Arithmetical investigations of a certain infinite product. (English) Zbl 0802.11027

Bundschuh, Peter; Väänänen, Keijo

Compos. Math. 91, No. 2, 175-199 (1994).

Let \(K\) be an algebraic number field, \(v\) any valuation of \(K\) and \(q\) an element of \(K\) satisfying \(| q|_ v>1\); assume also \(| q|_ w \neq 1\) for all archimedean valuations \(w\) of \(K\). Denote by \(C\) the complex field \(\mathbb{C}\) if \(v\) is archimedean, the \(p\)-adic field \(\mathbb{C}_ p\) if \(v\) is a \(p\)-adic valuation. This paper deals with arithmetic properties of values in \(C\) of the following \(q\)-analog of the exponential function \[ E_ q(z)= \prod_{j=1}^ \infty (1+zq^{- j}). \] In the archimedean case, A. V. Lototskij [Sur l’irrationalité d’un produit infini, Mat. Sb., Nov. Ser. 12(54), 262- 272 (1943)] started the qualitative study of the arithmetic properties of values of this function, and P. Bundschuh [Invent. Math. 6, 275-295 (1969; Zbl 0195.337)] started the quantitative one. Recently, J.-P. Bézivin [Manuscr. Math. 61, 103-129 (1988; Zbl 0644.10025)] proved that for \(\alpha\in K\) with \(\alpha\not\in \{0, -q, -q^ 2,\dots\}\), the numbers \(E_ q (\alpha), E'_ q (\alpha), E_ q^{\prime\prime} (\alpha),\dots\) are all linearly independent over the field of rational numbers. The authors use a different method which yields only a lower bound for the dimension of the \(\mathbb{Q}\)-vector space generated by \(k\) numbers \(E_ q (\alpha), E'_ q (\alpha), \dots, E_ q^{(k-1)} (\alpha)\), but which allows quantitative estimates. For instance, as a consequence of their main result they get an irrationality measure for \[ L_ q (\alpha)= E_ q (\alpha)/ E'_ q (\alpha)= \sum_{j=1}^ \infty (q^ j+ \alpha)^{-1}; \] the exponent of irrationality is bounded by \(4.311\) in the general case, by \(2.509\) in case \(\alpha=-1\). This is a quantitative refinement to Borwein’s irrationality result [P. B. Borwein, J. Number Theory 37, 253-259 (1991; Zbl 0718.11029)]. The authors also achieve an irrationality measure (not only over \(\mathbb{Q}\), but also over \(\mathbb{Q} (\sqrt{5})\)) for the number \(\sum_{n\geq 1} 1/F_ n\), where \(F_ n\) is the \(n\)th Fibonacci number; the exponent of irrationality here is \(8.621\). The irrationality of this number has been proved only recently by R. André-Jeannin [C. R. Acad. Sci., Paris, Sér. I 308, 539-541 (1989; Zbl 0682.10025)].

Reviewer: M.Waldschmidt (Paris)

MSC:

11J72	Irrationality; linear independence over a field
11J61	Approximation in non-Archimedean valuations
11J82	Measures of irrationality and of transcendence

Keywords:

infinite products; values of the \(q\)-analog of the exponential function; arithmetic properties; irrationality measure

Citations:

Zbl 0195.337; Zbl 0644.10025; Zbl 0718.11029; Zbl 0682.10025

References:

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