Adjoint Bernoulli’s Polynomials Coupling Gamma Functions and their Approximations

This paper introduces a novel relationship between adjoint Bernoulli polynomials and the gamma function, leading to the formation of a new class of linear positive operators denoted by {G̃_{r,λ}(.;.)}₁^∞. An in-depth investigation into the convergence behavior exhibited by this sequence of operators is carried out in a variety of function spaces. We obtain conclusions that pertain to approximation, which include explicit estimations of the order and rate of convergence, by utilizing Korovkin’s theorem, Voronovskaja-type asymptotic formulas, the modulus of continuity, and Peetre’s
K
-functional. The research expands to include a bivariate generalization of these operators, in which a comprehensive investigation of their uniform approximation features and convergence order in a variety of spaces is conducted.