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We consider sequences{ρj}∞0(ρ0=1,ρj≥1),{αj}∞0(α0=0,αj=1−(1/ρj)),{Wj(x)}∞0∈W,whereW={{Wj(x)}∞0/W0(x)≡1,Wj(x)>0,W′j(x)≤0,Wj(x)∈C∞[0,a]},C∞[0,a]is the class of functions of infinitely differentiable. For such sequenceswe introduce systems of operators{A∗a,nf}∞0,{ ̃A∗a,nf}∞0and functions{Ua,n(x)}∞0,{Φn(x,t)}∞0.For a certain class of functions a generalization ofTaylor–Maclaurin type formulae was obtained. We also introduce the conceptof〈ρj,Wj〉generalized completely monotone functions and establish a theoremon their representation.