The Littlewood-Salem-Izumi constant is the number such that the partial
sums of the series are uniformly bounded
below for , but not for (Zygmund 1988, p. 192; Arias de Reyna
and van de Lune 2009).
This constant is given by the unique solution for
of
The origin of the defining property for appeared in an unpublished result of Littlewood and
Salem and the equation defining is due to S. Izumi (Zygmund 1988, p. 379),
thus justifying the name Littlewood-Salem-Izumi constant (Arias de Reyna and van
de Lune 2009).
Arias de Reyna, J. and van de Lune, J. "High Precision Computation of a Constant in the Theory of Trigonometric Series." Math. Comput.78,
2187-2191, 2009. https://doi.org/10.1090/S0025-5718-09-02222-4.Askey,
R. Orthogonal
Polynomials and Special Functions. Philadelphia, PA: SIAM, 1975.Belov,
A. S. "Coefficients of Trigonometric Cosine Series with Nonnegative Partial
Sums." Translated in Proc. Steklov Inst. Math.1992, 1-18, 1992.
"Theory of Functions. (Amberd, 1987)." Trudy Mat. Inst. Steklov,
190, pp. 3-21, 1989.Boas, R. P. Jr. and Klema, C. "A
Constant in the Theory of Trigonometric Series." Math. Comput.18,
674, 1964.Brown, G.; Dai, F.; and Wang, K. "On Positive Cosine
Sums." Math. Proc. Cambridge Philos. Soc.142, 219-232, 2007.Brown,
G.; Wang, K.; and Wilson, D. C. "Positivity of Some Basic Cosine Sums."
Math. Proc. Cambridge Philos. Soc.114, 383-391, 1993.Church,
R. F. "On a Constant in the Theory of Trigonometric Series." Math.
Comput.19, 501, 1965.Finch, S. R. Mathematical
Constants. Cambridge, England: Cambridge University Press, 2003.Grandjot,
K.; Jarnik, V.; Landau, E.; and Littlewood, J. E. "Bestimmung einer absoluten
Konstanten aus der Theorie der trigonometrischen Reihen." Annali di Mat.6,
1-7, 1929.Koumandos, S. and Ruscheweyh, S. "Positive Gegenbauer
Polynomial Sums and Applications to Starlike Functions." Constr. Approx.23,
197-210, 2006.Luke, Y. L.; Fair, W.; Coombs, G.; and Moran, R.
"On a Constant in the Theory of Trigonometric Series." Math. Comput.19,
501-502, 1965.Sloane, N. J. A. Sequence A157957
in "The On-Line Encyclopedia of Integer Sequences."Zygmund,
A. G. Trigonometric
Series, Vols. 1-2, 2nd ed. New York: Cambridge University Press, 1988.
such that the partial
sums of the series 
are uniformly bounded
below for 
, but not for 
(Zygmund 1988, p. 192; Arias de Reyna
and van de Lune 2009).
of
is a generalized hypergeometric
function. Numerically, 
(OEIS A157957).
appeared in an unpublished result of Littlewood and
Salem and the equation defining 
is due to S. Izumi (Zygmund 1988, p. 379),
thus justifying the name Littlewood-Salem-Izumi constant (Arias de Reyna and van
de Lune 2009).
