A227424 Decimal expansion of 'mu', a Young-Fejér-Jackson constant linked to the positivity of certain sine sums.

8, 1, 2, 8, 2, 5, 2, 4, 2, 1, 0, 5, 5, 0, 7, 2, 6, 0, 7, 6, 0, 0, 8, 7, 1, 2, 3, 1, 1, 8, 3, 7, 0, 2, 9, 8, 6, 4, 7, 0, 1, 0, 1, 3, 4, 0, 5, 2, 8, 7, 0, 3, 4, 0, 6, 5, 7, 3, 6, 0, 0, 3, 9, 5, 8, 0, 7, 2, 7, 4, 7, 2, 6, 7, 9, 4, 0, 2, 2, 7, 2, 3, 8, 3, 9, 1, 2, 5, 2, 9, 4, 7, 9, 0, 9, 6, 4, 6, 7, 2, 9, 8, 2

Offset

0,1

Comments

Named after the English mathematician William Henry Young (1863-1942), the Hungarian mathematician Lipót Fejér (or Leopold Fejér, 1880-1959) and the American mathematician Dunham Jackson (1888-1946). - Amiram Eldar, Jun 24 2021

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 242.

Links

Table of n, a(n) for n=0..102.

Leopold Fejér, Lebesguessche Konstanten und divergente Fourierreihen, J. Reine Angew. Math., Vol.. 138 (1910), pp. 22-53; alternative link.

Dunham Jackson, Über eine trigonometrische Summe, Rendiconti del Circolo Matematico di Palermo (1884-1940), Vol. 32, No. 1 (1911), pp. 257-262.

W. H. Young, On a certain series of Fourier, Proc. London Math. Soc., Vol. 11 (1912), pp. 357-366.

Formula

Given lambda from A227423, mu is the unique positive solution of (1+lambda)*sin(mu*Pi) = mu*sin(lambda*Pi).

Example

0.81282524210550726076008712311837029864701013405287034065736003958072747...

Maple

Digits:= 127:
lambda:= solve((1+x)*Pi - tan(x*Pi), x):
mu:= fsolve((1+lambda)*sin(x*Pi)-x*sin(lambda*Pi), x, 0.1..1):
s:= convert(mu, string):
seq(parse(s[n+1]), n=1..Digits-10);  # Alois P. Heinz, Jul 11 2013

Mathematica

mu /. FindRoot[(1 + lambda)*Pi == Tan[lambda*Pi] && (1 + lambda)*Sin[mu*Pi] == mu* Sin[lambda*Pi], {lambda, 2/5}, {mu, 4/5}, WorkingPrecision -> 100] // RealDigits // First

Crossrefs

Cf. A227422, A227423, A227425.

Sequence in context: A140242 A156944 A371855 * A248965 A266152 A021127

Adjacent sequences: A227421 A227422 A227423 * A227425 A227426 A227427

Keyword

nonn,cons

Author

Jean-François Alcover, Jul 11 2013

Status

approved