International Journal of Analysis and Applications Volume 19, Number 3 (2021), 389-404 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-19-2021-389 ON A NEW APPROACH BY MODIFIED (p; q)-SZÁSZ-MIRAKYAN OPERATORS VISHNU NARAYAN MISHRA1,∗, ANKITA R. DEVDHARA2, KHURSHEED J. ANSARI3, SEDA KARATEKE4 1Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak 484 887, Madhya Pradesh, India 2Applied Mathematics and Humanities Department, SVNIT, Surat-395007, India 3 Department of Mathematics, College of Science, King Khalid University, 61413, Abha, Saudi Arabia 4Department of Mathematics and Computer Science, Faculty of Science and Letters, Istanbul Arel University, Istanbul-34537, Turkey ∗Corresponding author: vishnunarayanmishra@gmail.com Abstract. In this paper, we introduce a new type of (p; q) exponential function with some properties and a modified (p; q)-Szász-Mirakyan operators by virtue of this function by investigating approximation properties. We obtain moments of generalized (p; q)-Szász-Mirakyan operators. Furthermore, we derive direct results, rate of convergence, weighted approximation result, statistical convergence and Voronovskaya type result of these operators with numerical examples. Graphical representations reveal that modified (p; q)-Szász-Mirakyan operators have a better approximation to continuous functions than pioneer one. 1. Introduction Approximation theory is one of the oldest branches of mathematics. To approximate continuous functions with q-analogue of linear positive operators is significant application of q-calculus in approximation theory. Cieśliński [1] established alternative definition of q-exponential function. He defined q-exponential function Received November 29th, 2020; accepted January 4th, 2021; published April 9th, 2021. 2010 Mathematics Subject Classification. 46B28. Key words and phrases. (p; q)-Szász-Mirakyan operators; uniform convergence; Voronovskaya type result; statistical convergence. ©2021 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 389 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-389 Int. J. Anal. Appl. 19 (3) (2021) 390 using Cayley transformation. The main advantages of the new q-exponential function consist of better qual- itative properties i.e., its properties are more similar to properties of ez,z ∈ C [1]. Over the years, many research papers were developed on q-analogue of various linear positive operators and their approximation properties. Recently, in [2] research of Bernstein-Stancu operators on (p; q)- integers were performed and discussed uniform convergence and direct result of the operators. Eventually, in [3] (p; q)-analogue of Bernstein oper- ators was investigated and developed the same convergence. Acar [5] and Mursaleen et.al [4], [12] proposed (p; q)-generalization of Szász-Mirakyan operators and discussed uniform convergence, rate of convergence, Voronovskaya result in those papers. The motivation of recent work is developing a new type of (p; q) exponential function and utilizing this new exponential function to modify (p; q)-Szász-Mirakyan operators. We studied uniform convergence and statistical convergence of modified (p; q)-Szász-Mirakyan operators. In the first section, we discussed some sequences and rate of convergence of operators. We also proved Voronovskaya type result. In the last section, we present some graphical representations. Consider, 0 < q < p ≤ 1. The definition of (p; q)-integer is, {m}p,q = pm −qm p−q , m ∈ N,(1.1) {0}p,q = 0, and (p; q)−factorial is {m}p,q! = m∏ k=1 {k}p;q, m ∈ N {0}p;q! = 1.(1.2) (p; q)-exponential function is defined as [6] ep,q(z) = ∞∑ j=0 p j(j−1) 2 zj {j}p,q! ,(1.3) Ep,q(z) = ∞∑ j=0 q j(j−1) 2 zj {j}p,q! .(1.4) The (p; q)-exponential functions have following property: ep,q(z)Ep,q(−z) = Ep,q(z)ep,q(−z) = 1. Int. J. Anal. Appl. 19 (3) (2021) 391 Another way of defining two (p; q)-exponentials as infinite products is ep,q(z) = ∞∏ j=0 1 (pj −qj(p−q)z) ,(1.5) Ep,q(z) = ∞∏ j=0 (pj + qj(p−q)z).(1.6) 2. New type of (p; q)-exponential function New (p; q)-exponential function is determined as (2.1) Ep,q(z) = ep,q(z/2)Ep,q(z/2) = ∞∏ j=0 pj + qj(p−q)z 2 pj −qj(p−q)z 2 , ep,q(z),Ep,q(z) are usual (p; q)-exponential functions. Theorem 2.1. (p; q)-exponential function Ep,q(z) is analytic in |z| < Rp,q (2.2) Ep,q(z) = ∞∑ j=0 zj [j]p,q! , |z| < Rp,q where, Rp,q =   2 p−q , 0 < q < p < 1. 2q q−p, q > p. ∞, p = q = 1. (2.3) [j]p,q = pj −qj p−q . 2 pj−1 + qj−1 = {j}p,q. 2 pj−1 + qj−1 ,(2.4) and [j]p,q! = j∏ m=1 [m]p,q = j∏ m=1 {m}p,q 2 pj−1 + qj−1 = {j}p,q! 2j∏j−1 m=0(p m + qm) .(2.5) Proof. Since (1.5) and (1.6) are absolutely convergent in |z| < 1, multiplying (1.5) and (1.6), we obtain ep,q(z/2)Ep,q(z/2) = ∞∑ n=0 ∞∑ k=0 p n(n−1) 2 q k(k−1) 2 (z 2 )n+k {n}p,q!{k}p,q! = ∞∑ j=0 (z 2 )j {j}p,q! ∞∑ k=0 p (j−k)(j−k−1) 2 q k(k−1) 2 {j}p,q! {j −k}p,q!{k}p,q! .(2.6) Using formula for the (p; q)-binomial coefficients [7], we have (2.7) j−1∏ r=0 (pr + xqr) = j∑ k=0 p (j−k)(j−k−1) 2 q k(k−1) 2 {j}p,q! {j −k}p,q!{k}p,q! xk. In particular, (2.8) j∑ k=0 p (j−k)(j−k−1) 2 q k(k−1) 2 {j}p,q! {j −k}p,q!{k}p,q! = (1 + 1)(p + q)...