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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
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References:

[1] Adams, W.W. : Transcendental numbers in the p-adic domain , Amer. J. Math. 88 (1966) 279-308. · Zbl 0144.29301
[2] André-Jeannin, R. : Irrationalité de la somme des inverses de certaines suites récurrentes , C.R. Acad. Sci. Paris, Ser. I Math. 308 (1989) 539-541. · Zbl 0682.10025
[3] Bézivin, J.P. : Indépendance linéaire des valeurs des solutions transcendantes de certaines équations fonctionnelles , Manuscripta Math. 61 (1988) 103-129. · Zbl 0644.10025
[4] Borwein, P.B. : On the irrationality of \Sigma (1/(qn + r)) , J. Number Theory 37 (1991) 253-259. · Zbl 0718.11029
[5] Bundschuh, P. , Arithmetische Untersuchungen unendlicher Produkte , Inventiones Math. 6 (1969) 275-295. · Zbl 0195.33703
[6] Bundschuh, P. und Töpfer, T. : Über lineare Unabhängigkeit (submitted). · Zbl 0804.11044
[7] Erdös, P. : On arithmetical properties of Lambert series , J. Indian Math. Soc.(N.S.) 12 (1948) 63-66. · Zbl 0032.01701
[8] Exton, H. : q- Hypergeometric Functions and Applications , Ellis Horwood Ltd., Chichester (1983). · Zbl 0514.33001
[9] Gel’Fond, A.O. : Functions which take on integral values (Russian) , Mat. Zametki 1 (1967) 509-513; Engl. transl., Math. Notes 1, 337-340. · Zbl 0211.07103
[10] Lototsky, A.V. : Sur l’irrationalité d’un produit infini , Math. Sbornik 12 (54) (1943) 262-272. · Zbl 0063.03644
[11] Nesterenko, Yu. V. : On the linear independence of numbers (Russian) , Vestnik Moskov. Univ. Ser. I Math Mekh. 1 (1985) 46-49; Engl. transl., Moscow Univ. Math. Bull. 40, 69-74. · Zbl 0572.10027
[12] Popov, A. Yu. : Arithmetical properties of values of some infinite products (Russian) . In: Diophantine Approximations 2 , Collect. Artic., Moskva (1986) pp. 63-78. · Zbl 0648.10022
[13] Popov, A. Yu. : Approximation of values of some infinite products (Russian), Vestnik Moskov. Univ. Ser. I Mat. Mekh. 6 (1990) 3-6; Engl. transl., Moscow Univ. Math. Bull. 45, 4-6. · Zbl 0722.11036
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