Szász–Durrmeyer-type operators generated by Euler polynomials of negative order via a Poisson–Gamma representation

Mine Menekşe Yılmaz; Erkan Agyuz; Mine Menekşe Yılmaz; Erkan Agyuz

doi:10.3934/math.2026782

AIMS Mathematics

2026, Volume 11, Issue 6: 19217-19241. doi: 10.3934/math.2026782

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Szász–Durrmeyer-type operators generated by Euler polynomials of negative order via a Poisson–Gamma representation

Mine Menekşe Yılmaz ^{1
,
,},
Erkan Agyuz ²

1.
Department of Mathematics, Faculty of Arts and Science, Gaziantep University, 27310 Gaziantep, Türkiye
2.
Naci Topçuoğlu Vocational School, Gaziantep University, 27310 Gaziantep, Türkiye

Received: 05 May 2026 Revised: 09 June 2026 Accepted: 16 June 2026 Published: 30 June 2026
MSC : 41A10, 41A25, 41A36

We introduced a Szász–Durrmeyer-type family of positive linear operators generated by Euler polynomials of negative order $ -k $, where $ k\in\mathbb{N} $. The corresponding discrete kernel admits a finite binomial mixture of shifted Poisson probabilities, while the Durrmeyer integral part is described by a Gamma-type kernel. This Poisson–Gamma structure makes positivity and normalization transparent and provides a convenient basis for the computation of algebraic and central moments. Using these moments, we established quantitative estimates, weighted Korovkin-type convergence, and a compact-uniform Voronovskaya-type theorem. The asymptotic formula shows that the classical approximation order is preserved, whereas the negative-order Euler parameter affects only the drift term in the first-order asymptotic profile.
- Szász–Durrmeyer operators,
- Euler polynomials of negative order,
- positive linear operators,
- Poisson–Gamma representation,
- weighted Korovkin theorem,
- Voronovskaya theorem
Citation: Mine Menekşe Yılmaz, Erkan Agyuz. Szász–Durrmeyer-type operators generated by Euler polynomials of negative order via a Poisson–Gamma representation[J]. AIMS Mathematics, 2026, 11(6): 19217-19241. doi: 10.3934/math.2026782

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Abstract

We introduced a Szász–Durrmeyer-type family of positive linear operators generated by Euler polynomials of negative order $ -k $, where $ k\in\mathbb{N} $. The corresponding discrete kernel admits a finite binomial mixture of shifted Poisson probabilities, while the Durrmeyer integral part is described by a Gamma-type kernel. This Poisson–Gamma structure makes positivity and normalization transparent and provides a convenient basis for the computation of algebraic and central moments. Using these moments, we established quantitative estimates, weighted Korovkin-type convergence, and a compact-uniform Voronovskaya-type theorem. The asymptotic formula shows that the classical approximation order is preserved, whereas the negative-order Euler parameter affects only the drift term in the first-order asymptotic profile.

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