On ‘translated quasi-Cesàro’ summability

Abstract

Corresponding to a fixed sequence {μn}, the Hausdorff method of summability (H, μn) is defined by the sequence-to-sequence transformation†where we writeThe quasi-Hausdorff method (H*, μn) is defined by the transformationthus the matrix of the (H*, μn) transformation is the transpose of that of the (H*, μn) transformation. A method introduced by Ramanujan (9), which we will call‡ (S,μn) is given by the transformationThus the elements of row n of the matrix of the (S, μn) transformation are those of the corresponding row of the (H*, μn) transformation moved n places to the left.

On generalized translated quasi-Cesàro summability

“…3* Theorems* The following two theorems with β = 0 are Theorem 1' and Theorem 2' given by Kuttner [6]. The proof of Theorem 1 is similar to that of Theorem 1' in [6], and Theorem 2 follows from Lemma 1 and Lemma 2 of this paper.…”

“…and since a > 0, Q ^> β + 3, we see that the contribution to the expression on the left of (6) of the "0" term in (7) is Hence the result would follow if (corresponding to Lemma 2 of [6]) we could prove that the convergence of (3) implied that, for relevant 4, r,…”

“…From equations analogous to those of the last line and line 6 from bottom of p. 709 of [6], we find that where j>(ί) is a polynomial in ^ (which may be different for each term in the sum), and the sum is taken over those integers q, r which are such that q ;> 1, r Ξ> 1, q, r not both 1, q + r ^ Q .…”

“…As in [6], we may suppose that s 0 -0. Then the result would follow if, corresponding to equation (11) of [6], we proved that, if (3) converges, then uniformly for 0 ^ Θ < 1,…”

Generalised quasi-Nörlund summability

“…Kuttner [7] defined quasi-Cesaro summability and investigated its main properties as a quasi-Hausdorff transformation (see also Ramunujan [8] and White [11]. Thorpe [9] defined quasi-Nόrlund (quasi-Riesz) summability.…”