Classes of formulas and expansions of Taylor–Maclaurin type associated with differential operators of fractional order

This article contains new applications of functions of Mittag–Leffler type $E_\rho(z,\mu)=\sum_0^\infty\Gamma^{-1}\bigl(\mu+\frac k\rho\bigr)z^k$ in mathematical analysis. We construct an apparatus of formulas and expansions of Taylor–Maclaurin type in systems generated by a system of functions of the form $\bigl\{E_\rho\bigl(-\lambda_nx^{1/\rho},\frac1\rho\bigr)\bigr\}_0^\infty$ and an arbitrary sequence $\{\lambda_n\}_0^\infty$ ($0=\lambda_0\leqslant\lambda_n\leqslant\lambda_{n+1}$). In addition, in the article we introduce essentially new classes $\langle\rho,\lambda_n\rangle$ of absolutely monotone functions and establish theorems on their representations.
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