnÞn be a sequence satisfying (4.1). Then for all f 2 C2ðRþÞ, we have

st � limn

Tn f ; qn; �ð Þ � fk k2a ¼ 0; a > 1:

Proof. It is clear that

st � limn

Tn e0; qn; �ð Þ � e0k k2 ¼ 0: ð4:2Þ

Based on (3.4) we have

Dqn/nðxÞ ¼ �/nðqnxÞbn;0;0;qn

ðxÞ x 2 Rþ; n 2 N;

where

st � limn

bn;0;0;qnð0Þ

½n�qn

¼ 1: ð4:3Þ

Thus, by (3.2) and (3.10), we obtain

Tnðe1; qn; xÞ � e1ðxÞj j1þ x2 6 e1k k2

bn;0;0;qnð0Þ

½n�qn

� 1

����������:

Consequently,

Tn e1; qn; �ð Þ � e1k k2 612

bn;0;0;qnð0Þ

½n�qn

� 1

����������

and for any e > 0 we have dðAÞ 6 dðBÞ ¼ 0, where

2322 C. Radu / Applied Mathematics and Computation 215 (2009) 2317–2325

A ¼ n 2 N : Tn e1; qn; �ð Þ � e1k k2 P e�

;

B ¼ n 2 N :bn;0;0;qn

ð0Þ½n�qn

� 1

����������P 2e

( ):

Hence, we get

st � limn

Tn e1; qn; �ð Þ � e1k k2 ¼ 0: ð4:4Þ

The condition (3.4) implies that for any n 2 N, we have

D2qn

/nðxÞ ¼ �Dqn/nðqnxÞbn;1;0;qn

ðxÞ ð4:5Þ

or

D2qn

/nðxÞ ¼ /nðq2nxÞbn;1;1;qn

ðxÞ: ð4:6Þ

Case 1. If (4.5) holds true, then D2qn

/nð0Þ ¼ bn;0;0;qnð0Þbn;1;0;qn

ð0Þ. From (3.11) we get

Tnðe2; qn; xÞ � e2ðxÞj j 6bn;0;0;qn

ð0Þbn;1;0;qnð0Þ

qn½n�2qn

� 1

����������x2 þ

bn;0;0;qnð0Þ

½n�2qn

x:

By using the elementary inequality

XY � 1j j 6max X2 � 1��� ���; Y2 � 1

��� ���n o; X;Y 2 R;XY P 0;

we can write

Tnðe2; qn; xÞ � e2ðxÞj j1þ x2 6 e2k k2 max

k¼0;1

bn;k;0;qnð0Þ

qkn½n�qn

!2

� 1

������������

8<:

9=;þ e1k k2

bn;0;0;qnð0Þ

½n�2qn

6maxk¼0;1

bn;k;0;qnð0Þ

qkn½n�qn

� 1

���������� 2þ

bn;k;0;qnð0Þ

qkn½n�qn

� 1

����������

!( )þ 1

2bn;0;0;qn

ð0Þ½n�2qn

:

Since st � limnqn ¼ 1 we have

st � limn

1½n�qn

¼ 0: ð4:7Þ

From (3.5) and (4.7) we obtain

st � limn

Tn e2; qn; �ð Þ � e2k k2 ¼ 0: ð4:8Þ

Case 2. If (4.6) holds true, then D2qn

/nð0Þ ¼ bn;1;1;qnð0Þ. By using (3.11) we get

Tnðe2; qn; xÞ � e2ðxÞj j 6bn;1;1;qn

ð0Þqn½n�

2qn

� 1

����������x2 þ

bn;0;0;qnð0Þ

½n�2qn

x;

and

Tn e2; qn; �ð Þ � e2k k2 6bn;1;1;qn

ð0Þqn½n�

2qn

� 1

����������þ 1

2bn;0;0;qn

ð0Þ½n�2qn

:

