Analytical study of the pantograph equation using Jacobi theta functions

Abstract

The aim of this paper is to use the analytic theory of linear q-difference equations for the study of the functional-differential equation y′(x)=ay(qx)+by(x), where a and b are two non-zero real or complex numbers. When 0<q<1 and y(0)=1, the associated Cauchy problem admits a unique power series solution, ∑n≥0(−a/b;q)nn!(bx)n, that converges in the whole complex x-plane. The principal result obtained in the paper explains how to express this entire function solution into a linear combination of solutions at infinity with the help of integral representations involving Jacobi theta functions. As a by-product, this connection formula between zero and infinity allows one to rediscover the classic theorem of Kato and McLeod on the asymptotic behavior of the solutions over the real axis.