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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[edit]

Let

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

be a formal power series in z.

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

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

lim α → 0 B α y ( z ) {\displaystyle \lim _{\alpha \rightarrow 0}{\mathcal {B}}_{\alpha }y(z)} {\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 B 1 y ( z ) {\displaystyle {\mathcal {B}}_{1}y(z)} {\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

∫ 0 ∞ e − t B α y ( t α z ) d t {\displaystyle \int _{0}^{\infty }e^{-t}{\mathcal {B}}_{\alpha }y(t^{\alpha }z)\,dt} {\displaystyle \int _{0}^{\infty }e^{-t}{\mathcal {B}}_{\alpha }y(t^{\alpha }z)\,dt}

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

See also[edit]

References[edit]

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