Bernstein type semi-exponential operators

The concept of semi-exponential operators is an extension of the exponential operators in approximation theory. The analysis begins by considering the partial differential equation, ∂∂xWβ(r,x,t)=r(t−x)p(x)Wβ(r,x,t)−βWβ(r,x,t)with domain of definition Ω, for the kernel of sequence of operators Sr(f;x)=∑k∈ΩWβ(r,x,kr)f(kr)while satisfying the normalization condition ∑k∈ΩWβ(r,x,kr)=1. Carrying forward this notion, we examine these operators specifically for the function p(x)=x(1−x) and focus our attention on the semi-exponential Bernstein operators. Our primary focus is to explore the properties and characteristics of these Bernstein type semi-exponential operators. The paper delves into deriving a recurrence relation for the central moments of these operators, while also establishing a moment generating function for the same. Finally, we analyze the approximation of these operators using the Voronovskaya-type asymptotic result. The objective of this analysis is to characterize functions by the degree of approximation.