Arithmetic properties of infinite products of cyclotomic polynomials (Q2814336)

The paper considers infinite products of the type NEWLINE\[NEWLINE F_{d,l}(z) := \prod_{j \geq 0} \Phi_l(z^{d^j} \in \mathbb{Z} [[z]] NEWLINE\]NEWLINE where \(\Phi_l\) denotes the cyclotomic polynomial for \(l>1\) and \(d\geq 2\) an integer. Among these functions there are some generating functions of well-known number theoretic sequences. The authors show that with the exception of the case that \(d\) is a prime dividing \(l\), the functions \(F_{d,l}(z)\) are hypertranscendental, i.e., they do not satisfy any algebraic differential equation. Moreover, their values at algebraic points \(\alpha, \, 0<|\alpha |< 1\,\), are also transcendental (and -- under some obvious additional hypothesis -- even algebraically independent for fixed \(\alpha\) and \(d\)).NEWLINENEWLINEThe proofs of these strong theorems rely mainly on Mahler's method (see [\textit{K. Nishioka}, Mahler functions and transcendence. Berlin: Springer (1996; Zbl 0876.11034)]), on papers of \textit{K. K. Kubota} [Math. Ann. 227, 9--50 (1977; Zbl 0359.10030)], \textit{P. Corvaja} and \textit{U. Zannier} [Compos. Math. 131, No. 3, 319--340 (2002; Zbl 1010.11038)] and the first author [Springer Proceedings in Mathematics and Statistics 43, 143--156 (2013; Zbl 1325.11070)].