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On \(q\)-analogues of the Euler constant and Lerch’s limit formula. (English) Zbl 1114.33022

Summary: We introduce and study a \(q\)-analogue \(\gamma(q)\) of the Euler constant via a suitably defined \(q\)-analogue of the Riemann zeta function. We show, in particular, that the value \(\gamma(2)\) is irrational. We also present a \(q\)-analogue of the Hurwitz zeta function and establish an analogue of the limit formula of Lerch in 1894 for the gamma function. This limit formula can be regarded as a natural generalization of the formula of \(\gamma(q)\).

MSC:

33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
11M35 Hurwitz and Lerch zeta functions
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References:

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