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

\( y(z) = \sum_{k = 0}^\infty y_kz^k \)

be a formal power series in z.

Define the transform \( \scriptstyle \mathcal{B}_\alpha y\) of \( \scriptstyle y \) by

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

Then the Mittag-Leffler sum of y is given by

\( \lim_{\alpha\rightarrow 0}\mathcal{B}_\alpha y( z) \)

if each sum converges and the limit exists.

A closely related summation method, also called Mittag-Leffler summation, is given as follows (Sansone & Gerretsen 1960). Suppose that the Borel transform \( {\displaystyle {\mathcal {B}}_{1}y(z)} \) 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

\( \int_0^\infty e^{-t} \mathcal{B}_\alpha y(t^\alpha z) \, dt \)

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

See also

Mittag-Leffler function

Nachbin's theorem

References

"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

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