# 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:

- 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}

}

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