cp's OEIS Frontend

A242710 Decimal expansion of "beta", a Kneser-Mahler polynomial constant (a constant related to the asymptotic evaluation of the supremum norm of polynomials).

%I A242710 #25 Feb 16 2025 08:33:22
%S A242710 1,3,8,1,3,5,6,4,4,4,5,1,8,4,9,7,7,9,3,3,7,1,4,6,6,9,5,6,8,5,0,6,2,4,
%T A242710 1,2,6,2,8,9,6,3,7,2,6,2,2,3,9,0,7,0,5,6,0,1,9,8,7,6,4,8,4,5,3,0,0,5,
%U A242710 5,4,9,6,3,6,3,6,6,3,6,2,4,5,4,0,8,6,3,9,7,6,7,9,5,4,4,2,8,1,1,6
%N A242710 Decimal expansion of "beta", a Kneser-Mahler polynomial constant (a constant related to the asymptotic evaluation of the supremum norm of polynomials).
%D A242710 Steven R. Finch, Mathematical Constants, Cambridge, 2003; see Section 3.10, Kneser-Mahler polynomial constants, p. 232, and Section 5.23, Monomer-dimer constants, p. 408.
%H A242710 Jonathan M. Borwein, Armin Straub, James Wan, and Wadim Zudilin, <a href="http://dx.doi.org/10.4153/CJM-2011-079-2">Densities of Short Uniform Random Walks</a>, Canad. J. Math. 64(1) (2012), 961-990; see p. 978.
%H A242710 Henry Cohn, Richard Kenyon, and James Propp, <a href="https://arxiv.org/abs/math/0008220">A variational principle for domino tilings</a>, arXiv:math/0008220 [math.CO], 2000.
%H A242710 Henry Cohn, Richard Kenyon, and James Propp, <a href="https://doi.org/10.1090/S0894-0347-00-00355-6">A variational principle for domino tilings</a>, J. Amer. Math. Soc. 14(2) (2000), 297-346.
%H A242710 Kurt Mahler, <a href="https://web.archive.org/web/20150328133345/http://carmaweb.newcastle.edu.au/mahler/docs/153.pdf">A remark on a paper of mine on polynomials</a>. [In this paper, j is log(beta) = A244996.]
%H A242710 Kurt Mahler, <a href="https://projecteuclid.org/euclid.ijm/1256067451">A remark on a paper of mine on polynomials</a>, Illinois J. Math. 8(1) (1964), 1-4.
%H A242710 Francisco Santos, <a href="https://arxiv.org/abs/math/0312069">The Cayley trick and triangulations of products of simplices</a>, arXiv:math/0312069 [math.CO], 2004; see part (2) of Theorem 1 (p. 2, possible typo), Lemma 4.8 (p. 22), and Theorem 4.9 (p. 22).
%H A242710 Francisco Santos, <a href="http://dx.doi.org/10.1090/conm/374/06904">The Cayley trick and triangulations of products of simplices</a>, Cont. Math. 374 (2005), 151-177.
%H A242710 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/ClausensIntegral.html">Clausen's Integral</a>.
%H A242710 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/GiesekingsConstant.html">Gieseking's Constant</a>.
%H A242710 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/LobachevskysFunction.html">Lobachevsky's Function</a>.
%H A242710 Wikipedia, <a href="https://en.wikipedia.org/wiki/Clausen_function">Clausen function</a>.
%F A242710 beta = exp(G/Pi) = exp((PolyGamma(1, 4/3) - PolyGamma(1, 2/3) + 9)/(4*sqrt(3)*Pi)), where G is Gieseking's constant (cf. A143298) and PolyGamma(1,z) the first derivative of the digamma function psi(z).
%F A242710 Also equals exp(-Im(Li_2( 1/2 - (i*sqrt(3))/2))/Pi), where Li_2 is the dilogarithm function.
%e A242710 1.38135644451849779337146695685...
%t A242710 Exp[(PolyGamma[1, 4/3] - PolyGamma[1, 2/3] + 9)/(4*Sqrt[3]*Pi)] // RealDigits[#, 10, 100]& // First
%Y A242710 Cf. A122722, A130834, A143298, A229728, A244996.
%K A242710 nonn,cons
%O A242710 1,2
%A A242710 _Jean-François Alcover_, May 21 2014