On the irrationality of factorial series

Cantor series expansions of rational numbers

“…In some papers (see [13,45,7,11,24]), the case of Cantor series for which sequences (q k ) and (ε k ) are sequences of integers and the condition Z ∋ q k > 1 holds for all k ∈ N, is investigated. However the main problem of the present article is studied for the case of series (1) (e.g., see [2,3,22,29]) and still for the case of Cantor series of special type (e.g., see [15,17,16,13]). For example, in the papers [3,14,6], Ahmes series are considered.…”

“…In the papers [3,10,11,13,14,15,7,29,45], necessary and sufficient conditions for a rational (irrational) number to be representable by a Cantor series are studied, and sufficient conditions are investigated in the papers [7,3,13,22,29,45]. Although much research has been devoted to the problem on representations of rational (irrational) numbers by Cantor series for which sequences (q k ) and (ε k ) are sequences of special types (see [2,7,10,11,13,22,29,45]), little is known about necessary and sufficient conditions of the rationality (irrationality) for the case of an arbitrary sequence (q k ) (see [3,13,45]).…”

Certain Functions Defined in Terms of Cantor Series

“…xn K /n!, is a strictly increasing function of x > 0 which does not take rational values (see [4,Remark to Corollary 3.4]). Here K is an arbitrary positive integer and [y] is an integer part of y.…”

On the irrationality of factorial series II

“…the proof of their Theorem 4.3). Later the authors [8,9] pursued this idea by studying K -th order differences. This enabled them to derive a variety of results in cases where the integer sequence a n has polynomial growth.…”