Some theorems on absolute Cesàro summability

Source abstract

1. It is convenient to begin with a brief statement of the notation which will be used throughout this paper. Let k be any positive number and let where is the coefficient of x n in the formal expansion of (1 – x ) –k–1 , and let Then the series Σα n is said to be summable( C, k ) if is convergent, that is, if tends to a limit, and absolutely summable ( C, k ), or summable | C, k |, if is absolutely convergent.