Arithmetical properties of the solutions of certain functional equations (Q2710113)

The functional equations of the title are \(q\)-differential equations. The author surveys the arithmetic of \(q\) analogues of the classical functions where he has made many significant contributions. The theory contains general theorems on irrrationality over a field \(K\), which is traditionally either the rationals or an imaginary quadratic field. The examples include the original Tschakaloff function \(T_q(z)=\sum_{n=0}^\infty z^n/q^{n(n-1)/2}\) and another, \(F_q(x,y)=\prod_{j=1}^\infty(1+xq^{-j}+xyq^{-2j})\), from his own recent work [Analysis, München 19, 93-101 (1999; Zbl 0940.11028)]. Suppose \(q\) is an integer in \(K\). The following statements typify the results. If \(\alpha\) is a nonzero element of \(K\), then \(T_q(\alpha)\) is not in \(K\). If \(\alpha, \beta\) are nonzero elements of \(K\) and \(1\pm\alpha q^{-j}+\beta q^{-2j}\neq 0\) for any \(j\), then \(F_q(\alpha,\beta),F_q(-\alpha,-\beta)\) are not both in \(K\). The paper also contains a proof of a quantitative version of this result. The author conjectures that \(F_q(\alpha,\beta)\) is not in \(K\) without any assumptions on \(1-\alpha q^{-j}+\beta q^{-2j}\).NEWLINENEWLINENEWLINE\textit{J.-P. Bézivin} [Manuscr. Math. 61, 103-129 (1988; Zbl 0644.10025), and Acta Arith. 55, 233-240 (1990; Zbl 0712.11038)] has recently found a different approach showing that \(\alpha\) and \(T_q(\alpha)\) cannot both belong to any quadratic number field when \(\alpha\neq 0\) and \(q\) is rational. The author and \textit{R. Wallisser} [Abh. Math. Semin. Univ. Hamb. 69, 103-122 (1999; Zbl 0961.11022)] have recently given quantitative linear independence results is this more general setting.NEWLINENEWLINENEWLINETranscendence has proved much harder. Recently, using Nesterenko's theory, \textit{D. Duverney, Ke. Nishioka, Ku. Nishioka} and \textit{I. Shiokawa} [Proc. Japan Acad., Ser. A Math. Sci. 72, 202-203 (1996; Zbl 0884.11030)] showed the transcendence of \(T_q(1)\) and some related examples.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00053].