EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 5, 2017, 1067-1077 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global On a modification of Dunkl generalization of Szász Operators via q-calculus Vishnu Narayan Mishra1,∗, Shikha Pandey2, Idrees A. Khan3 1 Department of Mathematics, Indira Gandhi National Tribal University, Amarkantak, Madhya Pradesh, India. 2 Department of Applied Mathematics & Humanities, Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev Dumas Road, Surat -395 007 (Gujarat), India. 3 Department of Mathematics, Integral University, Lucknow-226026, Uttar Pradesh, India. Abstract. Theory of approximation is a very extensive field and study of approximation via q- calculus and (p, q)- calculus is of great mathematical interest with great practical importance. Positive approximation processes play an important role in approximation theory and appear in a very natural way dealing with approximation of continuous functions, especially one, which requires further qualitative properties such as monotonicity, convexity and shape preservation and so on. This paper deals with the q- form of Dunkl generalization of Szász - Beta type operators. Estimation of their moments and establishing basic approximation results which comprise weighted approximation and direct estimates in view of modulus of continuity is the aim of this paper. 2010 Mathematics Subject Classifications: Primary 41A25, 41A36, Secondary 33C45 Key Words and Phrases: Dunkl analogue, Szász-Beta operators, generalization of exponential function 1. Introduction and Preliminaries Approximation theory is the branch of mathematics where the focus of study is to work out a complicated function by easier to compute functions. In 1885, Weierstrass firstly obtained a significant result, which established the fact that the set of algebraic polynomials in the class of continuous real valued functions on a closed interval is dense. Weierstrass’s theorem has encouraged mathematicians over the years to give too much of their attention to pathological functions with a little bit of smoothness. This theorem was proved by various mathematicians such as Picard, Fejer, Landau and de la Vallee Poussin using singular integrals. Bernstein [5] gave the most effective proof using probabilistic method. In 1950, Szász ∗Corresponding author. Email addresses: vishnunarayanmishra@gmail.com, vishnu narayanmishra@yahoo.co.in (V.N. Mishra), sp1486@gmail.com (S. Pandey), idrees maths@yahoo.com (I.A. Khan) http://www.ejpam.com 1067 c© 2017 EJPAM All rights reserved. V.N. Mishra, S. Pandey, I.A. Khan / Eur. J. Pure Appl. Math, 10 (5) (2017), 1067-1077 1068 [21] proved that for a continuous function f defined in positive semi-axis, the following polynomial sequence converges to f(x), Sn(f ;x) := e−nx ∞∑ k=0 (nx)k k! f ( k n ) . (1) Thereafter mathematicians have introduced various operators which gives better approx- imation to continuous functions [See [2], [3],[17],[24], [25], [26] etc.] In 20th century, the study of quantum calculus began when Jackson [11] defined the q-integral in a systematic way. Later on De Sole and Kac [19] presented the integral representations of q-gamma and q-beta functions. q-calculus has important applications in number theory, combinatorics, orthogonal polynomials, hypergeometric functions, me- chanics, the theory of relativity and quantum theory. In approximation theory, appli- cation of q-calculus finds its way when Phillips [15] proposed q-Bernstein polynomials. Thereafter various mathematicians studied q-analogue of various operators. Different q- generalizations of Szász-Mirakjan operators were introduced and studied by Aral [4], Radu [16] and Mahmudov [13] for 0 < q ≤ 1 , [14] for q > 1. As a summation integral type of modification of q-Szász-Mirakyan operators, Gupta and Mahmudov [7] presented q-Szász-beta operators for 0 < q ≤ 1, f ∈ C[0,∞) as Bn,q(f ;x) = e−[n]qx ∞∑ k=0 ([n]qx)k [k]q! ∫ ∞/A 0 qk 2 tk Bq(k + 1, n)(1 + t)n+k+1 q f(t)dqt, A > 0, x ∈ [0,∞). (2) Adell et al. [10] had shown that linear positive operators having beta type probability distributions preserves shape properties (monotonocity and convexity), likewise other pos- itive linear operator, such as Bernstein, Szász and Baskakov operators. For polynomial approximation, Hermite polynomials forms a family of orthogonal polyno- mial sequence which is complete in the space of all polynomials. The generalized Hermite polynomials were defined by G. Szëgo in [[22], p380, Problem 25] as being orthogonal polynomials with respect to weight function |x|2µe−x2 , µ > −1/2 in (−∞,∞). In [18], M. Rosenblum has given the definition of generalized Hermite polynomial as, let Hµ n be the generalized Hermite polynomial of degree n, then for even values of n, Hµ 2m = (−1)m(2m)! Γ(µ+ 1 2) Γ(m+ µ+ 1 2) L µ− 1 2 m (x2) (3) and for odd values of n, Hµ 2m+1 = (−1)m(2m+ 1)! Γ(µ+ 3 2) Γ(m+ µ+ 3 2) xL µ+ 1 2 m (x2), (4) where Lγm is the γ−Laguerre polynomial of degree m. The generalized Hermite poly- nomials {Hµ n} have a generating function (2.5.8) of [18] which involves the generalized V.N. Mishra, S. Pandey, I.A. Khan / Eur. J. Pure Appl. Math, 10 (5) (2017), 1067-1077 1069 exponential function eµ defined by eµ(z) = ∞∑ m=0 zm γµ(m) , (5) where γµ(m) is a generalized factorial defined as γµ(2m) = 22mm!Γ(m+ µ+ 1 2) Γ(µ+ 1 2) = (2m)! Γ(m+ µ+ 1 2) Γ(µ+ 1 2) Γ(12) Γ(m+ 1 2) , γµ(2m+ 1) = 22m+1m!