Taking into account (3.5) and (4.7), the last inequality implies

st � limn

Tn e2; qn; �ð Þ � e2k k2 ¼ 0: ð4:9Þ

Finally, using (4.2), (4.4) and (4.8) or (4.9), the proof follows from Theorem 2 by choosing A ¼ C1, the Cesàro matrix oforder one and .1ðxÞ ¼ 1þ x2;.2ðxÞ ¼ 1þ x2a; x 2 Rþ;a > 1. h

5. Special cases of Tn operator

In this section we present two particular cases of the operator Tn given by (3.8), which turn into the well-known SzászMirakyan operator and the classical Baskakov operator, respectively, in the case q! 1�.

C. Radu / Applied Mathematics and Computation 215 (2009) 2317–2325 2323

5.1. A q-analogue of Szász Mirakyan operator

There are two important q-analogues of the exponential function

EqðxÞ ¼X1k¼0

qkðk�1Þ

2xk

½k�q!¼ 1þ ð1� qÞxð Þ1q ;

eqðxÞ ¼X1k¼0

xk

½k�q!¼ 1

1� ð1� qÞxð Þ1q:

Note that for q 2 ð0;1Þ the series expansion of eqðxÞ has radius of convergence 11�q. On the contrary, the series expansion of

EqðxÞ converges for every real x. The equality between the series expansion and the infinite product expansion of eqðxÞ andEqðxÞ, in the domain where both expansions converge, is proved in [16, pp. 29–32].

The q-exponential functions satisfy the following properties

DqEqðaxÞ ¼ aEqðaqxÞ; DqeqðaxÞ ¼ aeqðaxÞeqðxÞEqð�xÞ ¼ EqðxÞeqð�xÞ ¼ 1:

For some q 2 ð0;1Þ, let ðanÞn be a sequence of positive real numbers satisfying st � limnan ¼ 1 and let/nðxÞ :¼ Eqð�½n�qanxÞ; x 2 Rþ;n 2 N. Then, for all ðn; kÞ 2 N�N, we have /nð0Þ ¼ 1 and

Dkq/nðxÞ ¼ ð�1Þk½n�kqak

nqkðk�1Þ

2 Eq �½n�qanqkx �

; x P 0:

It is obvious that under the assumption made upon the sequence ðanÞn, the condition (3.3) is fulfilled. Furthermore, for allk 2 N we get

Dkþ1q /nðxÞ ¼ �Dk

q/nðqxÞbn;k;0;qðxÞ;

where bn;k;0;qðxÞ ¼ ½n�qanqk. Consequently,

bn;k;0;qð0Þ½n�qqk

¼ an; n 2 N;

and (3.4), (3.5) are also satisfied.In this case the operator Tn turns into S�n, given as follows.

S�n f ; q; xð Þ ¼X1k¼0

an½n�qx �k

½k�q!qkðk�1ÞEq �½n�qqkanx

�f

½k�q½n�qqk�1

!; ð5:1Þ

for f 2 C2ðRþÞ; x 2 Rþ;n 2 N; q 2 ð0;1Þ.

Remark 3. Choosing an ¼ 1;n 2 N, the operator S�n given by (5.1) reduces to the classical Szász Mirakyan operator, whenq! 1�.

Based on Lemma 1 we have

S�n e0; q; xð Þ ¼ 1;S�n e1; q; xð Þ ¼ anx;

S�n e2; q; xð Þ ¼ a2nx2 þ an

½n�qx;

for x 2 Rþ;n 2 N; q 2 ð0;1Þ.We point out that our q-generalization S�n;n 2 N, is different by q-analogue of Szász Mirakyan operator, recently intro-

duced by Aral (see [12]) as follows.

Sqnðf ; xÞ ¼ Eq �½n�q

xbn

� �X1k¼0

f½k�qbn

½n�q

! ½n�qx �k

½k�q!ðbnÞk;

where 0 6 x < bn1�qn ; bn is a sequence of positive numbers such that limnbn ¼ 1.

