# 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 ${\mathcal {B}}_{\alpha }y$ of $y$ 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 ${\mathcal {B}}_{1}y(z)$ 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.