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

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, 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, 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

Offset: 1

Author

Jean-François Alcover, May 21 2014

Keywords

• nonn
• cons

Examples

			1.38135644451849779337146695685...
		

References

• 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.

Links

• Jonathan M. Borwein, Armin Straub, James Wan, and Wadim Zudilin, Densities of Short Uniform Random Walks, Canad. J. Math. 64(1) (2012), 961-990; see p. 978.
• Henry Cohn, Richard Kenyon, and James Propp, A variational principle for domino tilings, arXiv:math/0008220 [math.CO], 2000.
• Henry Cohn, Richard Kenyon, and James Propp, A variational principle for domino tilings, J. Amer. Math. Soc. 14(2) (2000), 297-346.
• Kurt Mahler, A remark on a paper of mine on polynomials. [In this paper, j is log(beta) = A244996.]
• Kurt Mahler, A remark on a paper of mine on polynomials, Illinois J. Math. 8(1) (1964), 1-4.
• Francisco Santos, The Cayley trick and triangulations of products of simplices, 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).
• Francisco Santos, The Cayley trick and triangulations of products of simplices, Cont. Math. 374 (2005), 151-177.
• Eric Weisstein's MathWorld, Clausen's Integral.
• Eric Weisstein's MathWorld, Gieseking's Constant.
• Eric Weisstein's MathWorld, Lobachevsky's Function.
• Wikipedia, Clausen function.

Crossrefs

Cf. A122722, A130834, A143298, A229728, A244996.

Programs

Mathematica

Exp[(PolyGamma[1, 4/3] - PolyGamma[1, 2/3] + 9)/(4*Sqrt[3]*Pi)] // RealDigits[#, 10, 100]& // First

Formula

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).

Also equals exp(-Im(Li_2( 1/2 - (i*sqrt(3))/2))/Pi), where Li_2 is the dilogarithm function.