ON A GENERALIZATION OF TAYLOR–MACLOURIN FORMULA FOR CLASSES OF DZRBASHYAN FUNCTIONS $C_{\alpha}^{*(\infty)}$

Abstract

In the paper for any $\rho\geq 1$ and an arbitrary increasing sequence of positive numbers

$\{\lambda_j\}_0^\infty$, the systems of operators and functions are introduced: $$\{L_\infty^{\frac{n}{\rho}}\}_0^\infty,~\{\varphi_n(x)\}_0^\infty,~x \in [0, +\infty),~L_\infty^{\frac{0}{\rho}}f \equiv f,~L_\infty^{\frac{n}{\rho}}f \equiv \prod\limits_{j=0}^{n-1}\left(D_\infty^{\frac{1}{\rho}}+\lambda_j\right)f, n\geq1,$$ where $D_\infty^{\frac{n}{\rho}}f\equiv D_\infty^{\frac{1}{\rho}} D_\infty^{\frac{n-1}{\rho}} f\left(1-\alpha=\dfrac{1}{\rho}\right)$; $\varphi_0(x)=e^{-\lambda_0^\rho x}$, $\varphi_n(x)=\sum\limits_{k=0}^n C_k^{(n)}e^{-\lambda_k^\rho x}$, $C_k^{(n)}=\left( \prod\limits_{j=0, (j\neq k)}^n\left(\lambda_j-\lambda_k\right)\right)^{-1}$. 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.