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

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

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.

See also

• Mittag-Leffler function

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), vol. 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