Approximation by Kantorovich-Type Operators Involving Degenerate Hermite Polynomials and Riemann–Liouville Fractional Integrals

In the present paper, we introduce a new class of positive linear operators of Szász–Kantorovich type which are generated by degenerate Hermite polynomials, where the ordinary Kantorovich average is replaced by a Riemann–Liouville fractional integral of a positive order. When the order equals one, the operators reduce to the classical Kantorovich operators. We compute moments and central moments in terms of beta-function constants and use them to prove a Korovkin-type convergence theorem. The rate of convergence is estimated through the modulus of continuity, the second-order modulus of smoothness, and a Lipschitz-type class, and an asymptotic formula of Voronovskaja type is derived in weighted spaces. The formula indicates that the asymptotic bias depends on the order of the fractional integral via the coefficient of the first derivative. We also construct a bivariate extension and prove its convergence by a Volkov-type theorem, with error estimates in terms of the total modulus of continuity and a Lipschitz class in two dimensions. We use numerical examples to illustrate the convergence.