(pj−1 + qj−1). Int. J. Anal. Appl. 19 (3) (2021) 392 Substituting (2.8) into (2.6), we obtain (2.2), where [j]p,q defined as in (2.4). To get the radius of convergence, lim n→∞ ∣∣∣∣ zj+1[j + 1]p,q! ∣∣∣∣ ∣∣∣∣[j]p,q!zj ∣∣∣∣ = lim n→∞ ∣∣∣∣ z[j + 1]p,q ∣∣∣∣ =   (p−q)|z| 2 , for q < p, (q−p)|z| 2q , for q > p. (2.9) Applying d’Alembert’s test on (2.9), we obtain (p,q 6= 1) the radius of convergence (2.3). For p = q = 1, Ep,q(z) is ez, thus R1 = ∞. � Theorem 2.2. The Ep,q(z) satisfies the following properties: (2.10) 1. Ep,q(−z) = (Ep,q(z))−1, 2. |Ep,q(ix)| = 1. Proof. The first part of above equation (2.10) directly comes from the definition of Ep,q(z). That implies, Ep,q(z) = Ep,q(z̄). Then, |Ep,q(ix)|2 = Ep,q(ix) Ep,q(ix) = 1. � The above (p; q)-exponential function (2.1) has more improved properties similar to function ez. The definition of (p,q)-Szász-Mirakyan operators in [5] is (2.11) Am,p,q(f; x) = ∞∑ j=0 1 Ep,q({m}p,qx) q j(j−1) 2 {m}jp,qxj {j}p,q! f ( {j}p,q qj−2{m}p,q ) . Acar obtained moments, uniform convergence and Voronovskaya result of the above operators. We define a different sort of modified (p,q)-Szász-Mirakyan operators via new (p; q)-exponential function for f ∈ C[0,∞] in (2.1) is (2.12) Sn,p,q(f; x) = 1 Ep,q([n]p,qx) ∞∑ k=0 [n]kp,qx k [k]p,q! f ( [k]p,q [n]p,q ) , where 0 < q < p ≤ 1, n ∈ N, 0 ≤ x < 2 (p−q)[n]p,q = p n−1+qn−1 pn−qn . Remark 2.1. We choose an x between 0 and p n−1+qn−1 pn−qn because we want Ep,q([n]p,qx) to be convergent. Remark 2.2. From calculations for every k ∈ N; [k]p,q [n]p,q = (pk−qk)(pn−1+qn−1) (pn−qn)(pk−1+qk−1) , 0 ≤ [k]p,q [n]p,q < p n−1+qn−1 pn−qn , Int. J. Anal. Appl. 19 (3) (2021) 393 Then we consider (2.13) sn(p,q; x) = 1 Ep,q([n]p,qx) . [n]kp,qx k [k]p,q! . Clearly, sn(p,q; x) is positive for 0 < q < p ≤ 1, n ∈ N and every 0 ≤ x < 2(p−q)[n]p,q . The operator Sn,p,q is linear and positive. 3. Moments of Sn,p,q Here, we determine approximation moments of operators (2.12). Lemma 3.1. For n ∈ N and 0 < q < p ≤ 1. Below equalities are verified: Sn,p,q(1; x) = 1,(3.1) Sn,p,q(t; x) = x,(3.2) Sn,p,q(t 2; x) = x2 + x [n]p,q ,(3.3) Sn,p,q(t 3; x) = x3 + 3x2 [n]p,q + x [n]2p,q .(3.4) Proof. The result is obvious for Sn,p,q(1; x). Now Sn,p,q(t; x) = 1 Ep,q([n]p,qx) ∞∑ k=0 [n]kp,qx k [k]p,q! ( [k]p,q [n]p,q ) = 1 Ep,q([n]p,qx) ∞∑ k=1 [n]k−1p,q x k [k − 1]p,q! = x. and Sn,p,q(t 2; x) = 1 Ep,q([n]p,qx) ∞∑ k=0 [n]kp,qx k [k]p,q! ( [k]p,q [n]p,q )2 = 1 Ep,q([n]p,qx) ∞∑ k=1 [n]k−1p,q x k [k − 1]p,q! ( 1 [n]p,q ) + 1 Ep,q([n]p,qx) ∞∑ k=2 [n]k−2p,q x k [k − 2]p,q! = x [n]p,q + x2. Also Sn,p,q(t 3; x) = 1 Ep,q([n]p,qx) ∞∑ k=0 [n]kp,qx k [k]p,q! ( [k]p,q [n]p,q )3 = 1 Ep,q([n]p,qx) ∞∑ k=0 [n]kp,qx k [k]p,q! ( [k]3p,q − 3[k]2p,q + 2[k]p,q + 3[k]2p,q − 2[k]p,q [n]3p,q ) = x3 + 3x2 [n]p,q + x [n]2p,q . Int. J. Anal. Appl. 19 (3) (2021) 394 � Central moments are: Sn,p,q(t−x; x) = 0,(3.5) Sn,p,q((t−x)2; x) = x [n]p,q .