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)
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(2)
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(which he calls "the" dilogarithm), while Gordon and McIntosh (1997) and Loxton (1991, p. 287) define the Rogers -function as
(3)
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(4)
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(5)
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The function satisfies the concise reflection relation
(6)
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(Euler 1768), as well as Abel's functional equation
(7)
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(Abel 1988, Bytsko 1999). Abel's duplication formula for follows from Abel's functional equation and is given by
(8)
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The function has the nice series
(9)
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(Lewin 1982; Loxton 1991, p. 298).
In terms of , the well-known dilogarithm identities become
(10)
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(11)
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(12)
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(13)
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(14)
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(Loxton 1991, pp. 287 and 289; Bytsko 1999), where .
Khoi (2014) gave the identity
(15)
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where is the golden ratio (Khoi 2014, Campbell 2021).
Numbers which satisfy
(16)
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for some value of are called L-algebraic numbers. Loxton (1991, p. 289) gives a slew of identities having rational coefficients
(17)
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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.
1 | 1 | 1 |
1 | ||
1 | ||
1 | ||
1 | ||
1 | ||
3 | ||
, | ||
1 | ||
2 | ||
3 | ||
1 | ||
2 | ||
Bytsko (1999) gives the additional identities
(18)
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(19)
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(20)
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(21)
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(22)
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(23)
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(24)
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(25)
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(26)
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where
(27)
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(28)
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(29)
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with the positive root of
(30)
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and and the real roots of
(31)
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Here, (◇) and (◇) are special cases of Watson's identities and (◇) is a special case of Abel's duplication formula with (Gordon and McIntosh 1997, Bytsko 1999).
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
(32)
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with
(33)
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(34)
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(Gordon and McIntosh 1997).