Matrix methods of summation, regular for 𝑝-adic valuations

Abstract

A method of summation for sequences is defined to be regular if it sums) every convergent sequence (with respect to the absolute value) to its sum. In 1910-1913 Silverman and Toplitz discovered necessary and sufficient conditions for matrix methods of summation to be regular with respect to absolute value. The theory of divergent series has expanded and many useful methods have been investigated, Hardy [I]. Matrices with similar properties for sequences which converge with respect to the p-adic valuations are investigated in this paper. The principal result is a set of necessary and sufficient conditions such that the matrix Mpk defines a summation method which is regular with respect to the p-adic valuation 4p. Whenever it is unnecessary to distinguish between 4p and 4q, the symbol q5 will be used to denote a p-adic valuation, 45,, of the rational field with respect to a prime p. DEFINITION 1. A sequence {ti} obtained from the matrix [amn] and the sequence { Si } using the relationship tm= 1 amiSi is said to be the [amn] transform of {Si}. If the sequence {ti} converges to T, the matrix [amnn] is said to sum the sequence { Si } to the sum T. DEFINITION 2. The method of summation defined by the matrix [amn] is called regular in the p-adic field Q, if every convergent sequence { Si } is equal to its transform { ti } in Q,. Sequences { ti } and Si } are, of course, equal if limm-.1 4)(tm-Sm) 0. Clearly, if { Si } is p-convergent, then {ti} is p-convergent. The converse need not hold.