Mittag-Leffler summation method

A semi-continuous summation method for summing series of numbers and functions, defined by a sequence of functions

\[ g_k(\delta) = \frac{1}{\Gamma(1 + \delta k)}, \quad \delta > 0, \quad k = 0, 1, \dots, \]

where $ \Gamma(x) $ is the gamma-function. A series

\[ \sum_{k=0}^{\infty} u_k \]

is summable by the Mittag-Leffler method to a sum $s$ if

\[ \lim\limits_{\delta \to 0}\sum_{k=0}^{\infty} \frac{u_k}{\Gamma(1 + \delta k)} = s \]

and if the series under the limit sign converges. The method was introduced by G. Mittag-Leffler [1] primarily for the series

\[ \sum_{k=0}^{\infty} z^k . \]

A Mittag-Leffler summation method is regular (see Regular summation methods) and is used as a tool for the analytic continuation of functions. If $ f(z) $ is the principal branch of an analytic function, regular at zero and represented by a series

\[ \sum_{k=0}^{\infty}a_k z^k \]

for small $z$, then this series is summable by the Mittag-Leffler method to $ f(z) $ in the whole star of the function $ f(z) $ (cf. Star of a function element) and, moreover, uniformly in any closed bounded domain contained in the interior of the star.

For summation methods defined by transformations of sequences by semi-continuous matrices $ a_k(\omega) $ of the type

\[ a_k(\omega) = \frac{c_{k+1}\omega^{k+1}}{E(\omega)}, \]

where

\[ E(\omega) = \sum_{k=0}^{\infty} c_k \omega^k \]

is an entire function, Mittag-Leffler considered the case when

\[ E(\omega) = \sum_{k=0}^{\infty} \frac{\omega^k}{\Gamma(1+ak)} \]

A matrix $ a_k(\omega) $ with such an entire function is called a Mittag-Leffler matrix.

References

[1]	G. Mittag-Leffler, , Atti IV congress. internaz. , 1 , Rome (1908) pp. 67–85
[2]	G. Mittag-Leffler, "Sur la répresentation analytique d'une branche uniforme d'une fonction monogène" Acta Math. , 29 (1905) pp. 101–181
[3]	G.H. Hardy, "Divergent series" , Clarendon Press (1949)
[4]	R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)

Comments

The function $E(\omega)$ considered by Mittag-Leffler is called a Mittag-Leffler function.