Product Theorems for Certain Summability Methods in Non-archimedean Fields

In this paper, K denotes a complete, non-trivially valued, nonarchimedean field. Sequences and infinite matrices have entries in K. The main purpose of this paper is to prove some product theorems involving the methods M and (N, pn) in such fields K. AMS subject classification: 40, 46.

Throughout the present paper, K denotes a complete, non-trivially valued, non-archimedean field.Sequences and infinite matrices have entries in K.
Given an infinite matrix A = (a nk ), n, k = 0, 1, 2, • • • and a sequence x = {x k }, k = 0, 1, 2, • • • , by the A-transform of x = {x k }, we mean the sequence A(x) = {(Ax) n }, where l, we say that the matrix method A is regular.Necessary and sufficient conditions for A to be regular in terms of its entries are well-known (see [2]).
The (N, p n ) methods (or Nörlund methods) were introduced in K and some of their properties were studied earlier by Srinivasan (see [6]).A more detailed study of the (N, p n ) methods was taken up by the author later and published in a series of articles (for instance, see [4], [5]).
The (N, p n ) method is defined by the matrix A = (a nk ), where where The following result is known ([4], Theorem 1).
Theorem 1.1:The (N, p n ) method is regular if and only if Let {λ n } be a sequence in K such that lim In this context, we note that the M method reduces to the Y method of Srinivasan (see [6]), when K = Q p , the p-adic field for a prime p, λ We need the following definition in the sequel.
Definition: Two matrix methods A = (a nk ), B = (b nk ) are said to be consistent, if whenever x = {x k } is A-summable to s and B-summable to t, then s = t.
It is clear that the relation "matrices A and B are consistent" is an equivalence relation.
We now recall that a product theorem means the following: given regular methods A, B, does x = {x k } ∈ (A) imply B(x) ∈ (A), limits being the same, where (A) is the convergence field of A? i.e., does "A(x) converges" imply "A(B(x)) converges to the same limit"?
The main purpose of this paper is to prove some product theorems involving the M, (N, p n ) methods in K.In the sequel, we suppose that the (N, p n ) methods are regular.