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STUDIA MATHEMATICA  Issue no. 4 / 2010  
Article: 
ON THE COMBINATORIAL IDENTITIES OF ABELHURWITZ TYPE AND THEIR USE IN CONSTRUCTIVE THEORY OF FUNCTIONS. Authors: ELENA IULIA STOICA. 

Abstract: This paper is concerned with the problem of approximation of multivariate functions by means of the AbelHurwitzStancu type linear positive operators. Inspired by the work of D. D. Stancu [13], we continue the discussions of the approximation of trivariate functions by a class of AbelHurwitzStancu operators in the case of trivariate variables, continues on the unit cub K_{3} = [0, 1]^{3}. In this paper there are three sections. In Section 1, which is the Introduction, is mentioned the generalization given by N. H. Abel [1] in 1826, for the Newton binomial formula and then the very important extension of this formula given by A. Hurwitz in 1902, in the paper [3]. Here is mentioned an interesting combinatorial significance in a cyclefree directed graphes given by D. E. Knuth [5]. Then is presented a main result given in 2002 by D. D. Stancu [13], where is used a variant of the Hurwitz identity in order to construct and investigate a new linear positive operator, which was used in the theory of approximation univariate functions. In Section 2 is discussed in detail the trivariate polynomial operator of StancuHurwitz type associated to a function f Є C(K_{3}), where K_{3 }is the unit cub [0, 1]^{3}. Section 3 is devoted to the evaluation of the remainder term of the approximation formula (3.1) of the function
f(x, y, z) by means of the StancuHurwitz type operator Firstly is presented an integral form of this remainder, based on the PeanoMilneStancu result [12]. Then we give a Cauchy type form for this remainder. By using a theorem of T. Popoviciu [8] we gave an expression, using the divided differences of the first three orders. When the coordinates of the vectors (β), (γ), (λ) have respectively the same values we are in the case of the second operator of CheneySharma [2]. In this case we obtain an extension of the results from the papers [13] and [17]. Key words and phrases. AbelHurwitzStancu operators. 
