International Journal of Analysis and Applications ISSN 2291-8639 Volume 15, Number 2 (2017), 138-145 DOI: 10.28924/2291-8639-15-2017-138 ON THE (p,q)−STANCU GENERALIZATION OF A GENUINE BASKAKOV- DURRMEYER TYPE OPERATORS İSMET YÜKSEL∗, ÜLKÜ DİNLEMEZ KANTAR AND BİROL ALTIN Abstract. In this paper, we introduce a Stancu generalization of a genuine Baskakov- Durrmeyer type operators via (p, q)− integer. We investigate approximation properties of these operators. Fur- thermore, we study on the linear positive operators in a weighted space of functions and obtain the rate of these convergence using weighted modulus of continuity. 1. Introduction In the field of approximation theory, the quantum calculus has been studied for a long time. The generalization of (p,q)− calculus was introduced by Sahai and Yadav in [15]. Recently, a series of papers giving (p,q)− generalizations a sequence of linear positive operators have been published in [3, 4, 9–13]. Our aim is to give Stancu type generalization, via (p,q)− integer, defined by Agrawal and Thamer as follows Bn(f,x) = (n− 1) ∞∑ k=1 bn,k(x) ∞∫ 0 bn,k−1(t)f (t) dt + f (0) (1 + x) −n q , (1.1) where bn,k(x) = ( n + k − 1 k ) xk (1 + x)n+k . in [5]. We refer reader to [2] for unexplained terminologies and notations. 2. Preliminaries and notations Let’s give a table of some basic formulas, motivated from q−calculus, used in (p,q)−calculus as the following Table1 (p,q)−calculus Relation with q−calculus [n]p,q = pn−qn p−q [n]p,q = p n−1 [n]q/p [n]p,q ! = [1]p,q [2]p,q ... [n]p,q [n]p,q ! = p (n2) [n]q/p ! (a⊕ b)np,q= (a + b) (ap + bq)...(ap n−1 +bqn−1) (a⊕ b)np,q= p (n2) (a + b) n q/p dp,qf(x) = f(px) −f(qx) dqf(x) = f(x) −f(qx) Table 1 Recall that the beta function, introduced [14], in q− calculus is defined by Bq(n,k) = K(A,n) ∞/A∫ 0 tk−1 (1 + t)n+kq dqt, (2.1) Received 14th April, 2017; accepted 5th June, 2017; published 1st November, 2017. 2010 Mathematics Subject Classification. 41A25, 41A36. Key words and phrases. Baskakov-Durrmeyer operators; weighted approximation; rates of approximation, (p, q)−calculus. c©2017 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 138 BASKAKOV- DURRMEYER TYPE OPERATORS 139 where K(A,n) = An 1 + A ( 1 + 1 A )n q (1 + A) 1−s q , A > 0. (2.2) In the formula (2.2), K(A,n) = qn(n−1)/2 and K(A, 0) = 1 for n ∈ N. Inspiring the formula (2.1), we introduce (p,q)-beta functions Bp,q(n,k), as a generalization of Bq(n,k), A > 0 and n,k ∈ N\{0}, defined by Bp,q(k,n) = p (n2)q( k 2) ∞/A∫ 0 tk−1 (1 ⊕ t)n+kp,q dp,qt. (2.3) If p = 1 is replaced in (2.3), then the formula is reduced to (2.1). 