On irrationality measures of ∑l=0∞dl/∏j=1l

Tapani Matala-aho¹

Abstract

We prove that the number τ = ∞ l=0 d l / l j =1 (1 + d j r + d 2j s), where d ∈ Z, |d| > 1, and r, s ∈ Q, s = 0, are such that 1 + d j r + d 2j s = 0 for any j ∈ Z + , has an irrationality measure 7/3 or 7/2 depending on whether r = −d −h − sd h for some h ∈ N or r 2 4s. More generally, irrationality measures are given for τ in both the archimedean and p-adic valuations, and also when d, r, s are certain algebraic numbers. For example, we give an effective irrationality measure 7/3 for B d (d), where B q (z) is a q-analogue of the Bessel function, and we get effective irrationality measures 7/3 and 7/2