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).

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.