3. Genuine type Stancu generalization via (p,q)−integer Let’s start to give a Stancu type (p,q)−generalization of these operators in (1.1). For 0 ≤ α,β and 0 < q < p ≤ 1, these operators are defined as follows; Bα,βp,q,n(f,x) = [n− 1]p,q ∞∑ k=1 { bn,k(p,q,x)p (n−1)2+kqk(k−1) × ∞/A∫ 0 bn,k−1(p,q,t)f ( [n]p,qt + α [n]p,q + β ) dp,qt   +f ( α [n]p,q + β ) p( n 2)(1 ⊕x)−np,q , (3.1) where bn,k(p,q,t) = [ n + k − 1 k ] p,q tk (1 ⊕ t)n+kp,q . If one replaces p = q = 1 and α = β = 0 in (3.1), then the operators Bα,βp,q,n are reduced to the operators Bnin (1.1). Similar type operators studied in [1, 7, 16]. To obtain our main results, we need calculating the values of Korovkin monomial functions. Lemma 3.1. The following equalities are satisfied for em(t) = t m, m = 0, 1, 2 and n > 3 Bα,βp,q,n(1,x) = 1, Bα,βp,q,n(t,x) = [n]2 p,q pq([n]p,q+β)[n−2]p,q x + α [n] p,q +β , Bα,βp,q,n(t 2,x) = [n]3 p,q [n+1] p,q p2q4([n]p,q+β) 2 [n−2] p,q [n−1] p,q x2 + ( pn−4[2] p,q [n]3 p,q q3([n]p,q+β) 2 [n−2]p,q[n−1]p,q + 2α[n]2 p,q pq([n]p,q+β) 2 [n−2]p,q ) x + α2( [n]pn,qn + β )2 . Proof. By the definition (p,q)−beta functions in (2.3), we obtain the estimate, ∞/A∫ 0 bn,k−1(p,q; t)t mdp,qt = [ n + k − 2 k − 1 ] p,q ∞/A∫ 0 tk+m−1 (1 ⊕ t)n+k−1p,q dp,qt = [k + m− 1]p,q! [n−m− 2]p,q! [k − 1]p,q! [n− 1]p,q!p (n−m−12 )q( k+m 2 ) . (3.2) 140 YÜKSEL, DİNLEMEZ KANTAR AND ALTIN If we apply the operators in (3.1) to the equality (3.2) for m = 0, we get Bα,βp,q,n(1,x) = [n− 1]p,q ∞∑ k=1 { bn,k(p,q; x)p (n−1)2+kqk(k−1) × ∞/A∫ 0 bn,k−1(p,q; t)dp,qt   + p( n 2)(1 ⊕x)−np,q = [n− 1]p,q ∞∑ k=1 bn,k(p,q; x)) p(n−1) 2+kqk(k−1) [n− 1]p,q p (n−12 )q( k 2) +p( n 2)(1 ⊕x)−np,q = ∞∑ k=0 [n + k − 1]p,q! [k]p,q! [n− 1]p,q! p( n 2)q( k 2) (px) k (1 ⊕x)n+kp,q = ∞∑ k=0 p( n 2)q( k 2)bn,k(p,q; px) = 1. And the proof of (i) is finished. With the direct computation, we obtain (ii) as follows: Bα,βp,q,n(t,x) = [n− 1]p,q ∞∑ k=1 { bn,k(p,q; x)p (n−1)2+kqk(k−1) × ∞/A∫ 0 bn,k−1(p,q; t) ( [n]p,qt + α [n]p,q + β ) dp,qt   + α [n]p,q + β p( n 2)(1 ⊕x)−np,q = [n]p,q [n]p,q + β ∞∑ k=1 bn,k(p,q; x) p(n−1) 2+kqk(k−1)[k] p,q [n−2] p,q p (n−22 )q( k+1 2 ) + α [n]p,q + β Bα,βp,q,n(1,x) = [n]2p,qx