(3.6) Remark 3.1. From our choice of p and q, we know that limn→∞[n]p,q = 1 p−q . But, to get the uniform convergence and other results of approximation for Sn,p,q we suppose that sequences qn ∈ (0,pn); pn ∈ (qn, 1] such that qn, pn → 1 and pNn → a, qN ′ n → b as n tending to infinity, i.e., limn→∞ 1/[n]p,q = 0. Now, we have uniform convergence of new kind of operators for all f ∈ Cϑ[0,∞) where Cϑ[0,∞) = {f ∈ C[0,∞) : |f(t)| ≤ A(1 + t)ϑ} for A > 0, ϑ > 0 and ‖f‖ = supx≥0 |f(x)| 1+x2 . Theorem 3.1. Let (pn) and (qn) be the sequences such that pn → 1, qn → 1 and pNn → a, qN ′ n → b as n tending to infinity then for each f ∈ Cϑ[0,∞) (3.7) lim n→∞ ‖Sn,p,q(f) −f‖ϑ = 0. Proof. From Korovkin’s result, we put evidence that lim n→∞ ‖Sn,p,q(ti) −xi‖ϑ = 0, i = 0, 1, 2. Since Sn,p,q(1; x) = 1, the result is clear for i = 0. For i = 1 lim n→∞ ‖Sn,p,q(t) −x‖ϑ = lim n→∞ ‖x−x‖ϑ = 0. and for i = 2 lim n→∞ ‖Sn,p,q(t2) −x2‖ϑ = lim n→∞ ‖x2 + x [n]p,q −x2‖ϑ = 0. Hence Sn,p,q(f; x) is uniformly convergent to f ∈ Cϑ[0,∞]. � Example 3.1. For p = 0.99 and q = 0.96, sequences of Sn,p,q defined by (2.12) is convergent to f(x) = x2 −5x + 10 (Fig. 1) and g(x) = x3 −x + 1 (Fig. 2) with increasing values of n (n = 10, 20, 30) respectively. Int. J. Anal. Appl. 19 (3) (2021) 395 Figure 1. Approximation to f by Sn,p,q for n = 10, 20, 30. Figure 2. Approximation to g by Sn,p,q for n = 10, 20, 30. Int. J. Anal. Appl. 19 (3) (2021) 396 Example 3.2. For different choices of p and q, the sequence of operators Sn,p,q defined by (2.12) is conver- gent to f(x) = x2 − 5x + 10 (Fig. 3) and g(x) = x3 −x + 1 (Fig. 4) with n = 50 respectively. Figure 3. Approximation to f by Sn,p,q for n = 50. Figure 4. Approximation to g by Sn,p,q for n = 50. 4. Some consequences For this section, we provide several results on local approximation for Sn,p,q(f; x). Here, Cb[0,∞) is the set of bounded, continuous functions f on [0,∞). Attached norm on Cb[0,∞) is defined by‖f‖ = sup x∈[0,∞) |f(x)|. Peetre’s K-functional is given by K2(f,δ) = inf h∈W2 {‖f −h‖ + δ‖h′′‖} where W2 = {h ∈ Cb[0,∞) : h′,h′′ ∈ Cb[0,∞)}. From ( [9], p.177), there exists A > 0 such that K2(f,δ) ≤ Aω2(f,δ 1/2),δ > 0, where ω2(f,δ 1/2) = sup 0<η<δ1/2,x∈[0,∞) |f(x + 2η) − 2f(x + η) + f(x)| is the second order modulus of continuity of functions f in Cb[0,∞). The first order modulus of continuity of function f ∈ Cb[0,∞) is defined by ω(f,δ1/2) = sup 0<η<δ1/2,x∈[0,∞) |f(x + η) −f(x)|. Int. J. Anal. Appl. 19 (3) (2021) 397 Theorem 4.1. Let 0 < q < 1 and p ∈ (q, 1]. The operators Sn,p,q map from Cb into Cb. Also, the following inequality is satisfied. (4.1) ‖Sn,p,q(f; x)‖Cb ≤‖f‖Cb. Proof. From the definition of Sn,p,q(f; x), |Sn,p,q(f; x)| ≤ 1 Ep,q([n]p,qx) ∞∑ k=0 [n]kp,qx k [k]p,q! ∣∣∣∣f ( [k]p,q [n]p,q )∣∣∣∣. Applying supremum to both sides here sup x≥0 |Sn,p,q(f; x)| ≤ sup x≥0 |f(x)| 1 Ep,q Sn,p,q(1; x). One has ‖Sn,p,q(f; x)‖Cb ≤‖f‖Cb. � Theorem 4.2. Let (pn) and (qn) be the sequences such that pn → 1, qn → 1 and pNn → a, qN ′ n → b as n tending to infinity. Then for f ∈ Cb[0,∞), there exists A > 0 such that (4.2) |Sn,p,q(f; x) −f(x)| ≤ Aω2 ( f, √ x [n]p,q ) . Proof. For h ∈ W2, using Taylor’s expansion h(t) = h(x) + (t−x)h′(x) + ∫ t x (t−u)h′′(u)du. Now |Sn,p,qh(t) −h(x)| ≤ 12‖h ′′‖Sn,p,q((t−x)2; x). Also |Sn,p,q(f; x)| ≤ ‖f‖. Hence |Sn,p,q(f; x) −f(x)| ≤ |Sn,p,q((f −h)(x); x) − (f −h)(x)| + |Sn,p,q(h; x) −h(x)| ≤ 2‖f −h‖ + 1 2 ‖h′′‖Sn,p,q((t−x)2; x). Taking infimum of the right hand side of above inequality for all h ∈ W2, |Sn,p,q(f; x) −f(x)| ≤ 2K2 ( f; 1 4 x [n]p,q ) . Since ω2(f,λδ) ≤ (λ + 1)2ω2(f; δ), |Sn,p,q(f; x) −f(x)| ≤ Aω2 ( f, √ x [n]p,q ) . Int. J. Anal. Appl. 19 (3) (2021) 398 � 5. Rate of convergence Suppose that C[0,∞) is set of all continuous functions on [0,∞) and consider following sets: Cϑ[0,∞) = {f ∈ C[0,∞) : |f(t)| ≤ A(1 + t)ϑ} for A > 0, ϑ > 0 and C∗ϑ[0,∞) = {f ∈ Cϑ[0,∞) : limx→∞ |f(x)| 1+x2 < ∞}. The first order modulus of continuity on [0,a] is defined as ωa(f,δ) = sup |t−x|≤δ sup 0≤x≤a |f(t) −f(x)|. Theorem 5.1. Suppose that f ∈ C∗ϑ[0,∞). Let (pn) and (qn) be the sequences such that pn → 1, qn → 1 and pNn → a, qN ′ n → b as n tending to infinity and ωa+1(f,δ) be the modulus of continuity on [0,a + 1] ⊂ [0,∞). Then (5.1) ‖Sn,p,q(f; x) −f(x)‖ϑ ≤ 6Af (1 + a2) a [n]p,q + 2ωa+1 ( f; √ a [n]p,q ) . Proof. For 0 ≤ x ≤ a; 0 ≤ t,∞; From [8] |f(t) −f(x)| ≤ 6Af (1 + a2)(t−x)2 + ωa+1(f; δ) ( |t−x| δ + 1 ) . Combining above inequality and Cauchy-Schwarz inequality, ‖Sn,p,q(f; x) −f(x)‖ϑ ≤ Sn,p,q(|(f; x) −f(x)|; x) ≤ 6Af (1 + a2)Sn,p,q((t−x)2; x) + ωa+1(f; δ) ( 1 + 1 δ2 Sn,p,q((t−x)2; x) ) . From the central moments of operators and for 0 ≤ x ≤ a Sn,p,q((t−x)2; x) = x[n]p,q ≤ a [n]p,q = ζn. Taking δ = √ ζ ‖Sn,p,q(f; x) −f(x)‖ϑ ≤ 6Af (1 + a2) a[n]p,q + 2ωa+1 ( f; √ a [n]p,q ) . � 6. Weighted approximation result Here, we discuss weighted approximation of Sn,p,q through polynomial weight over the space CM defined below. Consider that w0(x) = 1, wM (x) = (1 + x M )−1, (x ≥ 0,M ∈ N), Int. J. Anal. Appl. 19 (3) (2021) 399 CM = {f ∈ C[0,∞) : wM (f) is continuous, bounded and uniformly convergent.