Convolution series and the generalized convolution Taylor formula

Abstract

In this paper, we discuss the convolution polynomials and series that are a far reaching generalization of the conventional polynomials and power series with both integer and fractional exponents including the Mittag-Leffler type functions. Special attention is given to the most interesting case of the convolution polynomials and series generated by the Sonine kernels. These kernels were recently employed for construction of the general fractional integrals and derivatives, which include the single-, the multi-order, and the distributed order fractional derivatives as their particular cases. In the first part of the paper, we formulate and prove the second fundamental theorem for the n-fold general fractional integrals and the n-fold general sequential fractional derivatives of both the Riemann-Liouville and Caputo types. These results are then employed for derivation of two different forms of a generalized convolution Taylor formula. It provides a 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 in the Riemann-Liouville and the Caputo sense, respectively. We also discuss the generalized Taylor series in form of convolution series and deduce the explicit formulas for their coefficients in terms of the n-fold general sequential fractional derivatives evaluated at the point zero.