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 [1] 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.

References

[1]	G. Mittag-Leffler, , Atti IV congress. internaz. , 1 , Rome (1908) pp. 67–85
[2]	G. Mittag-Leffler, "Sur la répresentation analytique d'une branche uniforme d'une fonction monogène" Acta Math. , 29 (1905) pp. 101–181
[3]	G.H. Hardy, "Divergent series" , Clarendon Press (1949)
[4]	R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)

Comments

The function considered by Mittag-Leffler is called a Mittag-Leffler function.