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

[math]\displaystyle{ y(z) = \sum_{k = 0}^\infty y_kz^k }[/math]

be a formal power series in z.

Define the transform [math]\displaystyle{ \scriptstyle \mathcal{B}_\alpha y }[/math] of [math]\displaystyle{ \scriptstyle y }[/math] by

[math]\displaystyle{ \mathcal{B}_\alpha y(t) \equiv \sum_{k=0}^\infty \frac{y_k}{\Gamma(1+\alpha k)}t^k }[/math]

Then the Mittag-Leffler sum of y is given by

[math]\displaystyle{ \lim_{\alpha\rightarrow 0}\mathcal{B}_\alpha y( z) }[/math]

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 [math]\displaystyle{ \mathcal{B}_1 y(z) }[/math] 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

[math]\displaystyle{ \int_0^\infty e^{-t} \mathcal{B}_\alpha y(t^\alpha z) \, dt }[/math]

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