Γ(m+ µ+ 3 2) Γ(µ+ 1 2) = (2m+ 1)! Γ(m+ µ+ 3 2) Γ(µ+ 1 2) Γ(12) Γ(m+ 3 2) . A recurrence relation holds for γµ, γµ(k + 1) = (k + 1 + 2µθk+1)γµ(k), k ∈ N0, where θk = { 0, if k ∈ 2N 1, if k ∈ 2N + 1 . It is apparent that e0(x) = ex and eµ is an entire function. The µ-binomial coefficient and µ-binomial expansion is also defined in [18] as( n k ) µ = γµ(n) γµ(k)γµ(n− k) , (x+ y)nµ = n∑ j=0 ( n k ) µ xjyn−j . (6) Using the generalized exponential function, Sucu [20] defined a Dunkl analogue of Szász operators as follows S∗n(f ;x) := 1 eµ(nx) ∞∑ k=0 (nx)k γµ(k) f ( k + 2µθk n ) , (7) where µ ≥ 0, n ∈ N, x ≥ 0, f ∈ C[0,∞). Since then Dunkl analogue of various operators has been studied [See [23],[8],[9] etc.]. G. İçöz and B. Çekim [8] presented the Dunkl generalization of Szász operators via q-calculus as Dn,q(f ;x) := 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx)k γµ,q(k) f ( [k + 2µθk]q [n]q ) , (8) where µ > 1 2 , n ∈ N, x ≥ 0, 0 < q < 1, f ∈ C[0,∞), eµ,q(x) = ∞∑ n=0 xn γµ,q(n) and γµ,q(n+ 1) = [n+ 1 + 2µθn+1]qγµ,q(n). V.N. Mishra, S. Pandey, I.A. Khan / Eur. J. Pure Appl. Math, 10 (5) (2017), 1067-1077 1070 Cheikh et al. [27] stated the definition of q-Dunkl analogue of exponential function as Eµ,q(x) = ∞∑ n=0 q n(n−1) 2 xn γµ,q(n) , and explicit formula for γµ,q(n) is γµ,q(n) = (q2µ+1, q2)[n+1 2 ](q 2, q2)[n 2 ] (1− q)n , where (a, q)0 = 1, (a, q)n := ∏n−1 k=0(1− aqk). Using the definitions of µ-binomial coefficient and µ−binomial expansion, we get q−analogue of equation (6)( n k ) µ,q = γµ,q(n) γµ,q(k)γµ,q(n− k) , (x+ y)nµ,q = n∑ j=0 ( n k ) µ,q xjyn−j . Thus the first few µ-binomial polynomials are 1, x+y, x2+ [2]q [2µ+1]q xy+y2, x3+ [3+2µ]q [1+2µ]q (x2y+ xy2) + y3, x4 + 4 1 [1+2µ]q (x3y + xy3) + y4. Furthermore, q-analogue of µ-beta and µ-gamma functions are defined as, Bµ,q(m,n) = γµ,q(m− 1)γµ,q(n− 1) γµ,q(m+ n− 1) , Γµ,q(t) = ∫ ∞ 0 xt−1Eµ,q(−qx)dqx, t > 0. Now in this paper, we propose the Dunkl generalization of Szász-Beta operators via q- calculus as Dn,q(f ;x) := 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx)k γµ,q(k) ∫ ∞/A 0 qk 2 tk Bµ,q(k + 1, n)(1 + t)n+k+1 µ,q f(t)dqt, (9) where A > 0, 0 < q < 1, f ∈ C[0,∞). 2. Approximation Properties In this section we analyze the convergence behaviour of the operators Dn,q(f ;x) via universal Korovkin theorem and weighted approximation theorem as in [6]. Consider the notation F q,µm (n) = ∏m i=1[n− i+ 2µθn−i]q. Lemma 1. The operators Dn,q given by (9) satisfies the following Dn,q(1;x) = 1, (10) V.N. Mishra, S. Pandey, I.A. Khan / Eur. J. Pure Appl. Math, 10 (5) (2017), 1067-1077 1071 Dn,q(t;x) = 1 F q,µ1 (n) [ [n]qx q2 + Cosh([n]qx) + q2µSinh([n]qx) qeµ,q([n]qx) + [2µ]q Eµ,q(−[n]qx) eµ,q([n]qx) ] , (11) Dn,q(t 2;x) = 1 F q,µ2 (n) [ ([n]qx)2 q6 + [2]q[n]qx q5 · eµ,q([n]qx) { (q + q2µ)Sinh([n]qx) + (1 + q2µ+1)Cosh([n]qx) } + [2]q q3 Cosh([n]qx) + q4µSinh([n]qx) eµ,q([n]qx) + [2]q[2µ]q q2 Cosh([n]qx)− q2µSinh([n]qx) eµ,q([n]qx) ] , (12) Dn,q((t− x)2;x) = x2 [ 1− 2[n]q q2F q,µ1 (n) + [n]2q q6F q,µ2 (n) ] +x [ [2]q[n]q q5F q,µ2 (n) (q + q2µ)Sinh([n]qx) + (1 + q2µ+1)Cosh([n]qx) eµ,q([n]qx) −2 Cosh([n]qx) + q2µSinh([n]qx) qeµ,q([n]qx)F q,µ1 (n) − 2[2µ]q F q,µ1 (n) Eµ,q(−[n]qx) eµ,q([n]qx) ] +[2]q Cosh([n]qx) + q4µSinh([n]qx) q3F q,µ2 (n)eµ,q([n]qx) + [2]q[2µ]q q2F q,µ2 (n) Cosh([n]qx)− q2µSinh([n]qx) eµ,q([n]qx) . (13) Proof. Using the definition of generalised exponential function in the q-Gamma and q-Beta functions in [19], we can obtain the following important equality: qk 2 ∫ ∞/A 0 tk+m Bµ,q(k + 1, n)(1 + t)n+k+1 µ,q dqt = γµ,q(m+ k)γµ,q(n−m− 1)q[2k 2−(k+m)(k+m+1)]/2 γµ,q(k)γµ,q(n− 1) . (14) For f(t) = 1, using (14) with m = 0 , we obtain Dn,q(1;x) = 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx)k γµ,q(k) ∫ ∞/A 0 qk 2 tk Bµ,q(k + 1, n)(1 + t)n+k+1 µ,q dqt = 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx)k γµ,q(k) qk(k−1)/2 = 1 eµ,q([n]qx) Eµ,q([n]qx) = 1. Next for f(t) = t, using (14) with m = 1 and θk+1 = θk+(−1)k, [n]q = [s]q+q s[n−s]q, 0 ≤ s ≤ n, we obtain Dn,q(t;x) = 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx)k γµ,q(k) ∫ ∞/A 0 qk 2 tk+1 Bµ,q(k + 1, n)(1 + t)n+k+1 µ,q dqt V.N. Mishra, S. Pandey, I.A. Khan / Eur. J. Pure Appl. Math, 10 (5) (2017), 1067-1077 1072 = 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx)k γµ,q(k) γµ,q(k + 1)γµ,q(n− 2)q(k 2−3k−2)/2 γµ,q(k)γµ,q(n− 1) = 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx)k γµ,q(k) [k + 2µθk + 1 + 2µ(−1)k]qq (k2−3k−2)/2 [n− 1 + 2µθn−1]q = e−1µ,q([n]qx) F q,µ1 (n) ∞∑ k=1 ([n]qx)k γµ,q(k − 1) ([k + 2µθk]q + qk+2µθk [1 + 2µ(−1)k]q) [k + 2µθk]q q(k 2−3k−2)/2 = e−1µ,q([n]qx) F q,µ1 (n) [ [n]qx q2 ∞∑ k=0 ([n]qx)k γµ,q(k) qk(k−1)/2 + 1 q ∞∑ k=0 ([n]qx)k γµ,q(k) qk(k−1)/2q2µθk { 1 + q[2µ]q(−1)k }] = 1 F q,µ1 (n) [ [n]qx q2 + Cosh([n]qx) + q2µSinh([n]qx) qeµ,q([n]qx) + [2µ]q Eµ,q(−[n]qx) eµ,q([n]qx) ] . Next for f(t) = t2, using (14) with m = 2 and θk+2 = θk we obtain Dn,q(t 2;x) = 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx)k γµ,q(k) ∫ ∞/A 0 qk 2 tk+2 Bµ,q(k + 1, n)(1 + t)n+k+1 µ,q dqt = 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx)k γµ,q(k) γµ,q(k + 2)γµ,q(n− 3)q(k 2−5k−6)/2 γµ,q(k)γµ,q(n− 1) = e−1µ,q([n]qx) F q,µ2 (n) ∞∑ k=0 ([n]qx)k γµ,q(k − 1) ([k + 2 + 2µθk+2]q[k + 1 + 2µθk+1]q) [k + 2µθk]q q(k 2−5k−6)/2 = e−1µ,q([n]qx) F q,µ2 (n) ∞∑ k=0 ([n]qx)k γµ,q(k − 1) ([k + 2µθk]q + qk+2µθk [2]q)([k + 1 + 2µθk+1]q) [k + 2µθk]q q (k2−5k−6) 2 = 1 F q,µ2 (n) [ ([n]qx)2 q6 + [2]q[n]qx q5 · eµ,q([n]qx) { (q + q2µ)Sinh([n]qx) + (1 + q2µ+1)Cosh([n]qx) } + [2]q q3 Cosh([n]qx) + q4µSinh([n]qx) eµ,q([n]qx) + [2]q[2µ]q q2 Cosh([n]qx)− q2µSinh([n]qx) eµ,q([n]qx) ] . Using the above results, we can get Dn,q((t− x)2;x) = Dn,q(t 2;x)− 2xDn,q(t;x) + x2Dn,q(1;x) = 1 F q,µ2 (n) [ ([n]qx)2 q6 + [2]q[n]qx q5 · eµ,q([n]qx) { (q + q2µ)Sinh([n]qx) + (1 + q2µ+1)Cosh([n]qx) } + [2]q q3 Cosh([n]qx) + q4µSinh([n]qx) eµ,q([n]qx) + [2]q[2µ]q q2 Cosh([n]qx)− q2µSinh([n]qx) eµ,q([n]qx) ] − 1 F q,µ1 (n) [ 2[n]qx 2 q2 + +2x Cosh([n]qx) + q2µSinh([n]qx) qeµ,q([n]qx) + [2µ]q2x Eµ,q(−[n]qx) eµ,q([n]qx) ] +x2 V.N. Mishra, S. Pandey, I.A. Khan / Eur. J. Pure Appl. Math, 10 (5) (2017), 1067-1077 1073 = x2 [ 1− 2[n]q q2F q,µ1 (n) + [n]2q q6F q,µ2 (n) ] + x [ [2]q[n]q q5F q,µ2 (n) (q + q2µ)Sinh([n]qx) + (1 + q2µ+1)Cosh([n]qx) eµ,q([n]qx) −2 Cosh([n]qx) + q2µSinh([n]qx) qeµ,q([n]qx)F q,µ1 (n) − 2[2µ]q F q,µ1 (n) Eµ,q(−[n]qx) eµ,q([n]qx) ] +[2]q Cosh([n]qx) + q4µSinh([n]qx) q3F q,µ2 (n)eµ,q([n]qx) + [2]q[2µ]q q2F q,µ2 (n) Cosh([n]qx)− q2µSinh([n]qx) eµ,q([n]qx) . Theorem 1. Let Dn,q be the operators given by (9). Then for any f ∈ C[0,∞) ∩ E, the following relation: lim n→∞ Dn,q(f ;x) = f(x) holds uniformly on each compact subset of [0,∞), where E := { f : x ∈ [0,∞), f(x) 1+x2 is convergent as x→∞ } . Proof. The proof is based on the well-known universal Korovkin-type theorem (see details in [1],[12]). By taking into account the Korovkin’s theorem, it is sufficient to show that lim n→∞ ‖Dn,q(t i;x)− ti‖ = 0, i = 0, 1, 2. Let (qn) denote a sequence such that 0 < qn ≤ 1. Since for fixed q with 0 < q ≤ 1, lim n→∞ [n]q = 1, to ensure the convergence properties we will assume q = qn as a sequence such that lim n→∞ qn = 1, and lim n→∞ qnn = c where c ∈ (0, 1). Therefore, we guarantee that lim n→∞ 1 [n]qn = 0. For example, if we choose (qn) = (1− 1 n ) then lim n→∞ qnn = e−1. Hence, we obtain lim n→∞ 1 [n]qn = 0. Besides, the other way is to take the sequence qn ∈ (0, 1) such that lim n→∞ qn = 1. Thus, lim n→∞ 1 [n]qn = 0. Taking q = (qn) as above we prove the following results. Using Lemma 1, result for i = 0 is trivial. For i = 1 result can be obtained as lim n→∞ ‖Dn,q(t;x)− x‖ = lim n→∞ ∥∥∥∥∥ ( [n]q q2[n− 1 + 2µθn−1]q − 1 ) x+ Cosh([n]qx) + q2µSinh([n]qx) qF q,µ1 (n)eµ,q([n]qx) + [2µ]q F q,µ1 (n) Eµ,q(−[n]qx) eµ,q([n]qx) ∥∥∥∥∥= 0 V.N. Mishra, S. Pandey, I.A. Khan / Eur. J. Pure Appl. Math, 10 (5) (2017), 1067-1077 1074 For i = 2, lim n→∞ ‖Dn,q(t 2;x)− x2‖ = lim n→∞ ∥∥∥∥∥ ( ([n]q) 2 q6[n− 1 + 2µθn−1]q[n− 2 + 2µθn−2]q − 1 ) x2 + 1 F q,µ2 (n) [ [2]q[n]qx q5 · eµ,q([n]qx) { (q + q2µ)Sinh([n]qx) + (1 + q2µ+1)Cosh([n]qx) } + [2]q q3 Cosh([n]qx) + q4µSinh([n]qx) eµ,q([n]qx) + [2]q[2µ]q q2 Cosh([n]qx)− q2µSinh([n]qx) eµ,q([n]qx) ]∥∥∥∥∥ = 0. One can easily get the limit using the fact that as n → ∞ we have 1 [n]qn → 0, q → 1, (Sinh([n]qx) + Cosh([n]qx)) = eµ,q([n]qx) and eµ,q(−[n]qx)→ 0. Thus using Korovkin’s result we can conclude that lim n→∞ ‖Dn,q(f(t);x)− f(x)‖ = 0. Recalling the weighted spaces of the functions which are defined on the positive semi- axis R+ = [0,∞) as follows: Bω(R+) = {f : |f(x)| ≤ Mfω(x)} , Cω(R+) = { f : f ∈ Bω(R+) ∩ C[0,∞) } , Ckω(R+) = { f : f ∈ Cω(R+) and lim x→∞ f(x) ω(x) = k (k is a constant) } , where ω(x) = 1 +x2 is a weight function andMf is a constant depending only on f . One can observe that Cω(R+) is a normed space with norm defined as, ‖f‖ω := supx≥0 |f(x)| ω(x) . Theorem 2. Let Dn,q be the operators given by (9). Then for any f ∈ Ckω(R+), we have: lim n→∞ ‖Dn,q(f ;x)− f(x)‖ω = 0. Proof. Using Lemma 1, one can easily prove the theorem. 3. Main results Here, we give the rate of convergence of the operators with the help of the usual and second order modulus of continuity and Lipschitz class functions. Lipschitz class of order α,LipM (α) (0 < α ≤ 1, M > 0), is defined as follows LipM (α) := {f : |f(x)− f(y)| ≤M |x− y|α, x, y ∈ [0,∞)}. REFERENCES 1075 Theorem 3. Let f ∈ LipM (α), then |Dn,q(f ;x)− f(x)| ≤M(δn(x))α/2, where δn(x) = Dn,q((t− x)2;x). Proof. For f ∈ LipM (α) and linearity behaviour of Dn,q, |Dn,q(f ;x)− f(x)| ≤ Dn,q(|f(t)− f(x)|;x) ≤ MDn,q(|t− x|α;x). Using Hölder inequality for integral and then for sum with p = α/2 and q = 1 − α 2 , we have |Dn,q(f ;x)− f(x)| ≤ M ( Dn,q((t− x)2;x )α/2 . Choosing δn(x) = Dn,q((t− x)2;x), then we get the desired result. Theorem 4. Consider Č[0,∞) is the space of uniformly continuous functions on [0,∞). Let f ∈ Č[0,∞) ∩ E,Dn,q operators verify the following |Dn,q(f ;x)− f(x)| ≤ (1 + √ µ) ω ( f ; 1 F q,µ2 (n) ) . Proof. |Dn,q(f ;x)− f(x)| ≤ Dn,q(|f(t)− f(x)|;x) ≤ ( 1 + 1 δ Dn,q(|t− x|;x) ) ω(f ; δ) ≤ ( 1 + 1 δ √ Dn,q((t− x)2;x) ) ω(f ; δ). Choosing δ = 1 F q,µ2 (n) and µ = 1 δ2 Dn,q((t− x)2;x), we can obtain the desired result. Acknowledgement The authors would like to thank the anonymous referees for their respective helpful discussions and suggestions in preparation of this article. References [1] F. Altomare and M. Campiti. Korovkin-Type Approximation Theory and Its Appli- cations. de Gruyter Studies in Mathematics, 17, alter de Gruyter & Co., Berlin, 1994. REFERENCES 1076 [2] Deepmala A.R. Gairola and L.N. Mishra. Rate of Approximation by Finite Iterates of q-Durrmeyer Operators. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. (April-June 2016), 86:229–234, 2016. [3] L.N. Mishra A.R. Gairola, Deepmala. On the q−derivatives of a certain linear positive operators. Iranian J. Sci. Tech., Transactions A: Science, DOI 10.1007/s40995-017- 0227-8, 2017. [4] A. Aral. A generalization of Szász-Mirakyan operators based on q-integers. Math. Comput. Model., 47:1052–1062, 2008. [5] S.N. Bernstein. Démonstration du théoréme de Weierstrass fondée sur le calcul des probabilités. Commun. Soc. Math. Kharkow, 2:1–2, 1912. [6] A.D. Gadzhiev. The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogous to that of P.P. Korovkin. Sov. Math. Dokl., 15:1433–1436, 1974. [7] V. Gupta and N.I. Mahmudov. Approximation properties