# Mittag-Leffler summation

In mathematics, **Mittag-Leffler summation** is any of several variations of the Borel summation method for summing possibly divergent formal power series, introduced by Mittag-Leffler (1908)

## Definition

Let

be a formal power series in *z*.

Define the transform of by

Then the **Mittag-Leffler sum** of *y* is given by

if each sum converges and the limit exists.

A closely related summation method, also called Mittag-Leffler summation, is given as follows (Sansone & Gerretsen 1960).
Suppose that the Borel transform converges to an analytic function near 0 that can be analytically continued along the positive real axis to a function growing sufficiently slowly that the following integral is well defined (as an improper integral). Then the **Mittag-Leffler sum** of *y* is given by

When *α* = 1 this is the same as Borel summation.

## See also

## References

- Hazewinkel, Michiel, ed. (2001) [1994], "Mittag-Leffler summation method",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Mittag-Leffler, G. (1908), "Sur la représentation arithmétique des fonctions analytiques d'une variable complexe",
*Atti del IV Congresso Internazionale dei Matematici (Roma, 6–11 Aprile 1908)*,**I**, pp. 67–86 - Sansone, Giovanni; Gerretsen, Johan (1960),
*Lectures on the theory of functions of a complex variable. I. Holomorphic functions*, P. Noordhoff, Groningen, MR 0113988