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}^\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 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.

References

Hazewinkel, Michiel, ed. (2001), "Mittag-Leffler summation method", Encyclopedia of Mathematics, Springer, 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

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Mittag-Leffler summation

Definition

See also

References