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

Details Analytic continuation of solutions of the pantograph equation by means of $θ$-modular formula Analytic continuation of solutions of the pantograph equation by means of $θ$-modular formula

2012-02-02

arxiv.org/abs/1202.0423v1

Mathematics

Classical Analysis and ODEs

29 pages

Scientific paper

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.

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