Abstract
Recall that Lebesgue’s singular function L(t) is defined as the unique solution to the equation L(t) = qL(2t) + pL(2t − 1), where p, q > 0, q = 1 − p, p ≠ q. The variables M n = ∫01tndL(t), n = 0,1,… are called the moments of the function The principal result of this work is \({M_n} = {n^{{{\log }_2}p}}{e^{ - \tau (n)}}(1 + O({n^{ - 0.99}}))\), where the function τ(x) is periodic in log2x with the period 1 and is given as \(\tau (x) = \frac{1}{2}1np + \Gamma '(1)lo{g_2}p + \frac{1}{{1n2}}\frac{\partial }{{\partial z}}L{i_z}( - \frac{q}{p}){|_{z = 1}} + \frac{1}{{1n2}}\sum\nolimits_{k \ne 0} {\Gamma ({z_k})L{i_{{z_k} + 1}}( - \frac{q}{p})} {x^{ - {z_k}}}\), \({z_k} = \frac{{2\pi ik}}{{1n2}}\), k ≠ 0. The proof is based on poissonization and the Mellin transform.
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References
Flajolet, P. and Sedgewick, R., Analytic Combinatorics, Cambridge University Press, 2008.
Lomnicki, Z. and Ulam, S.E., Sur la theorie de la mesure dans les espaces combinatoires et son application au calcul des probabilites. I. Variables independantes Fundam. Math., 1934, vol. 23, no. 1, pp. 237–278.
NIST Handbook of Mathematical Functions, Olver, F.W.J., Ed., Cambridge University Press, 2010.
Salem, R., On some singular monotonic functions which are strictly increasing, Trans. Am. Math. Soc., 1943, vol. 53, no. 3, pp. 427–439.
De Rham, G., On some curves defined by functional equations, in Classics on Fractals, Edgar, G.A., Ed., 1993, pp. 285–298
Flajolet, P., Gourdon, X., and Dumas, P., Mellin transforms and asymptotics: Harmonic sums, Theor. Comput. Sci., 1995, vol. 144, nos. 1–2, pp. 3–58.
Szpankowski, W., Average Case Analysis of Algorithms on Sequences, New York: John Wiley & Sons, 2001.
Gradstein, I.S. and Ryzhik, I.M., Table of Integrals, Series, and Products, Academic Press, 1994.
Prudnikov, A.P., Brychkov, Yu.A., and Marichev, O.I., Integrals and Series, vol. 3: More Special Functions, New York: Gordon & Breach Sci., 1990.
Timofeev, E.A., Bias of a nonparametric entropy estimator for Markov measures, J. Math. Sci., 2011, vol. 176, no. 2, pp. 255–269.
Timofeev, E.A., Asymptotic formula for the moments of Lebesgue’s singular function, Model. Anal. Inf. Sist., 2015, vol. 22, no. 5, pp. 723–730.
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Original Russian Text © E.A. Timofeev, 2016, published in Modelirovanie i Analiz Informatsionnykh Sistem, 2016, Vol. 23, No. 5, pp. 595–602.
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Timofeev, E.A. Polylogarithms and the Asymptotic Formula for the Moments of Lebesgue’s Singular Function. Aut. Control Comp. Sci. 51, 634–638 (2017). https://doi.org/10.3103/S0146411617070203
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DOI: https://doi.org/10.3103/S0146411617070203