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