Convolution series and the generalized convolution Taylor formula

In this paper, we deal with the convolution series that are a far reaching generalization of the conventional power series and the power series with the fractional exponents including the Mittag-Leffler type functions. Special attention is given to the most interesting case of the convolution series generated by the Sonine kernels. In this paper, we first formulate and prove the second fundamental theorem for the general fractional integrals and the $n$-fold general sequential fractional derivatives of both the Riemann-Liouville and the Caputo types. These results are then employed for derivation of two different forms of a generalized convolution Taylor formula for representation of a function as a convolution polynomial with a remainder in form of a composition of the $n$-fold general fractional integral and the $n$-fold general sequential fractional derivative of the Riemann-Liouville and the Caputo types, respectively. We also discuss the generalized Taylor series in form of convolution series and deduce the formulas for its coefficients in terms of the $n$-fold general sequential fractional derivatives.

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