2012/02/02 by Changgui Zhang, Zhang, Changgui
Engineering · Mathematics · #33E30 #34K06 #34M40 #Classical Analysis and ODEs (math.CA) #Control and Dynamics of Mobile Robots #FOS: Mathematics #Nonlinear Differential Equations Analysis
paper · pdf · doi:10.48550/arxiv.1202.0423
openalex publication_date 2012/02/02 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
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 θ-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.