On approximation measures of q-logarithms

Abstract

Using Padé approximations of the q-logarithmic series we obtain new approximation measures for values of q-logarithms and for the series , k = 1, 2, …, where (un) is a recurrence sequence, satisfying

Irrationality Measures for the Series of Reciprocals from Recurrence Sequences

“…In Bundschuh and V€ a a€ a an€ a anen [8] and Matala-aho and V€ a a€ a an€ a anen [14], (6) and (7) are generalized to arbitrary number fields and in case (7) Borwein's irrationality measure 8.667 is improved to 4.311 in [8] and to 3.947 in [14]. Andre´-Jeannin [1] proved the irrationality of the series W r;AE1 ; when a and b are given by (2) and r 2 Z=f0g; t 2 Z=f0g; jtjojaj;…”

“…Linear independence measures of q-exponential function and its derivatives in [8] imply an irrationality measure 8.621 for the Fibonacci case of (4). Approximation measure results in [14] give (4). Here we mention the following trancendental series [11,16]:…”

“…Let ðW n Þ be given by (1), where d; R and S satisfy [8,14,17], respectively, will be improved considerably. In all the three papers cited the approximation formulae have connections to Pade´approximations of the q-logarithm series…”

О линейной независимости значений некоторых $q$-рядов

“…П. РОЧЕВ Следствие 1.2 качественно усиливает теорему 3 работы [10], где соответству-ющий результат доказан при условии λ < 1.33943985 . .…”

“…. (в [10] оно не сформу-лировано явно; это область определения функции m(−λ) в работе [10]) и допол-нительном требовании, чтобы |q| v ̸ = 1 при всех v|∞. Однако в количественном аспекте следствие 1.2 слабее, чем [10, теорема 3], где оценка для |L q (α) − θ| w степенная по H.…”