pq([n]p,q+β)[n−2]p,q ∞∑ k=0 p( n+1 2 )q( k 2)bn+1,k(p,q; px) + α [n]p,q + β = [n]2 p,q pq([n]p,q+β)[n−2]p,q x + α [n]p,q + β . For (iii), Bα,βp,q,n(t 2,x) = [n− 1]p,q ∞∑ k=1 { bn,k(p,q; x)p (n−1)2+kqk(k−1) = × ∞/A∫ 0 bn,k−1(p,q; t) ( [n]p,qt + α [n]p,q + β )2 dp,qt   + ( α [n]p,q + β )2 p( n 2)(1 ⊕x)−np,q . BASKAKOV- DURRMEYER TYPE OPERATORS 141 Using the equality [k]p,q = q k−s[s]p,q + p s[k −s]p,q, 0 ≤ s ≤ k, (3.3) we have Bα,βp,q,n(t 2,x) = [n] 2 p,q( [n]p,q + β )2 ∞∑ k=2 { [k] p,q(p 2[k−1] p,q +qk−1[2] p,q)p n2+3n+2k−10 2 q k2−5k−2 2 [n−2]p,q[n−3]p,q × bn,k(p,q; x)} + 2α [n]p,q [n]p,q + β ∞∑ k=1 [k]p,q [n− 2]p,q p n2+n+2k−4 2 q k2−3k 2 bn,k(p,q; x) + α2( [n]p,q + β )2 Bα,βp,q,n(1,x) Then Bα,βp,q,n(t 2,x) = [n]3p,q[n+1]p,qx 2 p2q4([n]p,q+β) 2 [n−2]p,q[n−3]p,q ∞∑ k=0 p( n+2 2 )q( k 2)bn+2,k(p,q; px) + ( pn−4[2]p,q[n] 2 p,q q3([n]p,q+β) 2 [n−2] p,q [n−1] p,q + 2α[n]2p,q pq([n]p,q+β) 2[n−2] p,q ) x × ∞∑ k=0 p( n+1 2 )q( k 2)bn+1,k(p,q; x) + α2( [n]p,q + β )2 Bα,βp,q,n(1,x). And so we have completed the proof of (iii). � Now we consider that B[0,∞) denotes the set of all bounded functions from [0,∞) to R, B[0,∞) is a normed space with the norm ‖f‖B = sup{|f(x)| : x ∈ [0,∞)} and CB[0,∞) denotes the subspace of all continuous functions in B[0,∞). We denote first modulus of continuity on finite interval [0,b], b > 0 ω[0,b](f,δ) = sup 0 0 where W2∞ = {g ∈ CB[0,∞) : g′,g′′ ∈ CB[0,∞)} . By [6, p. 177, Theorem 2.4], there exists a positive constant C such that K2(f,δ) ≤ Cω2(f, √ δ) where ω2(f, √ δ) = sup 0 α [n]pn,qn +β f(x) , 0 ≤ x ≤ α [n]pn,qn +β where σ(α,β,pn,qn,n; x) = pnqn ( [n]pn,qn + β ) [n− 2]pn,qn [n] 2 pn,qn ( x− α [n]pn,qn + β ) . Notice that these operators B̃α,βpn,qn,n are defined from the space C ∗ ρ[0,∞) into Bρ[0,∞). To sat- isfy hypothesis of Korovkin’s Theorem, we assume that lim n→∞ pnn and lim n→∞ qnn are real numbers when lim n→∞ pn = 1 and lim n→∞ qn = 1 for 0 < qn < pn ≤ 1 and n > 3. On the other hand, since the operators B̃α,βpn,qn,n(f,x) are defined as f(x) on the interval [ 0, α [n] pn,qn +β ] , it is enough to examine the approx- imation properties of these operators at the interval ( α [n]pn,qn +β ,∞ ) . The following lemma can be obtained with the help of Lemma 3.1. Lemma 3.2. The operators B̃α,βpn,qn,n satisfy the following equalities for x > α [n]pn,qn +β and em(t) = t m, m = 0, 1, 2 B̃α,βpn,qn,n(1,x) = 