Lindelöf summation method

A semi-continuous method for summing series of numbers and functions (cf. Summation methods), defined by the system of functions

$$ g _ {0} ( \delta ) = 1 ,\ \ g _ {k} ( \delta ) = \mathop{\rm exp} ( - \delta k \mathop{\rm ln} k ) ,\ \ \delta > 0 ,\ k = 1 , 2 , . . . . $$

The series

$$ \sum _ {k=0} ^ \infty u _ {k} $$

is summable by the Lindelöf summation method to the sum $ s $ if

$$ \lim\limits _ {\delta \rightarrow 0 } \ \left [ u _ {0} + \sum _ {k=0} ^ \infty \mathop{\rm exp} ( - \delta k \mathop{\rm ln} k ) u _ {k} \right ] = s $$

and the series under the limit sign converges. The method was introduced by E. Lindelöf [1] for the summation of power series.

The Lindelöf 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 the origin and representable by a series

$$ \sum _ {k=0}^ \infty a _ {k} z ^ {k} $$

for small $ z $, then this series is summable by the Lindelöf summation method to $ f ( z) $ in the whole star of the function $ f ( z) $( cf. Star of a function element), and it is uniformly summable in every closed bounded domain contained in the interior of the star.

Of the summation methods determined by a transformation of a sequence into a sequence by semi-continuous matrices $ a _ {k} ( \omega ) $ of 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, Lindelöf considered the case when

$$ E ( \omega ) = \sum _ {k=0}^ \infty \left [ \frac \omega { \mathop{\rm ln} ( k + \beta ) } \right ] ^ {k} ,\ \ \beta > 1 . $$

A matrix $ \| a _ {k} ( \omega ) \| $ constructed from an entire function of this kind is called a Lindelöf matrix.

References

[1]	E. Lindelöf, J. Math. , 9 (1903) pp. 213–221
[2]	E. Lindelöf, "Le calcul des résidus et ses applications à la théorie des fonctions" , Gauthier-Villars (1905)
[3]	G.H. Hardy, "Divergent series" , Clarendon Press (1949)
[4]	R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)