Mittag-Leffler summation method
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
is an entire function, Mittag-Leffler considered the case when
A matrix with such an entire function is called a Mittag-Leffler matrix.
|||G. Mittag-Leffler, , Atti IV congress. internaz. , 1 , Rome (1908) pp. 67–85|
|||G. Mittag-Leffler, "Sur la répresentation analytique d'une branche uniforme d'une fonction monogène" Acta Math. , 29 (1905) pp. 101–181|
|||G.H. Hardy, "Divergent series" , Clarendon Press (1949)|
|||R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)|
The function considered by Mittag-Leffler is called a Mittag-Leffler function.
Mittag-Leffler summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mittag-Leffler_summation_method&oldid=12118