On the $\left\langle\rho_j,~W_j\right\rangle$ generalized completely monotone functions

We consider sequences $\{\rho_j\}_0^\infty$ $(\rho_0=1,~\rho_j\geq 1),$ $\left\{\alpha_j\right\}_0^\infty \big(\alpha_0=0, \alpha_j=1-(1/\rho_j)\big),$ $\{W_j(x)\}_0^\infty \in W,$ where
$$W=\left\{\left\{W_j(x)\right\}_0^\infty \big{/} W_0(x)\equiv 1,~W_j(x)> 0,~W_j^{\prime}(x)\leq 0,~W_j(x)\in C^\infty[0,a] \right\},$$
$C^\infty[0,a]$ is the class of functions of infinitely differentiable. For such sequences we introduce systems of operators $\left\{A_{a,n}^*f\right\}_0^\infty,$ $\left\{\tilde{A}_{a,n}^*f\right\}_0^\infty$ and functions $\left\{U_{a,n}(x)\right\}_0^\infty,$ $\left\{\Phi_n (x,t)\right\}_0^\infty.$ For a certain class of functions a generalization of Taylor–Maclaurin type formulae was obtained. We also introduce the concept of $\langle\rho_j,\ W_j\rangle$ generalized completely monotone functions and establish a theorem on their representation.