Mittag-Leffler summation explained

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 ^{inf ty} _{k =0} y ^{k} _{kz}

be a formal power series in z.

Define the transform

\scriptstyle l{B} _{\alpha} y of \scriptstyle y byl{B} _{\alpha} y(t) \equiv \sum ^{inf ty} _{k =0} (y _{k})/(\Gamma( 1+ \alpha k)) t ^{k}

Then the Mittag-Leffler sum of y is given by

\lim _{\alpha →} l{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

l{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 ^{inf ty} _{0} e ^{-t} l{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