of the q-Szász-Mirakyan- Beta operators. Indian J. Ind. Appl. Math., 3:41–53, 2012. [8] G. İçöz and B. Çekim. Dunkl generalization of Szász operators via q-calculus. J. Inequal. Appl., 2015:11 pages, 2015. [9] G. İçöz and B. Çekim. Stancu type generalization of Dunkl analogue of Szász- Kontorovich operators. Math. Meth. Appl. Sci., 39:1803–1810, 2016. [10] F.G. Bad́ıa J.A. Adell and J. de la Cal. Beta-type operators preserve shape properties. Stoch. Proc. Appl., 48:1–8, 1993. [11] F.H. Jackson. On a q-definite integrals. Quart. J. Pure Appl. Math., 41:193–203, 1910. [12] P.P. Korovkin. On convergence of linear positive operators in the space of continuous functions. Dokl. Akad. Nauk SSSR, 90:961–964, 1953. [13] N.I. Mahmudov. On q-parametric Szász-Mirakjan operators. Mediterr. J. Math., 7:297–311, 2010. [14] N.I. Mahmudov. Approximation by the q-Szász-Mirakjan Operators. Abstr. Appl. Anal., 2012:16 pages, 2012. [15] G.M. Phillips. Bernstein polynomials based on the q-integers. Ann. Numer. Math., 4:511–518, 1997. [16] C. Radu. On statistical approximation of a general class of positive linear operators extended in q-calculus. Appl. Math. Comput., 215:2317–2325, 2009. REFERENCES 1077 [17] Deepmala R.B. Gandhi and V.N. Mishra. Local and global results for modified Szász - Mirakjan operators. Math. Meth. Appl. Sci., 40(7):2491–2504, 2017. [18] M. Rosenblum. Generalized Hermite polynomials and the Bose-like oscillator calculus. Oper.Theory Adv. Appl., 73:369–396, 1994. [19] A. De Sole and V.G. Kac. On integral representation of q-gamma and q-beta functions. Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 16:11– 29, 2005. [20] S. Sucu. Dunkl analogue of Szász operators. Appl. Math. Comput., 244:42–48, 2014. [21] O. Szász. Generalization of S. Bernstein’s polynomials to the infinite interval. J. Res. Nat. Bur. Stand., 45:239–245, 1950. [22] G. Szëgo. Orthogonal polynomials. Amer. Math. Soc. Colloq. Publ., 23, Providence R.I., 1959. [23] B. Çekim Ü. Dinlemez and İ. Yüksel. Dunkl generalization of Szász-beta type oper- ators. Math. Meth. Appl. Sci., (accepted for publication). [24] L.N. Mishra V.N. Mishra, K. Khatri. Statistical approximation by Kan- torovich type Discrete q−Beta operators. Adv. Difference Equ., 2013:345, DOI: 10.1186/10.1186/1687–1847–2013–345, 2013. [25] L.N. Mishra V.N. Mishra, K. Khatri and Deepmala. Inverse result in simultaneous approximation by Baskakov-Durrmeyer-Stancu operators. J. Inequal. Appl., 2013:586, doi: 10.1186/1029–242X–2013–586, 2013. [26] L.N. Mishra V.N. Mishra, P. Sharma. On statistical approximation properties of q−Baskakov-Szász-Stancu operators. J. Egyptian Math. Soc., 24(3):396–401, 2016. [27] M. Gaied Y. Ben Cheikh and A. Zaghouani. q-Dunkl-classical q-Hermite type poly- nomials. Georgian Math. J., 21:125–137, 2014.