Borel summation method

A method for summing series of functions, proposed by E. Borel [1]. Suppose one is given a series of numbers

$$ \tag{* } \sum _ { k=0 } ^ \infty a _ {k} , $$

let $ S _ {n} $ be its partial sums and let $ S $ be a real number. The series (*) is summable by the Borel method ( $ B $- method) to the number $ S $ if

$$ \lim\limits _ {x \rightarrow \infty } \ e ^ {-x} \sum _ { k=0 } ^ \infty \frac{x ^ {k} }{k!} S _ {k} = S . $$

There exists an integral summation method due to Borel. This is the $ B ^ { \prime } $- method: If

$$ \int\limits _ { 0 } ^ \infty e ^ {-u} \sum _ { k=0 } ^ \infty \frac{a _ {k} u ^ {k} }{k!} du = S, $$

then one says that the series (*) is summable by the $ B ^ { \prime } $- method to the number $ S $. For conditions under which the two methods $ B $ and $ B ^ { \prime } $ are equivalent, cf. [2]. The $ B $- method originated in the context of analytic extension of a function regular at a point. Let

$$ f(z) = \sum _ { n=0 } ^ \infty a _ {n} z ^ {n} $$

be regular at the point $ O $ and let $ C $ be the set of all its singular points. Draw the segment $ OP $ and the straight line $ L _ {P} $ normal to $ OP $ through any point $ P \in C $. The set of points on the same side with $ O $ for each straight line $ L _ {P} $ is denoted by $ \Pi $; the boundary $ \Gamma $ of the domain $ \Pi $ is then called the Borel polygon of the function $ f(z) $, while the domain $ \Pi $ is called its interior domain. The following theorem is valid: The series

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

is summable by the $ B ^ { \prime } $- method in $ \Pi $, but not in the domain $ \Pi ^ {*} $ which is the complement of $ \Pi $[2].

References

[1]	E. Borel, "Mémoire sur les séries divergentes" Ann. Sci. École Norm. Sup. (3) , 16 (1899) pp. 9–131
[2]	G.H. Hardy, "Divergent series" , Clarendon Press (1949)

References

[a1]	W. Beekmann, K. Zeller, "Theorie der Limitierungsverfahren" , Springer (1970)