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A [[Semi-continuous summation method|semi-continuous summation method]] for summing series of numbers and functions, defined by a sequence of functions
 
A [[Semi-continuous summation method|semi-continuous summation method]] for summing series of numbers and functions, defined by a sequence of functions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064160/m0641601.png" /></td> </tr></table>
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\[ g_k(\delta) = \frac{1}{\Gamma(1 + \delta k)}, \quad \delta > 0, \quad k = 0, 1, \dots, \]
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064160/m0641602.png" /> is the gamma-function. A series
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where $ \Gamma(x) $ is the gamma-function. A series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064160/m0641603.png" /></td> </tr></table>
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\[ \sum_{k=0}^{\infty} u_k \]
  
is summable by the Mittag-Leffler method to a sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064160/m0641604.png" /> if
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is summable by the Mittag-Leffler method to a sum $s$ if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064160/m0641605.png" /></td> </tr></table>
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\[ \lim\limits_{\delta \to 0}\sum_{k=0}^{\infty} \frac{u_k}{\Gamma(1 + \delta k)} = s \]
  
 
and if the series under the limit sign converges. The method was introduced by G. Mittag-Leffler [[#References|[1]]] primarily for the series
 
and if the series under the limit sign converges. The method was introduced by G. Mittag-Leffler [[#References|[1]]] primarily for the series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064160/m0641606.png" /></td> </tr></table>
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\[ \sum_{k=0}^{\infty} z^k . \]
  
A Mittag-Leffler summation method is regular (see [[Regular summation methods|Regular summation methods]]) and is used as a tool for the [[Analytic continuation|analytic continuation]] of functions. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064160/m0641607.png" /> is the principal branch of an analytic function, regular at zero and represented by a series
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A Mittag-Leffler summation method is regular (see [[Regular summation methods|Regular summation methods]]) and is used as a tool for the [[Analytic continuation|analytic continuation]] of functions. If $ f(z) $ is the principal branch of an analytic function, regular at zero and represented by a series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064160/m0641608.png" /></td> </tr></table>
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\[ \sum_{k=0}^{\infty}a_k z^k \]
  
for small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064160/m0641609.png" />, then this series is summable by the Mittag-Leffler method to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064160/m06416010.png" /> in the whole star of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064160/m06416011.png" /> (cf. [[Star of a function element|Star of a function element]]) and, moreover, uniformly in any closed bounded domain contained in the interior of the star.
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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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064160/m06416012.png" /> of the type
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For summation methods defined by transformations of sequences by semi-continuous matrices $ a_k(\omega) $ of the type
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064160/m06416013.png" /></td> </tr></table>
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\[ a_k(\omega) = \frac{c_{k+1}\omega^{k+1}}{E(\omega)}, \]
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064160/m06416014.png" /></td> </tr></table>
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\[ E(\omega) = \sum_{k=0}^{\infty} c_k \omega^k \]
  
 
is an entire function, Mittag-Leffler considered the case when
 
is an entire function, Mittag-Leffler considered the case when
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064160/m06416015.png" /></td> </tr></table>
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\[ E(\omega) = \sum_{k=0}^{\infty} \frac{\omega^k}{\Gamma(1+ak)} \]
  
A matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064160/m06416016.png" /> with such an entire function is called a Mittag-Leffler matrix.
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A matrix $ a_k(\omega) $ with such an entire function is called a Mittag-Leffler matrix.
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064160/m06416017.png" /> considered by Mittag-Leffler is called a [[Mittag-Leffler function|Mittag-Leffler function]].
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The function $E(\omega)$ considered by Mittag-Leffler is called a [[Mittag-Leffler function|Mittag-Leffler function]].

Latest revision as of 02:56, 2 June 2013


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.

How to Cite This Entry:
Mittag-Leffler summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mittag-Leffler_summation_method&oldid=12118
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article