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

Jordanka Paneva-Konovska ()
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Jordanka Paneva-Konovska: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria

Mathematics, 2022, vol. 10, issue 22, 1-15

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 2 m -parametric multi-index Mittag–Leffler function. It turns out that this is expressed via the 3 m -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 2 m -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 ( 2 m -parametric) Mittag–Leffler functions. Their Taylor series are derived from the general case as corollaries, as well.

Keywords: Mittag–Leffler functions; multi-index Mittag–Leffler functions; integer order derivatives; estimates; asymptotics; Taylor series (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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