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)].


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


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.