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Rogers L-Function

المؤلف:  Abel, N. H.

المصدر:  Oeuvres Completes, Vol. 2 (Ed. L. Sylow and S. Lie). New York: Johnson Reprint Corp.,

الجزء والصفحة:  ...

13-8-2019

3201

Rogers L-Function

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

			(3)
			(4)
			(5)

The function  satisfies the concise reflection relation

(6)

(Euler 1768), as well as Abel's functional equation

(7)

(Abel 1988, Bytsko 1999). Abel's duplication formula for  follows from Abel's functional equation and is given by

(8)

The function has the nice series

(9)

(Lewin 1982; Loxton 1991, p. 298).

In terms of , the well-known dilogarithm identities become

			(10)
			(11)
			(12)
			(13)
			(14)

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

Numbers  which satisfy

(15)

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

(16)

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 relationalgorithms.

1	1	1
	1
		1
	1
		1
		1
		3
,
		1
		2
		3
		1
		2

Bytsko (1999) gives the additional identities

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

where

			(26)
			(27)
			(28)

with  the positive root of

(29)

and  and  the real roots of

(30)

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

(31)

with

			(32)
			(33)

(Gordon and McIntosh 1997).

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REFERENCES:

Abel, N. H. Oeuvres Completes, Vol. 2 (Ed. L. Sylow and S. Lie). New York: Johnson 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.

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.

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.