Başlık: The voronovskaja type asymptotic formula for q-derivative of integral generalization of q-bernstein operatorsYazar(lar):MISHRA, Vishnu Narayan; PATEL, PrashantkumarCilt: 67 Sayı: 2 Sayfa: 298-305 DOI: 10.1501/Commua1_0000000883 Yayın Tarihi: 2018 

C om mun. Fac. Sci. U niv. A nk. Ser. A 1 M ath. Stat. Volum e 67, N umb er 2, Pages 298–305 (2018) D O I: 10.1501/C om mua1_ 0000000883 ISSN 1303–5991

http://com munications.science.ankara.edu.tr/index.php?series= A 1

THE VORONOVSKAJA TYPE ASYMPTOTIC FORMULA FOR q-DERIVATIVE OF INTEGRAL GENERALIZATION OF

q-BERNSTEIN OPERATORS

VISHNU NARAYAN MISHRA AND PRASHANTKUMAR PATEL

Abstract. The Voronovskaja type asymptotic formula for function having q-derivative of the integral generalization Bernstein operators based on q-integer is discussed. The same formula for Stancu type generalization of this operators is mentioned.

1. Introduction

The classical Bernstein-Durrmeyer operators Dn introduced by Durrmeyer [1] is

associated with an integrable function f on the interval [0; 1] and is de…ned as Dn(f ; x) = (n + 1) n X k=0 pn;k(x) Z 1 0 pn;k(t)f (t)dt; x 2 [0; 1]; (1.1) where pn;k(x) = n k x k_{(1 x)}n k_.

These operators have been studied by Derriennic [2] and many others. For the last 30 years, q-calculus has been an active area of research in approximation theory. In 1987, the q-analogues of Bernstein operators was introduced by Lupas [3] and in [4], q-generalization of the operators (1.1) was introduced as

Dn;q(f ; x) = [n + 1]q n X k=0 q kpn;k(q; x) Z 1 0 f (t)pn;k(q; qt)dqt; (1.2) where pn;k(q; x) = n k _qx k_{(1 x)}n k q .

The rate of convergence of the operators (1.2) was discussed by Zeng et al. [5]. In 2014, Mishra and Patel [6, 7] introduced the generalization due to Stancu

Received by the editors: July 12, 2017, Accepted: September 09, 2017. 2010 Mathematics Subject Classi…cation. 41A25–41A35.

Key words and phrases. q-integers; q-Durrmeyer operators; q-derivative, asymptotic formula.

c 2 0 1 8 A n ka ra U n ive rsity. C o m m u n ic a tio n s Fa c u lty o f S c ie n c e s U n ive rs ity o f A n ka ra -S e rie s A 1 M a t h e m a tic s a n d S t a tis tic s . C o m m u n ic a tio n s d e la Fa c u lté d e s S c ie n c e s d e l’U n ive rs ité d ’A n ka ra -S é rie s A 1 M a t h e m a tic s a n d S t a tis t ic s .

and proved Voronovskaja type asymptotic formula and various other approxima-tion properties of the q-Durrmeyer-Stancu operators. Here, in this manuscript, we establish Voronovskaja type asymptotic formula for function having q-derivative.

2. Estimation of moments and Asymptotic formula In the sequel, we shall need the following auxiliary results:

Theorem 1 _{([8]). If m-th (m > 0; m 2 N) order moments of operator (1.2) is} de…ned as D_n;mq (x) = Dn;q(tm; x) = [n + 1]q n X k=0 q kpn;k(q; x) Z 1 0 pn;k(q; qt)tmdqt; x 2 [0; 1];

then Dq_n;0(x) = 1 and for n > m + 2, we have the following recurrence relation, [n+m+2]qDn;m+1q (x) = ([m+1]q+qm+1x[n]q)Dqn;m(x)+x(1 x)qm+1Dq(Dqn;m(x)):

To establish asymptotic formula for functions having q-derivative, it is necessary to compute moments of …rst to fourth degree. Using above theorem one can have …rst, second, third and fourth order moments. The …rst three moments of Lemma 1 was also established in [4].

