The series h_q(-r) = sum_(n = 1)^∞ 1/(q^n + r) for q an integer other than 0 and ± 1. h_q and the related series Ln_q(-r + 1) = sum_(n = 1)^∞ (-1)^n/(q^n + r), which is a q-analog of the natural logarithm of 2, are irrational for r a rational number other than 0 or -q^n. In fact, Amdeberhan and Zeilberger showed that the irrationality measures of both h_q(1) and Ln_q(2) are 4.80, improving the value of 54.0 implied by Borwein (1991, 1992).

harmonic series | irrationality measure