# 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

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

be a formal power series in z.

Define the transform ${\displaystyle \scriptstyle {\mathcal {B}}_{\alpha }y}$ of ${\displaystyle \scriptstyle y}$ by

${\displaystyle {\mathcal {B}}_{\alpha }y(t)\equiv \sum _{k=0}^{\infty }{\frac {y_{k}}{\Gamma (1+\alpha k)}}t^{k}}$

Then the Mittag-Leffler sum of y is given by

${\displaystyle \lim _{\alpha \rightarrow 0}{\mathcal {B}}_{\alpha }y(z)}$

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

${\displaystyle \int _{0}^{\infty }e^{-t}{\mathcal {B}}_{\alpha }y(t^{\alpha }z)\,dt}$

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