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L-Algebraic Number

An L-algebraic number is a number theta in (0,1) which satisfies

sum_(k=0)^nc_kL(theta^k)=0,

(1)

where L(x) is the Rogers L-function and c_k are integers not all equal to 0 (Gordon and Mcintosh 1997). Loxton (1991, p. 289) gives a slew of similar identities having rational coefficients

sum_(k=0)^n(e_k)/kL(theta^k)=0

(2)

instead of integers.

The only known L-algebraic numbers of order 1 are

(Loxton 1991, pp. 287 and 289; Bytsko 1999), where rho=(sqrt(5)-1)/2.

The only known rational L-algebraic numbers are 1/2 and 1/3:

L(1/(64))-2L(1/8)-6L(1/4)+2L(1)=0

(8)

L(1/9)-6L(1/3)+2L(1)=0

(9)

(Lewin 1982, pp. 317-318; Gordon and McIntosh 1997).

There are a number of known quadratic L-algebraic numbers. Watson (1937) found

L(alpha)-L(alpha^2)=1/7

(10)

2L(beta)+L(beta^2)=(10)/7

(11)

2L(gamma)+L(gamma^2)=8/7,

(12)

where alpha, -beta, and -1/gamma are the roots of

x^3+2x^2-x-1=0,

(13)

so that

(Loxton 1991, pp. 287-288). These are known as Watson's identities.

Higher-order algebraic identities include

5L(delta^3)-5L(delta)+L(1)=0

(17)

L(delta^(12))-2L(delta^6)-6L(delta^4)+4L(delta^3)+3L(delta^2)+4L(delta)

(18)

-4L(1)=0

(19)

3L(kappa^3)-9L(kappa^2)-9K(kappa)+7L(1)=0

(20)

3L(lambda^6)-6L(lambda^3)-27L(lambda^2)+18L(lambda)+2L(1)=0

(21)

3L(mu^6)-6L(mu^3)-27L(mu^2)+18L(mu)-2L(1)=0

(22)

2L(a^3)-2L(a^2)-11L(a)+3L(1)=0

(23)

2L(b^6)-4L(b^3)-15L(b^2)+22L(b)-6L(1)=0

(24)

2L(c^6)-4L(c^3)-15L(c^2)+22L(c)-4L(1)=0,

(25)

where

(Gordon and McIntosh 1997).

See also

Dilogarithm, Rogers L-Function, Watson's Identities

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References

Bytsko, A. G. "Two-Term Dilogarithm Identities Related to Conformal Field Theory." 9 Nov 1999. http://arxiv.org/abs/math-ph/9911012.Gordon, B. and McIntosh, R. J. "Algebraic Dilogarithm Identities." Ramanujan J. 1, 431-448, 1997.Lewin, L. "The Dilogarithm in Algebraic Fields." J. Austral. Math. Soc. Ser. A 33, 302-330, 1982.Lewin, L. (Ed.). Structural Properties of Polylogarithms. Providence, RI: Amer. Math. Soc., 1991.Loxton, J. H. "Special Values of the Dilogarithm Function." Acta Arith. 43, 155-166, 1984.Loxton, J. H. "Partition Identities and the Dilogarithm." Ch. 13 in Structural Properties of Polylogarithms (Ed. L. Lewin). Providence, RI: Amer. Math. Soc., pp. 287-299, 1991.Watson, G. N. "A Note on Spence's Logarithmic Transcendent." Quart. J. Math. Oxford Ser. 8, 39-42, 1937.

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L-Algebraic Number

Cite this as:

Weisstein, Eric W. "L-Algebraic Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/L-AlgebraicNumber.html

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