## Journal of the Mathematical Society of Japan

- J. Math. Soc. Japan
- Volume 70, Number 2 (2018), 711-731.

### A functional equation with Borel summable solutions and irregular singular solutions

#### Abstract

A Functional equation $\sum_{i=1}^{m}a_{i}(z)u(\varphi_{i}(z))=f(z)$ is considered. First we show the existence of solutions of formal power series. Second we study the homogeneous equation $(f(z)\equiv 0)$ and construct formal solutions containing exponential factors. Finally it is shown that there exists a genuine solution in a sector whose asymptotic expansion is a formal solution, by using the theory of Borel summability of formal power series. The equation has similar properties to those of irregular singular type in the theory of ordinary differential equations.

#### Article information

**Source**

J. Math. Soc. Japan, Volume 70, Number 2 (2018), 711-731.

**Dates**

First available in Project Euclid: 18 April 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.jmsj/1524038671

**Digital Object Identifier**

doi:10.2969/jmsj/07027491

**Mathematical Reviews number (MathSciNet)**

MR3787737

**Zentralblatt MATH identifier**

1395.30031

**Subjects**

Primary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX]

Secondary: 39B32: Equations for complex functions [See also 30D05] 44A10: Laplace transform

**Keywords**

Borel summable irregular singular functional equation

#### Citation

ŌUCHI, Sunao. A functional equation with Borel summable solutions and irregular singular solutions. J. Math. Soc. Japan 70 (2018), no. 2, 711--731. doi:10.2969/jmsj/07027491. https://projecteuclid.org/euclid.jmsj/1524038671