Analytic continuation of solutions of the pantograph equation by means of $\theta$-modular formula

TL;DR

This paper explores the analytic continuation of solutions to a specific functional-differential equation using theta-modular formulas, revealing explicit solutions, natural boundaries, and asymptotic behaviors related to special functions.

Contribution

It introduces a novel method using theta-modular relations to express solutions of the pantograph equation as linear combinations of fundamental solutions and analyzes their boundary properties.

Findings

Solutions can be expressed as linear combinations of fundamental solutions at infinity.

Explicit non-lacunary power series have a natural boundary.

Asymptotic behavior is characterized by the Lambert W-function.

Abstract

The aim of this paper is to treat the constant coefficients functional-differential equation $y'(x)=ay(qx)+by(x)$ with the help of the analytic theory of linear $q$-difference equations. When $ab\not=0$, the associated Cauchy problem with $y(0)=1$ admits a unique power series solution, which is the Hadamard product of a usual-hypergeometric series by a basic-hypergeometric series. By means of $\theta$-modular relation, it is proved that this entire function can be expressed as linear combination of all the elements of a system of canonical fundamental solutions at infinity. A family of power series related to values of Gamma function at vertical lines is then introduced, and what really surprises us is that these explicit non-lacunary power series possess a natural boundary. When $a\not=0$ and $b=0$, the asymptotic behavior of solutions will be formulated in terms of the Lambert $W$-function.