# Mittag-Leffler summation method

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A semi-continuous summation method for summing series of numbers and functions, defined by a sequence of functions where is the gamma-function. A series is summable by the Mittag-Leffler method to a sum if and if the series under the limit sign converges. The method was introduced by G. Mittag-Leffler  primarily for the series A Mittag-Leffler summation method is regular (see Regular summation methods) and is used as a tool for the analytic continuation of functions. If is the principal branch of an analytic function, regular at zero and represented by a series for small , then this series is summable by the Mittag-Leffler method to in the whole star of the function (cf. Star of a function element) and, moreover, uniformly in any closed bounded domain contained in the interior of the star.

For summation methods defined by transformations of sequences by semi-continuous matrices of the type where is an entire function, Mittag-Leffler considered the case when A matrix with such an entire function is called a Mittag-Leffler matrix.

How to Cite This Entry:
Mittag-Leffler summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mittag-Leffler_summation_method&oldid=12118
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article