In the paper for any ρ≥1 and an arbitrary increasing sequence of positive numbers
{λj}0∞, the systems of operators and functions are introduced:{L∞nρ}0∞, {φn(x)}0∞, x∈[0,+∞), L∞0ρf≡f, L∞nρf≡∏j=0n−1(D∞1ρ+λj)f,n≥1,where D∞nρf≡D∞1ρD∞n−1ρf(1−α=1ρ); φ0(x)=e−λ0ρx, φn(x)=∑k=0nCk(n)e−λkρx, Ck(n)=(∏j=0,(j≠k)n(λj−λk))−1. Some properties of these systems are investigated, as well as specific differential equations of fractional order are solved. Finally, for some classes of functions Taylor–McLaurens type formulas are obtained.