Lemma 1_{([4, 8]). For all x 2 [0; 1], n = 1; 2; : : : and 0 < q < 1, we have}

Dn;q (1; x) = 1; Dn;q (t; x) =1+qx[n]q_[n+2]q ; Dn;q (t2 ; x) =q3 x2 [n]q [n]q 1 +(1+q)2 qx[n]q +1+q [n+3]q [n+2]q ; Dn;q (t3 ; x) =q9 x3 [n]q [n 1]q [n 2]q +x2 q4[3]2q [n]q [n 1]q +xq[2]q [3]2q [n]q +[3]q [2]q [n+4]q [n+3]q [n+2]q ; Dn;q (t4 ; x) =q16 x4 [n]q [n 1]q [n 2]q [n 3]q +q9 x3[4]2q [n]q [n 1]q [n 2]q [n+5]q [n+4]q [n+3]q [n+2]q +q4 x2 [2]q [3] 2 q (1+q2 )2[n]q[n 1]q+qx[2]q[3]q[4]2q [n]q +[2]q [3]q [4]q [n+5]q [n+4]q [n+3]q [n+2]q

Lemma 2. _{For all x 2 [0; 1], n = 1; 2; : : : and 0 < q < 1, we have}

Dn;q (t x)q ; x =1 1+qn+1 x_[n+2]q ; Dn;q (t x)2_{q ; x} =qx2 [2]q [n] 2 q (1 q)2 q2 +[n]q (2q3 [3]q )+[3]q x[2]q ([3]q +q[n]q ( 1 q+q2 ))+[2]q [n+3]q [n+2]q ; Dn;q (t x)3_{q ; x} = q2 x3 ( q7 [n]q [n 1]q [n 2]q [n+2]q [n+3]q [n+4]q q2 [3]q [n]q [n 1]q [n+2]q [n+3]q + [2]q [n]q q[n+2]q [n+2]q [n+3]q [n+4]q ) + qx2 8 < : q3 [3]2_{q [n]q [n} 1]q [n+2]q [n+3]q [n+4]q [2]2_{q [3]q [n]q} [n+2]q [n+3]q+ [2]q [n+2]q 9 = ; + x[2]q [3]q _{[n+2]q [n+3]q [n+4]q}q[3]q [n]q [n+4]q + [3]q [2]q [n+2]q [n+3]q [n+4]q; Dn;q (t x)4_{q ; x} = q4 x4 8 < : q12 [n]q [n 1]q [n 2]q [n 3]q [n+5]q [n+4]q [n+3]q [n+2]q q5 [4]q [n]q [n 1]q [n 2]q [n+4]q [n+3]q [n+2]q + q [5]q +q2 [n]q[n 1]q [n+3]q [n+2]q [4]q [n]q [n+2]q + q2 9 = ; + x3 q2 8 < : q7 [4]2_{q [n]q [n} 1]q [n 2]q [n+5]q [n+4]q [n+3]q [n+2]q q2 [3]2_{q [4]q [n]q [n} 1]q [n+4]q [n+3]q [n+2]q+ [5]q +q2 [2]2_{q [n]q} [n+3]q [n+2]q q[4]q [n+2]q 9 = ; + qx2 8 < : q3 [2]q [3]2q (1+q2 )[n]q[n 1]q [n+5]q [n+4]q [n+3]q [n+2]q [2]q [3]2q [4]q [n]q [n+4]q [n+3]q [n+2]q+ [2]q [5]q +q2 [n+3]q [n+2]q 9 = ; + x [2]q [3]q [4]q q[4]q [n]q [n+5]q [n+5]q [n+4]q [n+3]q [n+2]q + [2]q [3]q [4]q [n+5]q [n+4]q [n+3]q [n+2]q.

