On the irrationality of generalized q-logarithm

Abstract

For integer p, |p|>1, and generic rational x and z, we establish the irrationality of the series p (x, z) = x ∞ n=1 z n p n − x. It is a symmetric (p (x, z) = p (z, x)) generalization of the q-logarithmic function (x = 1 and p = 1/q where |q| < 1), which in turn generalizes the q-harmonic series (x = z = 1). Our proof makes use of the Hankel determinants built on the Padé approximations to p (x, z).

A Determinantal Approach to Irrationality

“…It is expected that this q-Apéry constant is irrational for such values of q, but the corresponding q-approximations r n from [7] do not produce any irrationality result, because |r n | 1/n 2 ∼ 1 while δ 1/n 2 n ∼ |q| −9/π 2 as n → ∞, where δ n r n ∈ Zζ q (3) + Z. Though no arithmetic consequences come out from considering the Hankel determinants R n = det 0≤j,ℓ<n (r j+ℓ ), an analytic argument [17,Section 4] shows their (better than expected) behaviour |R n | 1/n 3 ∼ |q| 1/3 as n → ∞.…”

Linear independence of values of the $q$-exponential and related functions

“…The irrationality and linear independence of the values of the q-logarithm function have been extensively studied by various authors. We refer to [11] for a comprehensive history of the problem and investigation of the values of a generalization of the q-logarithm function.…”

LINEAR INDEPENDENCE OF VALUES OF THE q-EXPONENTIAL AND RELATED FUNCTIONS

“…The irrationality and linear independence of the values of the q -logarithm function have been studied extensively. We refer the reader to [10] for a comprehensive history of the problem and an investigation of the values of a generalisation of the q -logarithm function. The irrationality of when q is an integer was first obtained by Erdős [4].…”