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This paper deals with approximating properties of the newly defined q-generalization of the genuine Bernstein-Durrmeyer polynomials in the case q>1, which are no longer positive linear operators on C0,1. Quantitative estimates of the convergence, the Voronovskaja-type theorem, and saturation of convergence for complex genuine q-Bernstein-Durrmeyer polynomials attached to analytic functions in compact disks are given. In particular, it is proved that, for functions analytic in z∈ℂ:z<R, R>q, the rate of approximation by the genuine q-Bernstein-Durrmeyer polynomials q>1 is of order q−n versus 1/n for the classical genuine Bernstein-Durrmeyer polynomials. We give explicit formulas of Voronovskaja type for the genuine q-Bernstein-Durrmeyer for q>1. This paper represents an answer to the open problem initiated by Gal in (2013, page 115)

abstract and applied analysis

Approximation by Genuine<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math>-Bernstein-Durrmeyer Polynomials in Compact Disks in the Case<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>q</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>1</mml:mn></mml:math>

Approximation by Genuineq-Bernstein-Durrmeyer Polynomials in Compact Disks in the Caseq>1