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.
- Hazewinkel, Michiel, ed. (2001) , "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