On the asymptotic expansion of the q-dilogarithm
© Bouzeffour and Halouani 2016
Received: 25 August 2015
Accepted: 13 March 2016
Published: 9 May 2016
In this work, we study some asymptotic expansion of the q-dilogarithm at \(q=1\) and \(q=0\) by using the Mellin transform and an adequate decomposition allowed by the Lerch functional equation.
In Section 2.5, Corollary 10 of , Kirillov and Ueno and Nishizawa derived the asymptotic expansion (1.5) by using the Euler-Maclaurin summation formula; see also , an integral representation of Barnes type for the q-dilogarithm. Second, we use the Lerch functional equation to decompose the integrand and to apply the Cauchy theorem.
3 Asymptotic at \(q=1\)
The integral (2.28) will be used to derive asymptotic expansions of the q-dilogarithm. The contour of integration is moved at first to the left to obtain an asymptotic expansion at \(q=1\) and then to the right to get an asymptotic expansion at \(q=0\).
4 Asymptotic at \(q=0\)
The first author would like to extend his sincere appreciation to the Deanship of Scientific Research at King Saudi University for funding this Research group No. (RG-1437-020).
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