Composition of some positive linear integral operators (Q6622662)

Giving explicit expressions of the images of exponential functions with the generalized Baskakov operators is very useful for getting certain results concerning convergence of sequences of positive linear operators. But in this context, it is not easy to compute the moments of the operators. In this article, the authors construct sequences of integral operators having the same Voronovskaja formula as the generalized Baskakov operators and for which the moments and the central moments can be explicitly computed. They investigate the eigenstructure and the inverse of such an integral operator, describing the eigenstructure. The Voronovskaja type results for the newly introduced operators and for their inverses are also provided. Since the operators are compositions of other operators, they obtain a general Voronovskaja type result for such compositions. Differences between the new operators are investigated from the point of view of the asymptotic behavior, and the results are applied to obtain an alternative proof of a formula involving the Euler's gamma function.