for
an integer other than 0 and . and the related series

(2)

which is a q-analog of the natural logarithm of 2, are irrational for a rational number other
than 0 or
(Guy 1994). In fact, Amdeberhan and Zeilberger (1998) showed that the irrationality
measures of both
and
are 4.80, improving the value of 54.0 implied by Borwein (1991, 1992).

Amdeberhan and Zeilberger (1998) also show that the -harmonic series and q-analog
of
can be written in the more quickly converging forms

Amdeberhan, T. and Zeilberger, D. "q-Apéry Irrationality Proofs by q-WZ Pairs." Adv. Appl. Math.20,
275-283, 1998.Borwein, P. B. "On the Irrationality of ." J. Number Th.37,
253-259, 1991.Borwein, P. B. "On the Irrationality of Certain
Series." Math. Proc. Cambridge Philos. Soc.112, 141-146, 1992.Breusch,
R. "Solution to Problem 4518." Amer. Math. Monthly61, 264-265,
1954.Erdős, P. "On Arithmetical Properties of Lambert Series."
J. Indian Math. Soc.12, 63-66, 1948.Erdős, P. "On
the Irrationality of Certain Series: Problems and Results." In New
Advances in Transcendence Theory (Ed. A. Baker). Cambridge, England:
Cambridge University Press, pp. 102-109, 1988.Erdős, P. and
Kac, M. "Problem 4518." Amer. Math. Monthly60, 47, 1953.Guy,
R. K. "Some Irrational Series." §B14 in Unsolved
Problems in Number Theory, 2nd ed. New York:Springer-Verlag, p. 69,
1994.