Lehmer Cotangent Expansion

Lehmer (1938) showed that every positive irrational number x has a unique infinite continued cotangent representation of the form


where the b_ks are nonnegative and


Note that this growth condition on coefficients is essential for the uniqueness of Lehmer expansion.

The following table summarizes the coefficients b_k for various special constants.

eA0026682, 8, 75, 8949, 119646723, 15849841722437093, ...
Euler-Mascheroni constant gammaA0817820, 1, 3, 16, 389, 479403, 590817544217, ...
golden ratio phiA0062671, 4, 76, 439204, 84722519070079276, ...
Lehmer's constant xiA0020650, 1, 3, 13, 183, 33673, ...
piA0026673, 73, 8599, 400091364,371853741549033970, ...
Pythagoras's constant sqrt(2)A0026661, 5, 36, 3406, 14694817,727050997716715, ...

The expansion for the golden ratio phi has the beautiful closed form


where L_n is a Lucas number.

An illustration of a different cotangent expansion for phi that is not a Lehmer expansion because its coefficients grow too slowly is


where F_n is a Fibonacci number (B. Cloitre, pers. comm., Sep. 22, 2005).

See also

Cotangent, Inverse Cotangent, Lehmer's Constant

Explore with Wolfram|Alpha


Lehmer, D. H. "A Cotangent Analogue of Continued Fractions." Duke Math. J. 4, 323-340, 1938.Rivoal, T. "Propriétés diophantiennes du développement en cotangente continue de Lehmer.", J. "Predictable Regular Continued Cotangent Expansions." J. Res. Nat. Bur. Standards Sect. B 80B, 285-290, 1976.Sloane, N. J. A. Sequences A002065/M2961, A002666/M3983, A002668/M1900, A002667/M3171, A006267/M3699, and A081782 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Lehmer Cotangent Expansion

Cite this as:

Weisstein, Eric W. "Lehmer Cotangent Expansion." From MathWorld--A Wolfram Web Resource.

Subject classifications