Linear Independence of q-Analogues of Certain Classical Constants

Linear independence of certain Lambert and allied series

“…The case α = 1 of this theorem is Theorem 2 of [3]. We can also give a generalization of Corollary 1 of [3] and Theorem 1 and its corollary in [8].…”

“…We can also give a generalization of Corollary 1 of [3] and Theorem 1 and its corollary in [8]. For this, we introduce the q-logarithm, the first formula defining L q in |z| < |q| only:…”

“…Since a and b are linearly independent, we have d = 0. We now construct approximations to f (d, α) similarly to [3], see also [2]. For this we use the complex integral …”

“…Introduction. In our recent work [3] we considered linear independence of q-analogues of certain classical constants connected with Lambert series…”

“…In the present paper, our purpose is to extend the results of [3] to more general algebraic numbers q and to (1) f (a, α) := ∞ n=1 a n q n − α with α ∈ {1, −1}; note that f (a, 1) = F (a). The arithmetic part of the proof depends essentially on the value of α, and the new case α = −1 turns out to be much more interesting than the earlier α = 1.…”