Theorem 2. Let f be bounded and integrable on the interval [0; 1] and (qn) denote

a sequence such that 0 < qn< 1, qn ! 1 and qnn! c as n ! 1, where c is arbitrary

constant. Then we have for a point x 2 (0; 1), lim

n!1[n]qn[Dn;qn(f ; x) f (x)] = (1 2x) limn!1Dqnf (x) + x(1 x) limn!1D 2 qnf (x):

Proof: By q-Taylor formula [9] for f , we have f (t) = f (x) + Dqnf (x)(t x) + 1 [2]qn D_q2_nf (x)(t x)2_qn+ qn(x; t)(t x) 2 qn; for 0 < q < 1, where qn(x; t) = 8 > < > : f (t) f (x) Dqnf (x)(t x) 1 [2]qnD 2 qnf (x)(t x) 2 qn (t x)2 qn if x 6= t 0; if x = t: (2.1) We know that for n large enough

lim

t!x qn(x; t) = 0: (2.2)

That is for any > 0, there exists a > 0 such that

j qn(x; t)j ; (2.3)

for jt xj < and n su¢ ciently large. Using (2.1), we can write Dn;qn(f ; x) f (x) = Dqnf (x)Dn;qn((t x)qn; x)+ D2 qnf (x) [2]qn Dn;qn((t x) 2 qn; x)+E qn n (x); where E_nq(x) = [n + 1]qn n X k=0 q kpn;k(qn; x) Z 1 0 qn(x; t)pn;k(qn; qnt) (t x) 2 qndqnt: By Lemma 2, we have lim n!1[n]qnDn;qn((t x)qn; x) = (1 2x) and limn!1[n]qnDn;qn((t x) 2 qn; x) = 2x(1 x):

In order to complete the proof of the theorem, it is su¢ cient to show that limn!1[n]qnE

qn

n (x) = 0. We proceed as follows: Let

Pqn n;1(x) = [n]qn[n+1]qn n X k=0 qn kpn;k(qn; x) Z 1 0 qn(x; t)pn;k(qn; qnt) (t x) 2 qn x(t)dqnt and Pqn n;2(x) = [n]qn[n + 1]qn n X k=0 qn kpn;k(qn; x) Z 1 0 qn(x; t)pn;k(qn; qnt) (t x) 2 qn(1 x(t)) dqnt;

so that [n]qnE qn n (x) P qn n;1(x) + P qn n;2(x);

where _{x(t) is the characteristic function of the interval ft : jt xj < g.} It follows from (2.3) that

Pqn

n;1(x) = 2 x(1 x) as n ! 1:

If jt xj , then j qn(x; t)j M

2(t x)2, where M > 0 is a constant. Since

(t x)2 = t q2_nx + q_n2x x t q_n3x + q_n3x x = t q2nx t qn3x + x(qn3 1) t qn2x + x(qn2 1) t qn2x +x2(q_n2 1)(q_n2 q_n3) + x2(q_n2 1)(q_n3 1); we have jPqn n;2(x)j M 2 [n]qnDn;qn((t x) 4 qn; x) + x(2 q 2 n qn3)[n]qnDn;qn((t x) 3 qn; x) +x2(q_n2 1)2[n]qnDn;qn((t x) 2 qn; x) :

Using Lemma 2, we have Dn;qn((t x) 4 qn; x) C1 [n]3 qn ; Dn;qn((t x) 3 qn; x) C2 [n]2 qn and Dn;qn((t x) 2 qn; x) C3 [n]qn ; and the desired result is obtained.