}. The norm is defined by ‖f‖M = sup x≥0 wM (x)|f(x)|. Also, we refer some results associated to Steklov means. For h > 0 it is defined in [11] fh(x) = ( 2 h )2 ∫ h/2 0 ∫ h/2 0 [2f(x + s + t) −f(x + 2(s + t))]dsdt. We have f(x) −fh(x) = ( 2 h )2 ∫ h/2 0 ∫ h/2 0 ∆2s+tf(x)dsdt, f′′h (x) = h −2[8∆2 h/2 f(x) − ∆2hf(x)], and hence (6.1) ‖f −fh‖M ≤ w2M (f,h), ‖f ′′ h‖M ≤ 9h −2w2M (f,h) Theorem 6.1. Suppose that Sn,p,q(f; x) is defined as in (2.12), where (pn) and (qn) are the sequences such that pn → 1, qn → 1 and pNn → a, qN ′ n → b as n tending to infinity. Let M ∈ N∗, then for f ∈ CM , (6.2) wM|Sn,p,q(f; x) −f(x)| ≤ Nmw2M ( f; √ x [n]p,q ) ,Nm > 0. Proof. For M = 0, the result comes from Theorem 4.2. For f ∈ CM, M ∈ N, wM|Sn,p,q(f; x) −f(x)| ≤ wM|Sn,p,q(|f −fh|; x)| + wM|Sn,p,q(fh; x) −fh(x)| + wM|fh(x) −f(x)|. From Theorem 4.1 and the first property of Steklov means, wM|Sn,p,q(|f −fh|; x)| ≤ ‖Sn,p,q(f −fh)‖M ≤‖f −fh‖M ≤ w2M (f,h).(6.3) Also, by Taylor’s expansion wM|Sn,p,q(fh; x) −fh(x)| ≤ ‖f′h‖MSn,p,q((t−x); x) + 1 2 ‖f′′h‖MSn,p,q((t−x) 2; x). From moments of operators Sn,p,q and the second property of Steklov means wM|Sn,p,q(fh; x) −fh(x)| ≤ 9 2h2 w2M (f,h)Sn,p,q((t−x) 2; x) ≤ 9 2h2 w2M (f,h)Sn,p,q((t−x) 2; x) ≤ 9 2h2 w2M (f,h) x [n]p,q .(6.4) Int. J. Anal. Appl. 19 (3) (2021) 400 Setting h = √ x [n]p,q and from (6.1), (6.3), (6.4), we get wM|Sn,p,q(f; x) −f(x)| ≤ Nmw2M ( f; √ x [n]p,q ) . � 7. Statistical convergence In this section, we obtain statistical convergence for new modified (p; q)-Szász-Mirakyan operators. We need the following theorem [10] to prove statistical convergence of the operators on H′ and we set all real valued functions on real-valued functions on [0,∞) with condition |f(x) −f(y)| ≤ ω(|x−y|). Theorem 7.1. Let Mn be the sequence of positive linear operators from H ′ into Cb[0,∞) with three condi- tions st− lim n→∞ ‖Mn(tj; x) −xj‖Cb = 0, j=0,1,2. Then st− lim n→∞ ‖Mn(f; x) −f‖Cb = 0. Now, the result on statistical convergence of the operators defined in (2.12). Theorem 7.2. Suppose that Sn,p,q(f; x) is defined as in (2.12), where (pn) and (qn) are the sequences such that pn → 1, qn → 1 and pNn → a, qN ′ n → b as n tending to infinity. Then (7.1) st− lim n→∞ ‖Sn,p,q(f; x) −f‖Cb = 0. Proof. From above theorem, we only have to prove that st− lim n→∞ ‖Sn,p,q(tj; x) −xj‖Cb = 0, j = 0, 1, 2. From the moments of Sn,p,q(f; x), it is obvious that the result is true for j = 0, 1. For j = 2, ‖Sn,p,q(t2; x) −x2‖Cb ≤ 1 [n]p,q . But st− lim n→∞ 1 [n]p,q = 0. We define U = {n : ‖Sn,p,q(t2; x) −x2‖Cb ≥ �} U1 = {n : 1 [n]p,q ≥ �}. Int. J. Anal. Appl. 19 (3) (2021) 401 Clearly, U ⊆ U1. Then δ{k ≤ n : ‖Sn,p,q(t2; x) −x2‖Cb ≥ �}≤ δ{k ≤ n : 1 [n]p,q ≥ �} But the right hand side of the above inequality is zero because st− lim n→∞ 1 [n]p,q = 0. Hence, st− lim n→∞ ‖Sn,p,q(t2; x) −x2‖Cb = 0. The theorem is proved. � 8. Voronovskaya type result Theorem 8.1. Let (pn) and (qn) be the sequences such that pn → 1, qn → 1 and pNn → a, qN ′ n → b as n tending to infinity then for each function f, f′, f′′ ∈ C∗ϑ[0,∞) (8.1) lim n→∞ [n]p,q[Sn,p,q(f; x) −f(x)] = x 2 f′′(x), is uniformly convergent on [0,a], a > 0. Proof. Consider Taylor’s formula on f ∈ C∗ϑ[0,∞) f(t) = f(x) + (t−x)f′x + 1 2 (t−x)2f′′(x) + P(t,x)(t−x)2, where P(t,x) is Peano’s remainder, P(t,x) → 0 as t → x. Now [n]p,q[Sn,p,q(f; x) −f(x)] = [n]p,qf′(x)Sn,p,q((t−x); x) + [n]p,q f′′(x) 2 Sn,p,q((t−x)2; x) + [n]p,qSn,p,q(P(t,x)(t−x)2; x). From Cauchy-Schwarz inequality, Sn,p,q(P(t,x)(t−x)2; x) ≤ √ Sn,p,q((P2(t,x); x) √ Sn,p,q((t−x)4; x) is satisfied. Since P(t,x) ∈ C∗ϑ[0,∞) and P(x,x) = 0, lim n→∞ Sn,p,q((P 2(t,x); x) = P2(x,x) = 0, uniformly convergent for x ∈ [0,a]. So lim n→∞ [n]p,qSn,p,q(P(t,x)(t−x)2; x) = 0 is obtained. Also lim n→∞ [n]p,qSn,p,q((t−x); x) = 0. Int. J. Anal. Appl. 19 (3) (2021) 402 and lim n→∞ [n]p,qSn,p,q((t−x)2; x) = x. Hence, lim n→∞ [n]p,q[Sn,p,q(f; x) −f(x)] = x 2 f′′(x). � 9. Numerical Examples Example 9.1. We compute the absolute error (A.E.) of Sn,p,q with the function f(x) = x 2 − 5x + 10 and g(x) = x3 −x + 1 for the different values of n taking x = 1 and x = 2 in Table 1 and Table 2, respectively. Also, the absolute error can be seen graphically in the Figure 5 and 6. Table 1. A.E. of operators and function at x = 1 n |Sn,p,qf −f| |Sn,p,qg −g| 10 0.1252 0.3914 20 0.0798 0.2458 30 0.0673 0.2064 40 0.0633 0.1940 50 0.0631 0.1934 Table 2. A.E. of operators and function at x = 2 n |Sn,p,qf −f| |Sn,p,qg −g| 10 0.2505 1.5342 20 0.1596 0.9704 30 0.1346 0.8165 40 0.1267 0.7682 50 0.1263 0.7657 Int. J. Anal. Appl. 19 (3) (2021) 403 Figure 5. Absolute error of Sn,p,q at x = 1. Figure 6. Absolute error of Sn,p,q at x = 2. 10. Acknowledgements The second author would like to express his gratitude to King Khalid University, Abha, Saudi Arabia, for providing administrative and technical support. Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. References [1] J.L. Cieśliński, Improved q-exponential and q-trigonometric functions, Appl. Math. Lett. 24 (2011), 2110–2114. [2] M. Mursaleen, K.J. Ansari and A. Khan, On (p; q)-analogue of Bernstein operators, Appl. Math. Comput. 266 (2015), 874-882. [Erratum: Appl. Math. Comput. 278 (2016) 70-71]. [3] M. Mursaleen, K.J. Ansari and A. 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