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 Template:Harvs

Definition

Let

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

be a formal power series in z.

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

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

Then the Mittag-Leffler sum of y is given by

$\underset{\alpha \to 0}{lim}{ℬ}_{\alpha }y(z)$

if each sum converges and the limit exists.

A closely related summation method, also called Mittag-Leffler summation, is given as follows Template:Harv. Suppose that the Borel transform ${ℬ}_{1}y(z)$ 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

${\int }_0^{\mathrm{\infty }}{e}^{-t}{ℬ}_{\alpha }y(t^{\alpha }z)dt$

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

See also

• Mittag-Leffler distribution
• Mittag-Leffler function
• Nachbin's theorem

References

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