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

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. 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.