Geodesic
On linear independence of values of certain $q$-series
Izvestiya. Mathematics , Tome 75 (2011) no. 1, pp. 177-221.

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We obtain qualitative and quantitative results on the linear independence of the values of functions in a fairly wide class generalizing $q$-hypergeometric series and of their derivatives at algebraic points. The results are proved in both the Archimedean and $p$-adic cases.
Keywords: algebraic number field, absolute height of an algebraic number, $q$-series, $q$-exponential function, linear independence, linear independence measure, Hankel determinant
Mots-clés : $q$-logarithm, cyclotomic polynomial. 
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I. P. Rochev. On linear independence of values of certain $q$-series. Izvestiya. Mathematics , Tome 75 (2011) no. 1, pp. 177-221. http://geodesic.mathdoc.fr/item/IM2_2011_75_1_a6/

[1] M. Amou, K. Väänänen, “Linear independence of the values of $q$-hypergeometric series and related functions”, Ramanujan J., 9:3 (2005), 317–339 | DOI | MR | Zbl

[2] M. Amou, M. Katsurada, “Irrationality results for values of generalized Tschakaloff series. II”, J. Number Theory, 104:1 (2004), 132–155 | DOI | MR | Zbl

[3] J.-P. Bézivin, “Independance lineaire des valeurs des solutions transcendantes de certaines equations fonctionnelles”, Manuscripta Math., 61:1 (1988), 103–129 | DOI | MR | Zbl

[4] J.-P. Bézivin, “Sur les Propriétés Arithmétiques d'une Fonction Entière”, Math. Nachr., 190:1 (1998), 31–42 | DOI | MR | Zbl

[5] R. Choulet, “Des résultats d'irrationalité pour deux fonctions particulières”, Collect. Math., 52:1 (2001), 1–20 | MR | Zbl

[6] Ch. Krattenthaler, I. Rochev, K. Väänänen, W. Zudilin, “On the non-quadraticity of values of the $q$-exponential function and related $q$-series”, Acta Arith., 136:3 (2009), 243–269 | DOI | MR | Zbl

[7] K. Väänänen, “On linear independence of the values of generalized Heine series”, Math. Ann., 325:1 (2003), 123–136 | DOI | MR | Zbl

[8] K. Väänänen, W. Zudilin, “Baker-type estimates for linear forms in the values of $q$-series”, Canad. Math. Bull., 48:1 (2005), 147–160 | MR | Zbl

[9] O. Sankilampi, K. Väänänen, “On the values of Heine series at algebraic points”, Results Math., 50:1–2 (2007), 141–153 | DOI | MR | Zbl

[10] T. Matala-Aho, K. Väänänen, “On approximation measures of $q$-logarithms”, Bull. Austral. Math. Soc., 58:1 (1998), 15–31 | DOI | MR | Zbl

[11] M. Haas, “Über die lineare Unabhängigkeit von werten einer speziellen Reihe”, Arch. Math. (Basel), 56:2 (1991), 148–162 | DOI | MR | Zbl

[12] J.-P. Bézivin, “Irrationalité de certaines sommes de séries”, Manuscripta Math., 126:1 (2008), 41–47 | DOI | MR | Zbl

[13] G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge, 1934 | MR | MR | Zbl

[14] R. P. Stanley, Enumerative combinatorics, Wadsworth Brooks/Cole Math. Ser., Wadsworth Brooks, Monterey, CA, 1986 | MR | MR | Zbl | Zbl

[15] Y. Amice, Les nombres $p$-adiques, Presses Universitaires de France, Paris, 1975 | MR | Zbl

[16] G. Pólya, G. Szegő, Aufgaben und Lehrsätze aus der Analysis. Zweiter Band. Funktionentheorie, Nullstellen, Polynome, Determinanten, Zahlentheorie, Julius Springer, Berlin, 1925 | MR | MR | Zbl

[17] I. Rochev, New linear independence measures for values of $q$-hypergeometric series, arXiv: math/1006.5413v1

[18] J.-P. Bézivin, “Indépendance linéaire des valeurs des solutions transcendantes de certaines équations fonctionnelles. II”, Acta Arith., 55:3 (1990), 233–240 | MR | Zbl