Corollary 1. Let f be bounded and integrable on the interval [0; 1] and (qn) denote

a sequence such that 0 < qn< 1, qn ! 1 and qnn! c as n ! 1, where c is arbitrary

constant. Suppose that the …rst and second derivatives f0_{(x) and f}00_{(x) exist at a}

point x 2 (0; 1). Then, we have, for a point x 2 (0; 1) lim

n!1[n]qn[Dn;qn(f ; x) f (x)] = (1 2x)f

0_{(x) + x(1 x)f}00_(x):

3. Asymptotic formula for the Durrmeyer-Stancu Operators In the year 1968, Stancu [10] generalized Bernstein operators and discussed its approximation properties. After that many researchers gave Stancu type general-ization of several operators on …nite and in…nite intervals. We refer the readers to [11, 12, 13, 14, 15, 16, 17, 18, 19, 20] and the references there in. As mention in the introduction, Stancu generalization of q-Durrmeyer operators (1.2) was discussed by Mishra and Patel [6], which is de…ned as follows:

D_n;q; = [n + 1]q n X k=0 q kpn;k(q; x) Z 1 0 f [n]qt + [n]q+ pn;k(q; qt)dqt; (3.1)

where 0 and pn;k(q; x) as same as de…ned in (1.2). We shall need the

Lemma 3([7]). We have D ; n;q(1; x) = 1; Dn;q; (t; x) = [n]q+ [n+2]q+qx[n]2q [n+2]q([n]q+ ) ; D ; n;q(t2; x) = q3[n]3q([n]q 1)x2+((q(1+q)2+2 q4)[n]3q+2 q[3]q[n]2q)x ([n]q+ )2[n+2]q[n+3]q +_([n] 2 q+ )2 + (1+q+2 q3_)[n]2 q+2 [3]q[n]q ([n]q+ )2[n+2]q[n+3]q : Lemma 4([7]). We have Dn;q; (t x; x) = q[n]2q [n+2]q([n]q+ ) 1 x + [n]q+ [n+2]q [n+2]q([n]q+ ); D ; n;q((t x)2; x) = q4_[n]4 q q3[n]3q 2q[n]2q[n+3]q([n]q+ )+[n+2]q[n+3]q([n]q+ )2 ([n]q+ )2[n+2]q[n+3]q x 2 +q(1+q) 2_[n]3 q+2q [n]2q[n+3]q (2[n]q+2 [n+2]q)[n+3]q([n]q+ ) ([n]q+ )2[n+2]q[n+3]q x +(1+q)[n] 2 q+2 [n]q[n+3]q ([n]q+ )2[n+2]q[n+3]q :

Remark 1_{([7]). For all m 2 N [ f0g; 0} , we have the following recursive relation for the images of the monomials tm under D_n;q; in terms of Dn;q; j =

0; 1; 2; : : : ; m, as D_n;q; (tm; x) = m X j=0 m j [n]j q m j ([n]q+ )m Dn;q(tj; x):

Now, let us compute the moments and central moments of order 3 and 4 for the operators (3.1) in the following manner:

Dn;q; (t3; x) = q9_[n]4 q[n 1]q[n 2]q ([n]q+ )3[n + 4]q[n + 3]q[n + 2]q x3+q 4_[n]3 q[n 1]q [3]2q[n]q+ [n + 4]q ([n]q+ )3[n + 4]q[n + 3]q[n + 2]q x2 +q[n] 2 q [2]q[3]2q[n]2q+ [2]q2[n]q[n + 4]q+ 2[n + 4]q[n + 3]q ([n]q+ )3[n + 4]q[n + 3]q[n + 2]q x +[n] 3 q[3]q[2]q+ [2]q[n]2q[n + 4]q+ 2[n]q+ 3[n + 2]q [n + 4]q[n + 3]q ([n]q+ )3[n + 4]q[n + 3]q[n + 2]q : Also, D_{n;q (t}; 4; x) = q16 [n]5_{q [n} 1]q [n 2]q [n 3]q ([n]q + )4 [n + 5]q [n + 4]q [n + 3]q [n + 2]q x4+ q9 [n]4_{q [n} 1]q [n 2]q [4]2q [n]q + [n + 5]q ([n]q + )4 [n + 5]q [n + 4]q [n + 3]q [n + 2]q x3 + q4[n]3_{q [n} 1]q 8 < : [2]q [3]2q (1 + q2 )2[n]2q + [3]2q [n]q [n + 5]q + 2 [n + 4]q[n + 5]q ([n]q + )4 [n + 5]q [n + 4]q [n + 3]q [n + 2]q 9 = ;x 2 + q[n]2_q [2]q [3]q [4]2q [n]3q + [2]q [3]2q [n]2q [n + 5]q + [2]2q 2 [n]q[n + 4]q[n + 5]q + 3[n + 3]q[n + 4]q[n + 5]q ([n]q + )4 [n + 5]q [n + 4]q [n + 3]q [n + 2]q x +[4]q [3]q [2]q [n]4 + [3]q[2]q[n]3q [n + 5]q + 2 [2]q[n]2q [n + 4]q [n + 5]q ([n]q + )4 [n + 5]q [n + 4]q [n + 3]q [n + 2]q + 3 [n]q + 4[n + 2]q ([n]q + )4 [n + 2]q :

