L-Algebraic Number
المؤلف:
Bytsko, A. G.
المصدر:
"Two-Term Dilogarithm Identities Related to Conformal Field Theory." 9 Nov 1999. http://arxiv.org/abs/math-ph/9911012.
الجزء والصفحة:
...
10-8-2019
2392
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
(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
(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
(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.
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