Degenerate Appell Polynomials Connecting Beta Function as a Family of Operators and Their Approximations

This paper introduces a novel family of positive linear operators constructed by blending degenerate Appell polynomials with a classical Beta kernel in the Durrmeyer setting. The operators are defined as Hn(g;u)=∑ƿ=0∞hƿ(n+u;ƛ)∫01Kn(ƿ, ʈ)g(ʈ)dʈ, where hƿ(nu;ƛ) is derived from degenerate Appell polynomials (nu denotes the product of n and u) and Kn(ƿ, ʈ) is a Beta-type kernel. We establish the linearity and positivity of these operators and derive crucial moment estimates. Approximation properties are examined via Korovkin-type theorems, and the asymptotic behavior is investigated through a Voronovskaja-type theorem. The results extend and unify earlier work on Appell-based approximation operators and offer new tools for approximating functions in weighted spaces. Numerical examples and error estimates are provided to illustrate the efficacy of the proposed operators. In addition, the inherent symmetry in the structure of the proposed operators-arising from the symmetric nature of the Beta kernel and the generating functions of degenerate Appell polynomials is discussed. Such symmetry plays a key role in ensuring balanced approximation and convergence characteristics.