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Model. Anal. Inform. Sist., 2016, Volume 23, Number 5, Pages 595–602 (Mi mais526)  

Polylogarithms and the asymptotic formula for the moments of Lebesgues singular function

E. A. Timofeev

P.G. Demidov Yaroslavl State University, 14 Sovetskaya str., Yaroslavl 150003, Russia

Abstract: Recall the Lebesgue's singular function. We define a Lebesgue's singular function $L(t)$ as the unique continuous solution of the functional equation
$$ L(t) = qL(2t) +pL(2t-1), $$
where $p,q>0$, $q=1-p$, $p\ne q$. The moments of Lebesque' singular function are defined as
$$ M_n = \int_0^1t^n dL(t), \quad n = 0, 1, …$$
The main result of this paper is
$$ M_n = n^{\log_2 p} e^{-\tau(n)}(1 + \mathcal{O}(n^{-0.99})), $$
where
\begin{gather*} \tau(x) = \frac12\ln p + \Gamma'(1)\log_2 p +\frac1{\ln 2}\frac{\partial}{\partial z}.\mathrm{Li}_{z}(-\frac{q}{p})|_{z=1} +\frac1{\ln 2}\sum_{k\ne0} \Gamma(z_k)\mathrm{Li}_{z_k+1}(-\frac{q}{p}) x^{-z_k},
z_k = \frac{2\pi ik}{\ln 2}, k\ne 0. \end{gather*}
The proof is based on analytic techniques such as the poissonization and the Mellin transform.

Keywords: moments, self-similar, Lebesgues function, singular, Mellin transform, polylogarithm, asymptotic.

DOI: https://doi.org/10.18255/1818-1015-2016-5-595-602

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English version:
Automatic Control and Computer Sciences, 2017, 51:7, 634–638

Bibliographic databases:

UDC: 519.17
Received: 10.07.2016

Citation: E. A. Timofeev, “Polylogarithms and the asymptotic formula for the moments of Lebesgues singular function”, Model. Anal. Inform. Sist., 23:5 (2016), 595–602; Automatic Control and Computer Sciences, 51:7 (2017), 634–638

Citation in format AMSBIB
\Bibitem{Tim16}
\by E.~A.~Timofeev
\paper Polylogarithms and the asymptotic formula for the moments of Lebesgues singular function
\jour Model. Anal. Inform. Sist.
\yr 2016
\vol 23
\issue 5
\pages 595--602
\mathnet{http://mi.mathnet.ru/mais526}
\crossref{https://doi.org/10.18255/1818-1015-2016-5-595-602}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3569856}
\elib{http://elibrary.ru/item.asp?id=27202309}
\transl
\jour Automatic Control and Computer Sciences
\yr 2017
\vol 51
\issue 7
\pages 634--638
\crossref{https://doi.org/10.3103/S0146411617070203}


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