Quantitative linear independence of an infinite product and its derivatives. (English) Zbl 1181.11045

Let \(q\) be an integer with \(|q|>1\) and \[ H_q(z)=\prod_{n=1}^{\infty}\left(1+\frac{q^nz}{q^{2n}+1}\right), \qquad \Lambda_q=\prod_{n=1}^{\infty}\frac{q^{3n}}{(q^n-1)(q^{2n}+1)}. \] The authors establish a linear independence measure over \({\mathbb Q}\) for \(1, \Lambda_q\) and the values of \(H_q\) and its derivatives (up to some order) at a certain finite number of distinct non-zero rational points. This result is a quantitative and qualitative improvement of J.-P. Bézivin’s main theorem from [Manuscr. Math. 126, No. 1, 41-47 (2008; Zbl 1202.11039)]. As a consequence, the authors deduce irrationality measures for the values of the logarithmic derivative of \(H_q\) \[ \frac{H'_q(z)}{H_q(z)}=\sum_{n=1}^{\infty}\frac{q^n}{q^{2n}+q^nz+1} \] at suitable rational points.


11J82 Measures of irrationality and of transcendence
11J72 Irrationality; linear independence over a field
41A21 Padé approximation


Zbl 1202.11039
Full Text: DOI


[1] Amou M., Matala-aho T., Väänänen K.: On Siegel-Shidlovskii’s theory for q-difference equations. Acta Arith. 127, 309–335 (2007) · Zbl 1113.11042
[2] Bézivin J.-P.: Indépendance linéaire des valeurs des solutions transcendantes de certaines equations fonctionnelles. Manuscripta Math. 61, 103–129 (1988) · Zbl 0644.10025
[3] Bézivin J.-P.: Irrationalité de certaines sommes de séries. Manuscripta Math. 126, 41–47 (2008) · Zbl 1202.11039
[4] Bundschuh P., Shiokawa I.: A measure for the linear independence of certain numbers. Res. Math. 7, 130–144 (1984) · Zbl 0552.10021
[5] Bundschuh P., Väänänen K.: On the simultaneous diophantine approximation of new products. Analysis 20, 387–393 (2000) · Zbl 0966.11030
[6] Bundschuh P., Zudilin W.: Irrationality measures for certain q-mathematical constants. Math. Scand. 101, 104–122 (2007) · Zbl 1153.11034
[7] Duverney D.: Some arithmetical consequences of Jacobi’s triple product identity. Math. Proc. Cambridge Philos. Soc. 122, 393–399 (1997) · Zbl 0889.11026
[8] Katsurada M.: Linear independence measures for the values of Heine series. Math. Ann. 284, 449–460 (1989) · Zbl 0653.10031
[9] Koivula L., Sankilampi O., Väänänen K.: A linear independence measure for the values of Tschakaloff function and an application. JP J. Algebra Number Theory Appl. 6, 85–101 (2006) · Zbl 1173.11337
[10] Nesterenko, Yu.V.: Modular functions and transcendence questions. Mat. Sb. 187, 65–96 (1996); Engl. transl.: Sb. Math. 187, 1319–1348 (1996) · Zbl 0898.11031
[11] Rochev, I.: On linear independence of values of certain q-series (Russ.). Submitted · Zbl 1220.11089
[12] Smet, C., Van Assche, W.: Irrationality proof of a q-extension of {\(\zeta\)}(2) using little q-Jacobi polynomials. Submitted · Zbl 1226.11074
[13] Stihl T.: Arithmetische Eigenschaften spezieller Heinescher Reihen. Math. Ann. 268, 21–41 (1984) · Zbl 0533.10031
[14] Väänänen K., Zudilin W.: Linear independence of values of Tschakaloff functions with different parameters. J. Number Theory 128, 2549–2558 (2008) · Zbl 1173.11044
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