TOPICS

# L-Algebraic Number

An -algebraic number is a number which satisfies

 (1)

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

 (2)

instead of integers.

The only known -algebraic numbers of order 1 are

 (3) (4) (5) (6) (7)

(Loxton 1991, pp. 287 and 289; Bytsko 1999), where .

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

 (8)
 (9)

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

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

 (10)
 (11)
 (12)

where , , and are the roots of

 (13)

so that

 (14) (15) (16)

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

Higher-order algebraic identities include

 (17) (18) (19) (20) (21) (22) (23) (24) (25)

where

 (26) (27) (28) (29) (30) (31) (32)

(Gordon and McIntosh 1997).

## See also

Dilogarithm, Rogers L-Function, Watson's Identities

## 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.

## Referenced on Wolfram|Alpha

L-Algebraic Number

## Cite this as:

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