On complex singularity analysis for linear partial q-difference-differential equations using nonlinear differential equations

Abstract

We investigate the existence of local holomorphic solutions of linear q-difference-differential equations in two variables t; z whose coefficients have poles or algebraic branch points singularities in the variable t. These solutions are shown to develop poles or algebraic branch points along half q-spirals. We also give bounds for the rate of growth of the solutions near the singular points. We construct these solutions with the help of functions of infinitely many variables that satisfy functional equations that involve q-difference, partial derivatives and shift operators. We show that these functional equations have solutions in some Banach spaces of holomorphic functions in �� having sub-exponential growth.