ON A GENERALIZED FORMULA OF TAYLOR–MACLAURIN TYPE ON THE GENERALIZED COMPLETELY MONOTONE FUNCTIONS

Abstract

In the paper Taylor–Maclaurin type formulas for some classes of functions are obtained. The main result of this study introduces an idea of the generalized classes of $ \langle \rho_j \rangle $ completely monotone function. Under the various conditions $ \left( \sum\limits_{j=1}^{\infty} \dfrac{1}{\rho_j} < + \infty ,\ \sum\limits_{j=1}^{\infty} \dfrac{1}{\rho_j} = + \infty \right) $ the terms of their representation are obtained and some related theorems are proved.