# 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

$g_k(\delta) = \frac{1}{\Gamma(1 + \delta k)}, \quad \delta > 0, \quad k = 0, 1, \dots,$

where $\Gamma(x)$ is the gamma-function. A series

$\sum_{k=0}^{\infty} u_k$

is summable by the Mittag-Leffler method to a sum $s$ if

$\lim\limits_{\delta \to 0}\sum_{k=0}^{\infty} \frac{u_k}{\Gamma(1 + \delta k)} = s$

and if the series under the limit sign converges. The method was introduced by G. Mittag-Leffler  primarily for the series

$\sum_{k=0}^{\infty} z^k .$

A Mittag-Leffler summation method is regular (see Regular summation methods) and is used as a tool for the analytic continuation of functions. If $f(z)$ is the principal branch of an analytic function, regular at zero and represented by a series

$\sum_{k=0}^{\infty}a_k z^k$

for small $z$, then this series is summable by the Mittag-Leffler method to $f(z)$ in the whole star of the function $f(z)$ (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 $a_k(\omega)$ of the type

$a_k(\omega) = \frac{c_{k+1}\omega^{k+1}}{E(\omega)},$

where

$E(\omega) = \sum_{k=0}^{\infty} c_k \omega^k$

is an entire function, Mittag-Leffler considered the case when

$E(\omega) = \sum_{k=0}^{\infty} \frac{\omega^k}{\Gamma(1+ak)}$

A matrix $a_k(\omega)$ 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=29819
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article