zbMATH — the first resource for mathematics

On \(q\)-analogues of divergent and exponential series. (English) Zbl 1169.11031
Let \(\mathbb K\) be a number field. Assume that \(m\in \mathbb Z^+\). Suppose that \(a, q, \alpha_1, \dots , \alpha_m \in \mathbb K \setminus \{ 0\}\). Denote by \(f(t)\) each of the functions \(\sum_{n=0}^\infty a^n \prod_{j=1}^n (1-t^j)\), \(\sum_{n=0}^\infty t^n \prod_{j=1}^n (1-q^j)^{-1}\) and \(\prod_{n=0}^\infty (1-tq^n)\). Under the special conditions the author proves that the numbers \(1\), \(f(\alpha_1)\), …, \(f(\alpha_m)\) are linearly independent over \(\mathbb K\) and gives the bound for the measure. The proof makes use of Padé approximation.

11J82 Measures of irrationality and of transcendence
11J72 Irrationality; linear independence over a field
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
Full Text: DOI
[1] M. Amou, T. Matala-aho and K. Väänänen, On Siegel-Shidlovskii’s theory for \(q\)-difference equations, Acta Arith., 127 (2007), 309-335. · Zbl 1113.11042
[2] Y. André, Séries Gevrey de type arithmétique II, Transcendance sans transcendance, Ann. of Math., 151 (2000), 741-756. JSTOR: · Zbl 1037.11050
[3] G. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, 71 , Cambridge University Press, Cambridge, 1999. · Zbl 0920.33001
[4] B. C. Berndt, S. H. Chan, B. P. Yeap and Ae. J. Yee, A reciprocity theorem for certain \(q\)-series found in Ramanujan’s lost notebook, The Ramanujan J., 13 (2007), 27-37. · Zbl 1123.33012
[5] J.-P. Bézivin, Indépendance linéaire des valeurs des solutions trascendantes de certaines équations fonctionnelles, Manuscripta Math., 61 (1988), 103-129. · Zbl 0644.10025
[6] P. Bundschuh, Ein Satz über ganze Funktionen und Irrationalitätsaussagen, Invent. Math., 9 (1970), 175-184. · Zbl 0188.10801
[7] P. Bundschuh and K. Väänänen, Linear independence measures for infinite products, Manuscripta Math., 105 (2001), 253-263. · Zbl 0999.11036
[8] P. Bundschuh and R. Wallisser, Masse für die lineare Unabhängigkeit von Werten ganz transzendenter Lösungen gewisser Funktionalgleichungen, Abh. Math. Sem. Univ. Hamburg, 69 (1999), 103-122. · Zbl 0961.11022
[9] M. Katsurada, Linear independence measures for values of Heine series, Math. Ann., 284 (1989), 449-460. · Zbl 0653.10031
[10] A. V. Lototsky, Sur l’irrationalité d’un produit infini, Mat. Sbornik, 12 (1943), 262-272. · Zbl 0063.03644
[11] W. Maier, Potenzreihen irrationalen Grenzwertes, J. Reine Angew. Math., 156 (1927), 93-148. · JFM 53.0340.02
[12] T. Matala-aho, Remarks on the arithmetic properties of certain hypergeometric series of Gauss and Heine (Thesis), Acta Univ. Oul. A, 219 (1991), 1-112. · Zbl 1166.11342
[13] T. Matala-aho and K. Väänänen, On Diophantine approximations of Mock theta functions of third order, Ramanujan J., 4 (2000), 13-28. · Zbl 1001.11030
[14] O. Sankilampi and K. Väänänen, On the values of Heine series at algebraic points, Results Math., 50 (2007), 141-153. · Zbl 1206.11089
[15] Th. Stihl, Arithmetische Eigenschaften spezieller Heinescher Reihen, Math. Ann., 268 (1984), 21-41. · Zbl 0519.10024
[16] Th. Stihl and R. Wallisser, Zur Irrationalität und linearen Unabhängigkeit der Werte der Lösungen einer Funktionalgleichung von Poincaré, J. Reine Angew. Math., 341 (1983), 98-110. · Zbl 0497.10027
[17] K. Väänänen, On linear independence of the values of generalized Heine series, Math. Ann., 325 (2003), 123-136. · Zbl 1025.11023
[18] K. Väänänen and W. Zudilin, Baker-Type estimates for linear forms in the values of \(q\)-series, Can. Math. Bull., 48 (2005), 147-160. · Zbl 1064.11054
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.