Some Approximation Properties of -Baskakov-Beta-Stancu Type Operators

This paper deals with new type -Baskakov-Beta-Stancu operators defined in the paper. First, we have used the properties of -integral to establish the moments of these operators. We also obtain some approximation properties and asymptotic formulae for these operators. In the end we have also presented better error estimations for the -operators.


Introduction
In the recent years, the quantum calculus ( -calculus) has attracted a great deal of interest because of its potential applications in mathematics, mechanics, and physics. Due to the applications of -calculus in the area of approximation theory, -generalization of some positive operators has attracted much interest, and a great number of interesting results related to these operators have been obtained (see, for instance, [1][2][3]). In this direction, several authors have proposed the -analogues of different linear positive operators and studied their approximation behaviors. Also, Aral and Gupta [4] defined -generalization of the Baskakov operators and investigated some approximation properties of these operators. Subsequently, Finta and Gupta [5] obtained global direct error estimates for these operators using the second-order Ditzian Totik modulus of smoothness. To approximate Lebesgue integrable functions on the interval [0, ∞), modified Beta operators [6] are defined as where , ( ) = (1/ ( + 1, ))( /(1 + ) + +1 ) and , ( ) = ( + +1 ) ( /(1 + ) + ). The discrete -Beta operators are defined as Recently, Maheshwari and Sharma [7] introduced theanalogue of the Baskakov-Beta-Stancu operators and studied the rate of approximation and weighted approximation of these operators. Motivated by the Stancu type generalization of -Baskakov operators, we propose the -analogue of the operators ( , ) , recently introduced and studied for special values = = 0 by Gupta and Kim [8] as where , ( ) = ( ( −1)/2 / ( + 1, ))( /(1 + ) + +1 ) and , ( ) = ( + −1 ) ( ( −1)/2 / ( + 1, ))( /(1 + ) + +1 ). . We mention that ,0,0 ≡ (see [8]). Very recently, Gupta et al. [9] introduced some direct results in simultaneous approximation for Baskakov-Durrmeyer-Stancu operators. The aim of this paper is to study the approximation properties of a new generalization of the Baskakov type Beta Stancu operators based onintegers. We estimate moments for these operators. Also, we study asymptotic formula for these operators. Finally, we give better error estimations for the operator (3). First, we recall some definitions and notations of -calculus. Such notations can be found in [10,11]. We consider as a real number satisfying > 0.
The -binomial coefficients are given by The -derivative D of a function is given by The -analogues of product and quotient rules are defined as The -Jackson integrals and the -improper integrals are defined as [12,13] (9) provided that the sums converge absolutely. Using (9), De Sole and Kac [14] defined the -analogue of Beta functions of second kind ( , ) = ∫ ∞ 0 ( −1 /(1 + ) + ) as follows: where ( , ) = (1/(1+ )) (1+1/ ) (1+ ) 1− . This function is -constant in ; that is, ( , ) = ( , ). It was observed in [14] that ( , ) is independent of ; this is because from the integral and ( , ) the term cancels out. In particular for any positive integer , we have Also, we have In [8], Gupta and Kim obtained recurrence formula for the moments of the operators as follows.
Theorem 1 (see [8]). If one defined the central moments as then, for > + 2, one has the following recurrence relation:

Direct Result and Asymptotic Formula
Let the space [0, ∞) of all real-valued continuous bounded functions be endowed with the norm ‖ ‖ = sup{| ( )| : ∈ [0, ∞)}. Further, let us consider the following -functional: is the second-order modulus of smoothness of ∈ [0, ∞). Also we set We conclude by Remark 6 that where 4 1 = > 0. This proves the theorem.
Our next result in this section is an asymptotic formula.
where is bounded and lim → 0 ( ) = 0. Applying the operator , , to the above relation, we get which completes the proof.
As the operators , , introduced in (3) preserve only the constant functions, further modification of these operators is proposed to be made so that the modified operators preserve the constant as well as linear functions. For this purpose, the modification of , , is as follows: Theorem 11. Let ∈ ( , ) and = ∈ (0, 1) such that → 1 as → ∞. Then for all ∈ , and > 3, there exists an absolute constant > 0 such that