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Title: The Hurwitz Product, p-Adic Topology on ℤ, and Fundamental Solution to Linear Differential Equation in the Ring ℤ[[x]]

Abstract

We study the linear inhomogeneous first order differential equation by′ + f(x) = y in the ring of formal power series with integer coefficients. Using the p-adic topology on the ring of integers, we construct a counterpart of the Hurwitz product of the Euler series and an arbitrary formal power series with integer coefficients. It is shown that the Euler series can be interpreted as the fundamental solution to the equation under consideration. Bibliography: 8 titles.

Authors:
;  [1]
  1. Karazin Kharkiv National University (Ukraine)
Publication Date:
OSTI Identifier:
22771584
Resource Type:
Journal Article
Journal Name:
Journal of Mathematical Sciences
Additional Journal Information:
Journal Volume: 228; Journal Issue: 6; Other Information: Copyright (c) 2018 Springer Science+Business Media, LLC, part of Springer Nature; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1072-3374
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; DIFFERENTIAL EQUATIONS; MATHEMATICAL SOLUTIONS; POWER SERIES; TOPOLOGY

Citation Formats

Gefter, S. L., E-mail: gefter@univer.kharkov.ua, and Goncharuk, A. B., E-mail: a-ryska@ukr.net. The Hurwitz Product, p-Adic Topology on ℤ, and Fundamental Solution to Linear Differential Equation in the Ring ℤ[[x]]. United States: N. p., 2018. Web. doi:10.1007/S10958-017-3651-6.
Gefter, S. L., E-mail: gefter@univer.kharkov.ua, & Goncharuk, A. B., E-mail: a-ryska@ukr.net. The Hurwitz Product, p-Adic Topology on ℤ, and Fundamental Solution to Linear Differential Equation in the Ring ℤ[[x]]. United States. doi:10.1007/S10958-017-3651-6.
Gefter, S. L., E-mail: gefter@univer.kharkov.ua, and Goncharuk, A. B., E-mail: a-ryska@ukr.net. Thu . "The Hurwitz Product, p-Adic Topology on ℤ, and Fundamental Solution to Linear Differential Equation in the Ring ℤ[[x]]". United States. doi:10.1007/S10958-017-3651-6.
@article{osti_22771584,
title = {The Hurwitz Product, p-Adic Topology on ℤ, and Fundamental Solution to Linear Differential Equation in the Ring ℤ[[x]]},
author = {Gefter, S. L., E-mail: gefter@univer.kharkov.ua and Goncharuk, A. B., E-mail: a-ryska@ukr.net},
abstractNote = {We study the linear inhomogeneous first order differential equation by′ + f(x) = y in the ring of formal power series with integer coefficients. Using the p-adic topology on the ring of integers, we construct a counterpart of the Hurwitz product of the Euler series and an arbitrary formal power series with integer coefficients. It is shown that the Euler series can be interpreted as the fundamental solution to the equation under consideration. Bibliography: 8 titles.},
doi = {10.1007/S10958-017-3651-6},
journal = {Journal of Mathematical Sciences},
issn = {1072-3374},
number = 6,
volume = 228,
place = {United States},
year = {2018},
month = {2}
}