An Inclusion Relation for Abel, Borel, and Lambert Summability

Abstract

In this paper a new type of inclusion theorem concerning Abel, Borel and Lambert summability is established. To state our results we need some definitions and notations. With a formal series ∑k=0∞ak, ak∈ C, and its partial sums snwe associate the seriesThen ∑k=0∞ ak is said to be summable to the value s(a) by Abel's method, if (1.1) is convergent for |v| > 1 and limv→1+A(v)= s,(b) by Lambert's method, if (1.2) is convergent for |v| > 1 and limv→1+L(v)= s,(c) by Borel's method, if (1.3) is convergent for all x ∈ R and limx→+∞B(x)= s,