q-Harmonic Series

The series

(1)

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

			(3)
			(4)
			(5)
			(6)

where  is a q-binomial coefficient and  is a -Pochhammer symbol.

REFERENCES:

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. Monthly 61, 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. Monthly 60, 47, 1953.

Guy, R. K. "Some Irrational Series." §B14 in Unsolved Problems in Number Theory, 2nd ed. New York:Springer-Verlag, p. 69, 1994.