Zygmund (1988, p. 192) noted that there exists a number  such that for each , the partial sums of the series  are uniformly bounded below, whereas for , they are not (Arias de Reyna and van de Lune 2009).

This constant is given by the unique solution for  of

			(1)
			(2)

where  is a generalized hypergeometric function, which is given by  (OEIS A157957).

The origin of the defining property for  appeared in an unpublished result of Littlewood and Salem and the equation defining  is due to S. Izumi (Zygmund 1988, p. 379), thus justifying the name Littlewood-Salem-Izumi constant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).

REFERENCES:

Arias de Reyna, J. and van de Lune, J. "High Precision Computation of a Constant in the Theory of Trigonometric Series." Math. Comput. Published electronically, February 9, 2009.

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.; Wang, K.; and Wilson, D. C. "Positivity of Some Basic Cosine Sums." Math. Proc. Cambridge Philos. Soc. 114, 383-391, 1993.

Brown, G.; Dai, F.; and Wang, K. "On Positive Cosine Sums." Math. Proc. Cambridge Philos. Soc. 142, 219-232, 2007.

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.

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.

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.

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.