Mehmet Ali Özarslan

In this study, a generalization of the Hermite based second kind Peters-type Simsek polynomials is introduced. By means of the generating function, an explicit formula, summation formulas, a recurrence relation and a determinant representation are obtained. Using differential and integral operators, relations with the Stirling, Cauchy and Peters-type Simsek numbers are established. Two subfamilies, namely the three-variable and the parametric ones, are also defined, and their graphs and real roots are analyzed.

In this study, a generalization of the Hermite based second kind Peters-type Simsek polynomials is introduced. By means of the generating function, an explicit formula, summation formulas, a recurrence relation and a determinant representation are obtained. Using differential and integral operators, relations with the Stirling, Cauchy and Peters-type Simsek numbers are established. Two subfamilies, namely the three-variable and the parametric ones, are also defined, and their graphs and real roots are analyzed.

In this paper, we introduce a new class of multiple Stancu‐type Bernstein operators defined via Beta function based integral modification. We study their approximation properties and establish uniform convergence by means of Korovkin′s theorem under suitable conditions on the parameter α _η . Rates of convergence are obtained in terms of the modulus of continuity and Peetre′s κ ‐functional, and a Voronovskaja‐type asymptotic formula is derived. Furthermore, we obtain representations of the operator in terms of special functions, which, combined with its moment properties, lead to finite summation formulas for the generalized hypergeometric function ₃ F ₂ and coefficient identities associated with the confluent hypergeometric function ₁ F ₁ . Moreover, the operators unify several classical constructions: They reduce to the multiple Bernstein operators as α ⟶0 ⁺ , and yield a multiple Szász‐type operator under an appropriate scaling, recovering a result of (Varma, 2013). Numerical examples and graphical simulations are provided to illustrate the approximation behavior, confirm the theoretical findings, and highlight the crucial role of the parameter α _η in controlling the approximation behavior.

In this paper, by using Jain-Appell operators and classical Bernstein-Chlodowsky operators, we introduce Hybrid Chlodowsky-Jain-Appell operators. We investigate approximation properties of these new operators by using modulus of continuity, partial moduli of continuities, weighted modulus of continuity and bivariate Lipschitz class functionals. We introduce new operators of GBS type by using new defined Hybrid Chlodowsky-Jain-Appell operators. Furthermore, we investigate the approximation properties for new GBS version of the operators by using mixed modulus of continuity and bivariate Lipschitz class functional.

In this paper, we introduce multiple binomial‐type operators (multiple Bernstein–Sheffer operators) and investigate their approximation properties. These properties are analyzed by means of a Korovkin‐type theorem. Furthermore, by using the first and second modulus of continuity together with Peetre’s κ ‐functional, we establish the rate of convergence for continuous functions and study the approximation behavior on Lipschitz classes. Moreover, we show that, depending on the choice of the generating function H , the operator yields different special forms. In particular, when H is chosen to be linear, the operator reduces to a multiple Bernstein form, which can be regarded as a generalization of the classical Bernstein operator of degree 2 η . In this case, the operators achieve an improved rate of convergence, with an improvement of order compared to the classical Bernstein operators. Then, for the chosen H , the Korovkin conditions cannot be satisfied, and hence, the operator does not converge. Finally, with an appropriate choice of H , the operator can be represented as a Bell‐type operator.

In recent years, discrete Mittag-Leffler (ML) functions have been investigated intensively since they are connected with the discrete fractional calculus and have interesting applications in applied sciences. In this study, inspired by the five-parameter ML function by Özarslan et al. [16], we define its discrete version, namely five-parameter discrete ML function. We explore various characteristics, such as the double fractional sum and difference, of the discrete ML function with five parameters. Then, using the mentioned discrete function, we introduce the discrete double fractional operator having this function through the semigroup property. We also obtain the left inverse operator which is the corresponding discrete difference operator for the discrete double fractional operator.