## 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.

### MSC:

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

Zbl 1202.11039
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### References:

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