If
denotes the usual dilogarithm, then there are two
variants that are normalized slightly differently, both called the Rogers -function (Rogers 1907). Bytsko (1999) defines

(1)

(2)

(which he calls "the" dilogarithm), while Gordon and McIntosh (1997) and Loxton (1991, p. 287) define the Rogers -function as

In terms of ,
the well-known dilogarithm identities become

(10)

(11)

(12)

(13)

(14)

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

Khoi (2014) gave the identity

(15)

where
is the golden ratio (Khoi 2014, Campbell 2021).

Numbers
which satisfy

(16)

for some value of
are called L-algebraic numbers. Loxton (1991,
p. 289) gives a slew of identities having rational coefficients

(17)

instead of integers, where is a rational number, a
corrected and expanded version of which is summarized in the following table. In
this table, polynomials denote the real root of . Many more similar identities can be found using integer
relation algorithms.

Rogers (1907) obtained a dilogarithm identity in variables with terms which simplifies to Euler's identity for and Abel's functional
equation for
(Gordon and McIntosh 1997). For , it is equivalent to

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Reprint Corp., pp. 189-192, 1988.Bytsko, A. G. "Fermionic
Representations for Characters of , , and Minimal Models and Related Dilogarithm and Rogers-Ramanujan-Type
Identities." J. Phys. A: Math. Gen.32, 8045-8058, 1999.Bytsko,
A. G. "Two-Term Dilogarithm Identities Related to Conformal Field Theory."
9 Nov 1999. http://arxiv.org/abs/math-ph/9911012.Campbell,
J. M. "Some Nontrivial Two-Term Dilogarithm Identities." Irish
Math. Soc. Bull., No. 88, 31-37, 2021.Euler, L. Institutiones
calculi integralis, Vol. 1. Basel, Switzerland: Birkhäuser, pp. 110-113,
1768.Gordon, B. and McIntosh, R. J. "Algebraic Dilogarithm
Identities." Ramanujan J.1, 431-448, 1997.Khoi,
V. T. "Seifert Volumes and Dilogarithm Identities." J. Knot Th.
Ram.23, 1450025, 11, 2014.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. "Partition Identities and the Dilogarithm." Ch. 13 in Structural
Properties of Polylogarithms (Ed. L. Lewin). Providence, RI: Amer. Math.
Soc., pp. 287-299, 1991.Rogers, L. J. "On Function Sum
Theorems Connected with the Series ." Proc. London Math. Soc.4,
169-189, 1907.Watson, G. N. "A Note on Spence's Logarithmic
Transcendent." Quart. J. Math. Oxford Ser.8, 39-42, 1937.