The approximation function S�nðf ; q; �Þ is defined on ½0;1� for each n 2 N, while the domain of Sqnðf ; �Þ depends on n. More-

over, in the case an ¼ 1; n 2 N, since jS�nðe2; q; xÞ � e2ðxÞj ¼ 1½n�q

x and jSqnðe2; xÞ � e2ðxÞj ¼ ð1� qÞx2 þ bn

½n�qx, the behavior of

S�nð�; q; �Þ on e2 is better than Sqn on e2.

2324 C. Radu / Applied Mathematics and Computation 215 (2009) 2317–2325

5.2. A q-analogue of classical Baskakov operator

In order to give the second particular case of the operator Tn, we consider the next lemma. The proof follows immediatelyfrom (2.1)–(2.3), (see [16, pp. 106–107]).

Lemma 2. Let t; s; a 2 R

Dq 1þ axð Þtq ¼ t½ �qa 1þ aqxð Þt�1q ; ð5:2Þ

ð1þ xÞsþtq ¼ ð1þ xÞsq 1þ qsxð Þtq; ð5:3Þ

ð1þ xÞ�tq ¼

11þ q�txð Þtq

: ð5:4Þ

By using (5.2) and the identity ½�n�q ¼�½n�q

qn ; n 2 N, it is easy to see that

Dq 1þ axð Þ�nq ¼

�½n�qqn a 1þ aqxð Þ�n�1

q : ð5:5Þ

Let /nðxÞ :¼ ð1þ qnþ1xÞ�nq ; x 2 Rþ;n 2 N. Taking into account (5.4), (5.5) and the definition of the high q-derivatives, we

obtain

Dkq/nðxÞ ¼ ð�1Þk½n�q � . . . � nþ k� 1½ �qqk 1þ qnþkþ1x

� ��n�k

q ¼ ð�1Þknþ k� 1½ �q!

n� 1½ �q!qk 1

1þ qxð Þnþkq

;

for all x 2 Rþ; ðn; kÞ 2 N�N; q 2 ð0;1Þ. One can see that (3.3) is satisfied.Consequently, by using (5.3) we can write

Dkþ1q /nðxÞ ¼ ð�1Þkþ1 nþ k½ �q!

n� 1½ �q!qkþ1 1þ qnþkþ2x

� ��n�k�1

q ¼ ð�1Þkþ1 nþ k½ �q!

n� 1½ �q!qkþ1 1þ qnþkþ2x

� ��n�1

q 1þ qkþ1x� ��k

q

¼ ð�1ÞkDq/nðqkxÞbn;k;k�1;qðxÞ;

where bn;k;k�1;qðxÞ ¼½nþk�q !

½n�q !qkð1þ qkþ1xÞ�k

q , and bn;k;k�1;qð0Þ½n�kqqk ¼ ½nþk�q !

½n�kq ½n�q !.

Since, for 0 < q < 1, we have limn½nþk�q½n�q¼ 1, (3.4) and (3.5) are also verified with ik ¼ k� 1.

In this case the operator Tn turns into V�n, given as follows.

V�nðf ; q; xÞ ¼X1k¼0

nþ k� 1k

� �q

qkðk�1Þ

2 ðqxÞk 1

ð1þ qxÞnþkq

f½k�q½n�qqk�1

!; ð5:6Þ

for f 2 C2ðRþÞ; x 2 Rþ;n 2 N; q 2 ð0;1Þ.We mention that the first generalization in q-Calculus of Baskakov operators has been achieved by Aral and Gupta [19].

Remark 4. The operator V�n given by (5.6) becomes the classical nth Baskakov operator in the case q! 1�.

Based on Lemma 1 we have

V�nðe0; q; xÞ ¼ 1;V�nðe1; q; xÞ ¼ qx;

V�nðe2; q; xÞ ¼nþ 1½ �q½n�q

qx2 þ q½n�q

x; x 2 Rþ; n 2 N:

References

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