Hausdorff methods of summation and continued fractions

Abstract

where {IS,,} is an infinite sequence of numbers. If, in particular, we are concerned with the infinite series E,0u,, then sn =Efou,. Let I = (amn) represent the triangular matrix of the transformation (1.1). Then, the sequence tm } is called the transform of { s, } by the matrix !K and is represented in symbolic form by tm = 2[ { S, }. If the matrix 2[ is regular, in the sense of Silverman [I] and Toeplitz [2], then the matrix 2 defines a regular method of summation (summability). In this paper our attention is focused on a class of regular and permutable matrices { 2[ }, known as Hausdorff matrices ( [3] or [4]), which we shall presently define. Let Cm } be an infinite sequence of numbers defined by the Stieltjes integrals