The series
(1)
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for an integer other than 0 and . and the related series
(2)
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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)
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(4)
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(5)
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(6)
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where is a q-binomial coefficient and is a -Pochhammer symbol.