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

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)].

### MSC:

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

### Citations:

Zbl 0195.337; Zbl 0644.10025; Zbl 0718.11029; Zbl 0682.10025
Full Text:

### References:

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