Now, using the identity (t

x)

3

q

= t

3

[3]

q

xt

2

+ q[2]

q

x

2

t

q

3

x

3

and linear

properties of the operators D

n;q;

, we get

D_n;q; (t x)3_{q ; x} = q2 2 4 q7 [n]4q [n 1]q [n 2]q ([n]q + )3 [n + 4]q [n + 3]q [n + 2]q q2 [3]q [n]3q [n 1]q ([n]q + )2 [n + 2]q [n + 3]q + [2]q [n] 2 q [n + 2]q [n]q + q 3 5 x3 +q 2 4q3 [n]3q [n 1]q [n]q [3] 2 q + [n + 4]q ([n]q + )3 [n + 4]q [n + 3]q [n + 2]q [3]q [2]2_{q + 2 q}3 [n]3_{q + 2 [3]q [n]}2_q ([n]q + )2 [n + 2]q [n + 3]q +[2]q ([n]q + [n + 2]q ) [n + 2]q [n]q + 3 5 x2 + 2 4q[n]2q [2]q [3] 2 q [n]2q + [2]2q [n]q [n + 4]q + 2 [n + 4]q[n + 3]q ([n]q + )3 [n + 4]q [n + 3]q [n + 2]q [3]q 2 ([n]q + )2 (1 + q + 2 q3 )[3]q [n]2q + 2 [3]2q [n]q ([n]q + )2 [n + 2]q [n + 3]q 3 5 x + [n]3_{q [3]q [2]q +} [2]q [n]2q [n + 4]q ([n]q + )3 [n + 4]q [n + 3]q [n + 2]q + [n]q 2 + 3[n + 2]q [n + 2]q ([n]q + )3:

Finally, using identity (t

x)

4_q

= t

4

[4]

q

xt

3

+ q [5]

q

+ q

2

x

2

t

2

q

3

x

3

[4]

