Linear independence of \(q\)-analogues of certain classical constants (Q1781903)

The paper deals with some results concerning the linear independence of the Lambert series \(\sum_{n=1}^\infty \frac{c_n}{q^n-1}\) and the series \(\sum_{n=1}^\infty \frac{q^n}{(q^n-1)^2}\) where \(q\) is the suitable rational number. They prove e.g. that the numbers \(1\), \(\sum_{n=1}^\infty \frac{1}{q^n-1}\) and \(\sum_{n=1}^\infty \frac{(-1)^{n-1}}{(q^n-1)^2}\) are linearly independent over the rationals. The proofs are based on the calculation of the suitable integrals.