• Source: 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
    )
    =

    ∑

    k
    =
    0


    ∞



    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}

    of




    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}}


    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)}


    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)}

    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}


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


    See also


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


    References



    "Mittag-Leffler summation method", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
    Mittag-Leffler, G. (1908), "Sur la représentation arithmétique des fonctions analytiques d'une variable complexe", Atti del IV Congresso Internazionale dei Matematici (Roma, 6–11 Aprile 1908), vol. I, pp. 67–86, archived from the original on 2016-09-24, retrieved 2012-11-02
    Sansone, Giovanni; Gerretsen, Johan (1960), Lectures on the theory of functions of a complex variable. I. Holomorphic functions, P. Noordhoff, Groningen, MR 0113988

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