Remarks on irrationality of q-harmonic series

Heine's basic transform and a permutation group for q-harmonic series

“…Therefore the estimates (17) are consequences of the corresponding results for the quantity I(n; m) (see [Zu1,Lemmas 6 and 7]), and the lemma follows.…”

“…The first condition in (14) allows us to write a * 1 = a 1 , a * 2 = a 2 in (10), (11), and to apply Lemma 3 from [Zu1] to the quantity B 2 (p) after interchanging the summations in (11):…”

“…Thus using the fact that min{M 1 , M 2 } = M 2 under the hypothesis (14), we arrive at (15). To prove the second part of the lemma, we adopt the construction and results of [Zu1]. There we consider the family of series…”

“…In [Zu1] we made a mistake in computing the exponent of p when applying the identity of [Zu1,Lemma 3]. The correct application of the identity leads to…”

“…The works [BV], [MV], and [Ass] contain estimates for irrationality exponents of the q-logarithm values. Finally, introducing a q-arithmetic approach in [Zu1] resulted in a slight improvement of the estimate (3) and of the estimate in [Ass] for a q-analogue of log 2 (the result for µ(h p (1)) in [Zu1] is wrong due to a computational error; see the remark at the end of Section 3 below).…”