Mittag-Leffler summation

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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)



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[edit]


  • "Mittag-Leffler summation method", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  • 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, archived from the original on 2016-09-24, retrieved 2012-11-02
  • Sansone, Giovanni; Gerretsen, Johan (1960), Lectures on the theory of functions of a complex variable. I. Holomorphic functions, P. Noordhoff, Groningen, MR 0113988