Analytical study of the pantograph equation using Jacobi theta functions

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 Sigma(n >= 0) (-a/b; q)n/n! (bx)(n), that converges in the whole complex x-plane. The principal result series solution, 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.(c) 2023 Elsevier Inc. All rights reserved.