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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
