On a generalization of Taylor–Maclourin formula for classes of Dzrbashyan functions $C^{*\infty}_{\alpha}$

In the paper for any $\rho\geq1$ and an arbitrary increasing sequence of positive numbers $\{\lambda_j\}^{\infty}_0$, the systems of operators and functions are introduced: $\{L^{n/p}_{\infty}\}^{\infty}_0$, $\{\varphi_n(x)\}^{\infty}_0$, $x\in[0, +\infty)$; $L^{0/p}_{\infty}f\equiv f$, $L^{n/p}_{\infty}f\equiv\displaystyle\prod^{n-1}_{j=0}(D^{1/\rho}_{\infty}+\lambda_j)f$, $n\geq 1$, where $L^{1/p}_{\infty}f\equiv\dfrac{d}{dx}D^{\alpha}_{\infty} f$, $D^{-\alpha}_{\infty} f\equiv\dfrac{1}{\Gamma(\alpha)}\displaystyle\int_x^{\infty}(t-x)^{\alpha-1}f(t)dt$, $D^{n/\rho}_{\infty} f\equiv D^{1/\rho}_{\infty} D^{(n-1)/\rho}_{\infty}f(1-\alpha=1/\rho)$; $\varphi_0(x)=\exp(-\lambda^{\rho}_0x)$, $\varphi_n(x)=\displaystyle\sum_{k=0}^nC_k^{(n)}\exp(-\lambda^{\rho}_nx)$, $C_k^{(n)}=\left(\displaystyle\prod^n_{j=0,(j\neq k)}(\lambda_j-\lambda_k)\right)^{-1}.$
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.