q-Harmonic Series

The series


for q an integer other than 0 and +/-1. h_q and the related series


which is a q-analog of the natural logarithm of 2, are irrational for r a rational number other than 0 or -q^n (Guy 1994). In fact, Amdeberhan and Zeilberger (1998) 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).

Amdeberhan and Zeilberger (1998) also show that the q-harmonic series and q-analog of ln2 can be written in the more quickly converging forms

=sum_(n=1)^(infty)(1-q^n-q^(2n))/((q^n-1)(2n; n)_q(q)_n)
=sum_(n=1)^(infty)((-1)^(n-1)(q)_n(1-q^(3n)))/((1-q^n)^2(2n; n)_q(q^2)_n),

where (n; k)_q is a q-binomial coefficient and (q)_n is a q-Pochhammer symbol.

See also

Harmonic Series, Irrationality Measure

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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 sum1/(q^n+r)." 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.

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q-Harmonic Series

Cite this as:

Weisstein, Eric W. "q-Harmonic Series." From MathWorld--A Wolfram Web Resource.

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