On the irrationality of factorial series II

Abstract

In this paper we give irrationality results for numbers of the form ∞ n=1 an n! where the numbers a n behave like a geometric progression for a while. The method is elementary, not using differentiation or integration. In particular, we derive elementary proofs of the irrationality of π and e m for Gaussian integers m = 0.

Cantor series expansions of rational numbers

“…In some papers (see [13], [48], [7], [11], [26]), 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], [24], [31]) and still for the case of Cantor series of a special type (e.g., see [15], [17], [16], [13]). For example, in the papers [3], [14], [6], [21], Ahmes series are considered.…”

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.…”

Metrical irrationality results related to values of the Riemann $ζ$-function

“…is irrational for each n ∈ N. By contrast, the value of ζ at odd integers largely remains a mystery, and not many results were known until 1979 when Apéry [1] published a proof that ζ(3) is irrational. Rivoal [16] subsequently proved that infinitely many odd ζ-values are irrational; and Zudilin [21] proved that at least one of ζ(5), ζ (7), ζ (9) and ζ( 11) is irrational.…”