Linear independence of certain infinite series with monomial orders

Let b [Formula: see text] be an integer. In this paper, we show linear independence over [Formula: see text] of the numbers 1, [Formula: see text] [Formula: see text], [Formula: see text] [Formula: see text], where [Formula: see text] are sequences of integers with suitable non-vanishing and growth conditions. As an application, we obtain the linear independence over [Formula: see text] of the four numbers 1, [Formula: see text], [Formula: see text], [Formula: see text] for arbitrary fixed positive integers [Formula: see text] with [Formula: see text].