q

t +

q

6

x

4

, we have

D_n;q; (t x)4_{q ; x} = q4 2 4 q12 [n]5q [n 1]q [n 2]q [n 3]q ([n]q + )4 [n + 5]q [n + 4]q [n + 3]q [n + 2]q q5 [4]q [n]4q [n 1]q [n 2]q ([n]q + )3 [n + 4]q [n + 3]q [n + 2]q + q [5]q + q2 [n]3_{q [n} 1]q ([n]q + )2 [n + 2]q [n + 3]q [4]q [n]2q [n + 2]q [n]q + + q2 3 5 x4 +q2 2 4q7 [n]4q [n 1]q [n 2]q [4] 2 q [n]q + [n + 5]q ([n]q + )4 [n + 5]q [n + 4]q [n + 3]q [n + 2]q q2 [4]q [n]3q [n 1]q [3]2q [n]q + [n + 4]q ([n]q + )3 [n + 4]q [n + 3]q [n + 2]q + [5]q + q 2 _[2]2 q + 2 q3 [n]3q + 2 [3]q [n]2q ([n]q + )2 [n + 2]q [n + 3]q q[4]q [n]q + [n + 2]q [n + 2]q [n]q + 3 5 x3 +q 2 4q3[n]3_{q [n} 1]q 8 < : [2]q [3]2q (1 + q2 )2[n]2q + [3]2q [n]q [n + 5]q + 2 [n + 4]q[n + 5]q ([n]q + )4 [n + 5]q [n + 4]q [n + 3]q [n + 2]q 9 = ; [4]q [n]2q [2]q [3]2q [n]2q + [2]q [n]q [n + 4]q + 2 [n + 4]q[n + 3]q ([n]q + )3 [n + 4]q [n + 3]q [n + 2]q + 2 _{[5]q + q}2 ([n]q + )2 + [5]q + q2 (1 + q + 2 q3 )[n]2_{q + 2 [3]q [n]q} ([n]q + )2 [n + 2]q [n + 3]q 3 5 x2 + 2 4q[n]2q [2]q [3]q [4] 2 q [n]3q + [2]q [3]2q [n]2q [n + 5]q + 2 [2]2q [n]q [n + 4]q [n + 5]q + 3 [n + 3]q[n + 4]q[n + 5]q ([n]q + )4 [n + 5]q [n + 4]q [n + 3]q [n + 2]q [n]3_{q [4]q [3]q [2]q} ([n]q + )3 [n + 4]q [n + 3]q [n + 2]q [4]q [2]q [n]2q ([n]q + )3 [n + 3]q [n + 2]q [4]q [n]q 2 + 3[4]q[n + 2]q [n + 2]q ([n]q + )3 3 5 x +[4]q [3]q [2]q [n]4 + [3]q[2]q[n]3q [n + 5]q + 2 [2]q[n]2q [n + 4]q [n + 5]q ([n]q + )4 [n + 5]q [n + 4]q [n + 3]q [n + 2]q + 3 [n]q + 4[n + 2]q ([n]q + )4 [n + 2]q :

Theorem 3.

Let f be bounded and integrable on the interval [0; 1] and let

(q

n

) denote a sequence such that 0 < q

n

< 1, q

n

! 1 and q

nn

! c as n ! 1,

where c is arbitrary constant. Then, we have, for a point x 2 (0; 1)

lim

n!1

[n]

qn

[D

; n;qn

(f ; x) f (x)] = (1+

(2+ )x) lim

_n!1

D

qn

f (x)+x(1 x) lim

n!1

D

2 qn

f (x):

The proof of the above lemma follows along the lines of the proof of

Corollary 2

([6]).

Let f be bounded and integrable on the interval [0; 1]

and let (q

n

) denote a sequence such that 0 < q

n

< 1, q

n

! 1 and q

nn

! c

as n ! 1, where c is arbitrary constant. Suppose that the …rst and second

derivatives f

0

(x) and f

00

_{(x) exist at a point x 2 (0; 1). Then, we have, for a}

point x 2 (0; 1),

lim

n!1

[n]

qn

[D

; n;qn

(f ; x)

f (x)] = (1 +

(2 + )x)f

0

_{(x) + x(1}

_x)f

00

_(x):

Remark 2.

Theorem 2 and Theorem 3, give asymptotic formula for

q-Durrmeyer operators and q-q-Durrmeyer-Stancu operators respectively. If f

has …rst and second derivatives, then lim

n!1

D

qn

f (x) = f

0

_{(x) and lim}

n!1

D

2

qn

f (x) =

f

00

(x). We obtain the results of Mishra and Patel [6, Theorem 5], which are

mentioned in Corollary 2. So our results are more general than the existing

ones.

Acknowledgments

We thank the two anonymous referees for valuable comments which led

to improvement of the paper and the editor for giving us an opportunity to

submit the revised version.

References

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de Bernstein modi…es, J. Approx. Theory, 32 (1981) 325–343.