1, B̃α,βpn,qn,n(t,x) = x, B̃α,βpn,qn,n(t 2,x) = [n+1] pn,qn [n−2] pn,qn q2n[n−1]pn,qn [n]pn,qn x2 + ( −2α[n+1]pn,qn [n−2]pn,qn q2n([n]pn,qn +β)[n−1]pn,qn [n]pn,qn + pn−3n [2]pn,qn [n]pn,qn q2n([n]pn,qn +β)[n−1]pn,qn + 2α ([n]pn,qn +β) ) x− pn−3n [2]pn,qn [n] pn,qn α q2n([n]pn,qn +β) 2 [n−1]pn,qn . We need computing the second moment before giving our main results . Lemma 3.3. We have the following inequality B̃α,βpn,qn,n((t−x) 2 ,x) ≤ ( p3n −p3nq2n q2n + 10(α+β+1) q2n([n]pn,qn +β) ) x(x + 1) for x > α [n]pn,qn +β . Proof. By Lemma 3.2, we write the second moment as B̃α,βpn,qn,n((t−x) 2 ,x) = ( [n+1] pn,qn [n−2] pn,qn q2n[n−1]pn,qn [n]pn,qn − 1 ) x2 + ( −2α[n+1]pn,qn [n−2]pn,qn q2n([n]pn,qn +β)[n−1]pn,qn [n]pn,qn + pn−3n [2]pn,qn [n] pn,qn q2n([n]pn,qn +β)[n−1]pn,qn + 2α ([n]pn,qn +β) ) x − pn−3n [2]pn,qn [n] pn,qn α q2n([n]pn,qn +β) 2 [n−1] pn,qn ≤ ∣∣∣ [n+1]pn,qn [n−2]pn,qnq2n[n−1]pn,qn [n]pn,qn − 1 ∣∣∣x2 + ( pn−3n [2]pn,qn [n] pn,qn q2n([n]pn,qn +β)[n−1]pn,qn + 2α ([n]pn,qn +β) ) x. Considering the following equalities [n + 1]pn,qn = p 3 n [n− 2]pn,qn + q n−3 n [3]pn,qn , [n]pn,qn = p 2 n [n− 2]pn,qn + q n−2 n [2]pn,qn , [n− 1]pn,qn = pn [n− 2]pn,qn + q n−1 n [1]pn,qn , BASKAKOV- DURRMEYER TYPE OPERATORS 143 we get B̃α,βpn,qn,n((t−x) 2 ,x) ≤ (( p3n −p3nq2n ) q2n + [3]pn,qn [n]pn,qn ) x2 + ( pn−3n [2]pn,qn [n]pn,qn q2n([n]pn,qn +β)[n−1]pn,qn + 2α ([n]pn,qn +β) ) x ≤ ( p3n −p3nq2n q2n + 10(α+β+1) q2n([n]pn,qn +β) ) x(x + 1). And the proof of the Lemma 3.3 is now finished. � Thus we are ready to give direct results. Lemma 3.4. We have the inequality for every x ∈ [0,∞) and f′′ ∈ CB[0,∞)∣∣∣B̃α,βpn,qn,n(f,x) −f(x)∣∣∣ ≤ δα,βpn,qn,n(x)‖f′′‖B , where δα,βpn,qn,n(x) := ( p3n −p3nq2n q2n + 10(α+β+1) q2n([n]pn,qn +β) ) x(x + 1). Proof. Using Taylor’s expansion f(t) = f(x) + (t−x)f′(x) + t∫ x (t−u)f′′(u)du and the Lemma 3.2, we have the following equality B̃α,βpn,qn,n(f,x) −f(x) = B̃ α,β pn,qn,n   t∫ x (t−u)f′′(u)du; x   . On the other hand, combining the inequality∣∣∣∣∣∣ t∫ x (t−u)f′′(u)du ∣∣∣∣∣∣ ≤‖f′′‖B (t−x) 2 2 , and Lemma 3.3, we get ∣∣∣B̃α,βpn,qn,n(f,x) −f(x)∣∣∣ = ∣∣∣∣∣∣B̃α,βpn,qn,n   t∫ x (t−u)f′′(u)du,x   ∣∣∣∣∣∣ ≤ ‖f′′‖B 2 B̃α,βpn,qn,n((t−x) 2,x) ≤ ‖f′′‖B 2 ( p3n −p3nq2n q2n + 10(α+β+1) q2n([n]pn,qn +β) ) x(x + 1), as desired. � Theorem 3.1. We have the following inequality∣∣∣B̃α,βpn,qn,n(f,x) −f(x)∣∣∣ ≤ 2Cω2 ( f, √ δ α,β pn,qn,n(x) ) , where δα,βpn,qn,n(x) = ( p3n −p3nq2n q2n + 10(α+β+1) q2n([n]pn,qn +β) ) x(x + 1). Proof. For any g ∈ W2∞, we obtain the inequality∣∣∣B̃α,βpn,qn,n(f,x) −f(x)∣∣∣ ≤ ∣∣∣B̃α,βpn,qn,n(f −g,x) − (f −g) (x) + B̃α,βpn,qn,n(g,x) −g(x)∣∣∣ . 