A Tauberian theorem for a generalized power series method Original Research Article

The power series summability method is one of the most important and general methods in Summability Theory. In this work we will extend the power series method by considering the Bürmann series method (i.e. View the MathML sourcefs(zn)=1f(zn)∑k=0∞skbk[h(zn)]k, where View the MathML sourcef(zn)=∑k=0∞bk[h(zn)]k and |h(zn)|<1|h(zn)|<1). Using the generalization we obtain a Tauberian theorem which contains, as special cases, some standard theorems for Abel, Borel, and Sonnenschein methods. We use the Abel and Borel matrix methods to illustrate this theorem.