[3] Lupas, A., A q-analogue of the Bernstein operator, University of Cluj-Napoca, Seminar on numerical and statistical calculus, 9 (1987), 85-92, Calculus (Cluj-Napoca, 1987), Preprint, 87-9 Univ. Babes-Bolyai, Cluj. MR0956939 (90b:41026).

[4] Gupta, V., Some approximation properties of q-Durrmeyer operators, Appl. Math. Comp., 2008, 191(1), (2008) 172-178.

[5] Zeng, X. M., Lin, D. and Li, L., A note on approximation properties of q-Durrmeyer operators, Appl. Math. Comp., 216(3) (2010) 819–821.

[6] Mishra, V. N. and Patel, P., A short note on approximation properties of Stancu generaliza-tion of q-Durrmeyer operators, Fixed Point Th. Appl., 84(1) (2013) 5 pages.

[7] Mishra, V. N. and Patel, P., On generalized integral Bernstein operators based on q-integers, Appl. Math. Comp., 242 (2014) 931-944.

[8] Gupta, V. and Sharma, H. Recurrence formula and better approximation for q-Durrmeyer operators, Lobachevskii J. Math., 32(2) (2011) 140–145.

[9] De Sole, A.and Kac, V., On integral representations of q-gamma and q-beta functions, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei, (9) Mat. Appl., 16(1) (2005) 11-29.

[10] Stancu, D. D., Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pures Appl., 13(8) (1968) 1173-1194.

[11] Mohapatra R.N. and Walczak, Z., Remarks on a class of Szász-Mirakyan type operators, East J. Approx. 15(2) (2009) 197-206.

[12] Içöz, G.and Mohapatra, R. N., Approximation properties by q-Durrmeyer-Stancu operators. Anal. Theory Appl. 29(4) (2013) 373–383.

[13] Mishra, V. N. and Patel, P., Approximation by the Durrmeyer-Baskakov-Stancu operators, Lobachevskii J. Math., 34(3) (2013) 272–281.

[14] Mishra V. N. and Patel, P., The Durrmeyer type modi…cation of the q-Baskakov type oper-ators with two parameter and , Numerical Algorithms, 67(4) (2014) 753-769.

[15] Yurdakadim, T., Some Korovkin type results via power series method in modular spaces, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 65(2) (2016) 65-76.

[16] Karaisa, A. and Aral, A., Some approximation properties of Kontorovich variant of Chlodowsky operators based on q-integers, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 65(2) (2016) 97-119.

[17] Içöz, G. and Mohapatra, R. N., Weighted approximation properties of Stancu type modi…ca-tion of q-Szász-Durrmeyer operators, Commun. Ser. A1 Math. Stat, 65(1) (2016) 87-103. [18] Içöz, G. and Bayram, C., q-analogue of Mittag-Le- er operators, Miskolc Mathematical Notes

18(1), (2017), 211-221.

[19] Mishra, V. N., Khatri, K., Mishra, L.N. and Deemmala, Inverse result in simultaneous approx-imation by Baskakov-Durrmeyer-Stancu operators, Journal of Inequalities and Applications, 2013, (2013) 586. doi:10.1186/1029-242X-2013-586.

[20] Mishra, V. N., K Khatri, and Mishra, L. N., Statistical approximation by Kantorovich-type discrete q-Beta operators, Advances in Di¤ erence Equations, 345(1) (2013) doi:10.1186/1687-1847-2013-345.

Current address : Vishnu Narayan Mishra (Corresponding author): Department of Mathemat-ics, Indira Gandhi National Tribal University, Lalpur, Amarkantak 484 887, Madhya Pradesh, India

E-mail address : vishnu_narayanmishra@yahoo.co.in; vishnunarayanmishra@gmail.com ORCID Address: http://orcid.org/0000-0002-2159-7710

Current address : Prashantkumar Patel: Department of Mathematics, St. Xavier’s College (Autonomous), Ahmedabad-382350 (Gujarat), India

E-mail address : prashant225@gmail.com