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On Strong Nörlund Summability Fields

Published online by Cambridge University Press:  20 November 2018

Brian Kuttner  and
Brian Thorpe
Brian Kuttner
Affiliation:
University of Birmingham, Birmingham, England
Brian Thorpe
Affiliation:
University of Western Ontario, London, Ontario
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Let p denote the sequence {pn} and set wherever this series converges. (Where no limits are stated, sums are throughout to be taken from n = 0 to n = ∞.) We use a similar notation with other letters in place of p. Given two sequences p, q, the convolution p*q is defined by

it is familiar, and easily verified, that the operation of convolution is commutative and associative. We write Pn = (p*l)n (where 1 denotes the sequence {1} ), and take P-1 to mean 0. If, for all n ≦ 0, Pn ≠ 0, then we define the Nörlund mean (N, p) of the sequence s as σn, where

and (σ-1 = 0. If σn → λ as n → ∞, then 5 is said to be limitable (N, p) to the number λ

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 3 , June 1972 , pp. 390 - 399
Copyright
Copyright © Canadian Mathematical Society 1972

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