144 YÜKSEL, DİNLEMEZ KANTAR AND ALTIN Then, Lemma 3.4, we have∣∣∣B̃α,βpn,qn,n(f,x) −f(x)∣∣∣ ≤ 2‖f −g‖B + δα,βpn,qn,n(x)‖g′′‖B . Taking infimum over g ∈ W2∞ on the right side of the above inequality and using the inequality (3.5), we reach the desired result. � Theorem 3.2. For every f ∈ C∗ρ[0,∞), we have the following the limit lim n→∞ ∥∥∥B̃α,βpn,qn,n(f) −f∥∥∥ ρ = 0. Proof. From Lemma 3.2 , it is obvious that ∥∥∥B̃α,βpn,qn,n(e0) −e0∥∥∥ ρ = 0 and ∥∥∥B̃α,βpn,qn,n(e1) −e1∥∥∥ ρ = 0. We have ∥∥∥B̃α,βpn,qn,n(e2) −e2∥∥∥ ρ = B̃α,βpn,qn,n((t−x) 2 ,x) 1 + x2 ≤ sup x∈[0,∞)   ( p3n −p3nq2n q2n + 10(α+β+1) q2n([n]pn,qn +β) ) x(x + 1) 1 + x2   ≤ 2 ( p3n −p3nq2n q2n + 10(α+β+1) q2n([n]pn,qn +β) ) . Then we get lim n→∞ ∥∥∥B̃α,βpn,qn,n(e2) −e2∥∥∥ ρ = 0. Thus, from A. D. Gadzhiev’s Theorem in [8], we obtain the proof of Theorem 3.2. � Lemma 3.5. We assume that α [n] pn,qn +β < b. Then we have the following inequality∥∥∥B̃α,βpn,qn,n(f; x) −f(x)∥∥∥ C[0,b] ≤ N { (1 + b)2δα,βpn,qn,n(b) + ω[0,b+1](f; √ δ α,β pn,qn,n(b)) } , where δα,βpn,qn,n(x) = ( p3n −p3nq2n q2n + 10(α+β+1) q2n([n]pn,qn +β) ) x(x + 1), for every f ∈ C∗ρ[0,∞). Proof. Let x ∈ [ α [n]pn,qn +β ,b ] and t > b + 1. Since t−x > 1, we have |f(t) −f(x)| ≤ Kf (2 + (t−x + x)2 + x2) ≤ 3Kf (1 + b)2(t−x)2. (3.6) Let x ∈ [ α [n]pn,qn +β ,b ] , t < b + 1 and δ > 0. Then, we have |f(t) −f(x)| ≤ ( 1 + |t−x| δ ) ω[0,b+1](f,δ). (3.7) Due to (3.6) and (3.7), we can write |f(t) −f(x)| ≤ 3Kf (1 + b)2(t−x)2 + ( 1 + |t−x| δ ) ω[0,b+1](f,δ). After, using Cauchy- Schwarz’ s inequality , we get∣∣∣B̃α,βpn,qn,n(f; x) −f(x)∣∣∣ ≤ 3Kf (1 + b)2Bα,βpn,qn,n ( (t−x)2,x ) +ω[0,b+1](f; δ) [ 1 + 1 δ ( B̃α,βpn,qn,n ( (t−x)2,x ))1/2] ≤ 3Kf (1 + b)2δα,βpn,qn,n(x) + ω[0,b+1](f; δ) [ 1 + 1 δ ( δα,βpn,qn,n(x) )1/2] . BASKAKOV- DURRMEYER TYPE OPERATORS 145 Considering Lemma 3.3 and choosing δ2 := δα,βpn,qn,n(b) and N = max{3Kf, 2}. We reach the proof of Lemma 3.5. � Theorem 3.3. For every f ∈ C∗ρ[0,∞) and γ > 0 , we have the limit lim n→∞ sup x≥0 ∣∣∣B̃α,βpn,qn,n(f,x) −f(x)∣∣∣ 1 + x2+γ = 0. Proof. For γ > 0, f ∈ C∗ρ[0,∞) and b ≥ 1, using ( 3.4) the following inequality is satisfied sup x≥0 ∣∣∣B̃α,βpn,qn,n(f,x) −f(x)∣∣∣ 1 + x2+γ ≤ sup 0≤x