On linear independence of values of certain $ q$-series

Abstract

We prove, in a quantitative form, linear independence results for values of a certain class of q -series, which generalize classical q -hypergeometric series. These results refine our recent estimates.

Irrationality Measures of Numbers Related to Some $q$-Basic Hypergeometric Series

“…We note that there are a lot of works considering arithmetic properties of different type of q-series, see e.g. [14] for a survey of such results, and [3], [4], [5], [6], [8], [10], [11], [12] and [15] for some more recent results, but only a few study functions (2), see [7] and [10]. In [7] Chirskii considered the case K = Q with finite p and proved the following result.…”

On the irrationality of generalized q-logarithm

“…Independently, J.-P. Bézivin [3] in 1988 and P. Borwein [7] in 1991 pushed forward different analytical methods to establish the irrationality of the series for rational z and q of the form q = 1/p with p ∈ Z \ {0, ±1}, though Bézivin's paper did not mention L q (z) specifically-it was observed only later [9] that the general results from [3] imply the irrationality. The advantage of Borwein's method [7,8], in which he uses the Padé approximations to (2), is that it allows one to "measure" the irrationality of the numbers in question; this quantitative counterpart is absent in Erdős' method [18] and it was also absent in Bézivin's original method [3,4] until the recent work [30] of I. Rochev (see [31] for a further development). The Padé approximation technique as originated in [7,8] was significantly generalized and extended in later works (for example, [1,[9][10][11][12][13]17,22,24,25,32,34,36,37] to list a few) to sharpen the irrationality measures of the values of q-harmonic and q-logarithm series as well as to prove the irrationality and linear independence results for some close relatives of the series.…”

“…where the derivative is with respect to z. Though the method from [3,4] underwent modifications and generalizations in the later works of Bézivin himself and other authors [5,6,14,23,30,31], it essentially serves the original class of functions.…”

On the irrationality of generalized $q$-logarithm

“…Independently, J.-P. Bézivin [3] in 1988 and P. Borwein [7] in 1991 pushed forward different analytical methods to establish the irrationality of the series for rational z and q of the form q = 1/p with p ∈ Z \ {0, ±1}, though Bézivin's paper did not mention L q (z) specifically -it was observed only later [9] that the general results from [3] imply the irrationality. The advantage of Borwein's method [7,8], in which he uses the Padé approximations to (2), is that it allows one to "measure" the irrationality of the numbers in question; this quantitative counterpart is absent in Erdős' method [18] and it was also absent in Bézivin's original method [3,4] until the recent work [30] of I. Rochev (see [31] for a further development). The Padé approximation technique as originated in [7,8] was significantly generalized and extended in later works (for example, [1,9,10,11,12,13,17,22,24,25,32,34,36,37] to list a few) to sharpen the irrationality measures of the values of q-harmonic and qlogarithm series as well as to prove the irrationality and linear independence results for some close relatives of the series.…”

“…where the derivative is with respect to z. Though the method from [3,4] underwent modifications and generalizations in the later works of Bézivin himself and other authors [5,6,14,23,30,31], it essentially serves the original class of functions.…”