Taylor Series for the Mittag–Leffler Functions and Their Multi-Index Analogues

Abstract

It has been obtained that the n-th derivative of the 2-parametric Mittag–Leffler function is a 3-parametric Mittag–Leffler function, with exactness to a constant. Following the analogy, the author later obtained the n-th derivative of the 2m-parametric multi-index Mittag–Leffler function. It turns out that this is expressed via the 3m-parametric Mittag–Leffler function. In this paper, upper estimates of the remainder terms of these derivatives are found, depending on n. Some asymptotics are also found for “large” values of the parameters. Further, the Taylor series of the 2 and 2m-parametric Mittag–Leffler functions around a given point are obtained. Their coefficients are expressed through the values of the corresponding n-th order derivatives at this point. The convergence of the series to the represented Mittag–Leffler functions is justified. Finally, the Bessel-type functions are discussed as special cases of the multi-index (2m-parametric) Mittag–Leffler functions. Their Taylor series are derived from the general case as corollaries, as well.