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

$y(z)=\sum _{k=0}^{\mathrm{\infty }}{y}_k{z}^k$

be a formal power series in z.

Define the transform ${{ℬ}}_{\alpha }y$ of $y$ by

${{ℬ}}_{\alpha }y(t)\equiv \sum _{k=0}^{\mathrm{\infty }}\frac{y_k}{\mathrm{\Gamma }(1+\alpha k)}{t}^k$

Then the Mittag-Leffler sum of y is given by

$\underset{\alpha \to 0}{\lim }{{ℬ}}_{\alpha }y(z)$

if each sum converges and the limit exists.

A closely related summation method, also called Mittag-Leffler summation, is given as follows (Sansone Gerretsen). Suppose that the Borel transform ${{ℬ}}_{1}y(z)$ 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

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-t}{{ℬ}}_{\alpha }y(t^{\alpha }z)dt$

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

See also

• Mittag-Leffler distribution
• Mittag-Leffler function
• Nachbin's theorem

References

• 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, https://www.encyclopediaofmath.org/index.php?title=Mittag-Leffler_summation_method 
• 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, http://www.mathunion.org/ICM/ICM1908.1/, 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