Parametric summability and its applications to maximizing of the summability domain

In this paper, we study parametric summability based on parameterized double sequences of complex constants as it is defined in "Linear Operators, General Theory" by N. Dunford and J. T. Schwartz.We define "power double sequences" or infinite "power matrices" as certain generalizations of double sequences and power series.We show that the parameter dependence of the summability of power double sequences is similar to convergence of power series and we introduce the radius of summability.That opens a way to maximize the summability domain using the radius of summability.While others do investigate "power matrices, " their definitions, as far as we were able to find, differ from ours.Using our approach, we find new summability results for double sequences of constants in the case of power double sequences.We will give some applications to both standard summability theory and analytic functions.In section 7, we provide some examples to demonstrate the main results of this paper obtained in sections 5 and 6.Finally, to conclude this paper, in the last section, we give some ideas related to parametric summability for further study.