Rogers L-Function

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RogersLFunction

If Li_2(x) denotes the usual dilogarithm, then there are two variants that are normalized slightly differently, both called the Rogers L-function (Rogers 1907). Bytsko (1999) defines

L(x) = [画像:6/(pi^2)[Li_2(x)+1/2lnxln(1-x)]]

(1)

= [画像:6/(pi^2)[sum_(n=1)^(infty)(x^n)/(n^2)+1/2lnxln(1-x)],]

(2)

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

L_R(x) = Li_2(x)+1/2lnxln(1-x)

(3)

= [画像:(pi^2)/6L(x)]

(4)

= [画像:[sum_(n=1)^(infty)(x^n)/(n^2)+1/2lnxln(1-x)].]

(5)

The function L(x) satisfies the concise reflection relation

L(x)+L(1-x)=1

(6)

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

[画像: L(x)+L(y)=L(xy)+L((x(1-y))/(1-xy))+L((y(1-x))/(1-xy)) ]

(7)

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

[画像: 1/2L(x^2)=L(x)-L(x/(1+x)). ]

(8)

The function has the nice series

[画像: sum_(k=2)^inftyL(1/(k^2))=1 ]

(9)

(Lewin 1982; Loxton 1991, p. 298).

In terms of L(x), the well-known dilogarithm identities become

L(0) =

(10)

L(1-rho) = 2/5

(11)

L(1/2) = 1/2

(12)

L(rho) = 3/5

(13)

L(1) = 1

(14)

(Loxton 1991, pp. 287 and 289; Bytsko 1999), where rho=(sqrt(5)-1)/2.

Khoi (2014) gave the identity

[画像: L_R(1/(phi(sqrt(phi)+phi)))-L_R(phi/(sqrt(phi)+phi))=-(pi^2)/(20), ]

(15)

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

Numbers theta in (0,1) which satisfy

[画像: sum_(k=0)^nc_kL(theta^k)=0 ]

(16)

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

[画像: sum_(k=0)^n(e_k)/kL(theta^k)=c ]

(17)

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

theta e_k c

1 1 1

1/2 1 1/2

1/2 -1,6,3,0,0,-3 1/2

1/3 3,-1 1

1/2(sqrt(5)-1) 1 3/5

1/2(sqrt(5)-1) 1,-1,-12,0,0,6 -3/5

(sqrt(5)-2)^(1/3) 2,-1 1

sqrt(2)-1 2,-1 3/4

sqrt(2)-1 1,2,0,-1 5/8

3-2sqrt(2) 5,-2 1

1/2(sqrt(3)-1) 2,1,-1 5/6

sqrt(3)-1 2,-3,-1,0,0,1 1/2

2-sqrt(3) 4,1,0,-1 5/4

2-sqrt(3) 5,-3,-1,0,0,1 4/3

5-2sqrt(6) 23,-15,-3,0,0,3 3

1/2(sqrt(13)-3) 4,-2,-2,0,0,1 7/6

1/6(sqrt(13)-1) 3,1,-3,0,0,1 4/3

1/6(sqrt(13)+1) 3,-4,-3,0,0,2 2/3

4-sqrt(15) 15,2,-3,-2 5/2

1/2(5-sqrt(21)) 7,-1,-3,0,0,1 5/3

1/2sec(2/7pi), 1,-2 1/7

1/2sec(1/7pi) 1,1 5/7

2cos(3/7pi) 1,1 4/7

1/2sec(1/9pi) 1,2,-1 7/9

1/2sec(2/9pi) 1,-3,-1,0,0,1 -1/9

2cos(4/9pi) 1,-3,-1,0,0,1 1/9

x^3+2x-1 1,5,0,-4 1

x^3+2x-1 3,1,12,0,0,-6 2

2x^3+x-1 2,1,3,-2 3/2

x^3+x-1 2,6,3,0,0,-3 3

x^3-3x^2+4x-1 5,-9,-6,0,0,6 1

x^3+x^2-1 1,6,6,0,0,-6 2

x^3+x^2+x-1 1,1,-3 1/2

x^3+x^2+x-1 2,3,0,-2 3/2

Bytsko (1999) gives the additional identities

L(lambda^(-2))+L((lambda^2-1)^(-2))=4/7

(18)

L(lambda^(-2))+L((1+lambda)^(-1))=5/7

(19)

[画像:L(1-1/(sqrt(2)))+L(sqrt(2)-1)=3/4 ]

(20)

[画像:L(sqrt(rho))+L(1/(1+sqrt(rho)))=(13)/(11) ]

(21)

L(1/2-1/2rho)+L(2rho-1)=1/2

(22)

L(1-1/2rho-1/2sqrt(7rho-3))+L(1/2sqrt(28rho+45)-2rho-5/2)=2/5

(23)

L(1-delta^2)+L((1+delta)^(-2))=2/5

(24)

L(3/2-1/2sqrt(2)-1/2sqrt(2sqrt(2)-1))+L((3/2+sqrt(2))sqrt(2sqrt(2)-1)-3/2-3/2sqrt(2))=1/2

(25)

L(nu)-L(mu^(-1))=1/7

(26)

where

lambda = 2cos(pi/7)

(27)

rho = (sqrt(5)-1)/2

(28)

delta = 1/2(sqrt(3+2sqrt(5))-1),

(29)

with delta the positive root of

delta^4+delta^3-delta-1=0

(30)

and 0<nu<1 and mu>1 the real roots of

t^6-7t^5+19t^4-28t^3+20t^2-7t+1=0.

(31)

Here, (◇) and (◇) are special cases of Watson's identities and (◇) is a special case of Abel's duplication formula with x=1/sqrt(2) (Gordon and McIntosh 1997, Bytsko 1999).

Rogers (1907) obtained a dilogarithm identity in m variables with m^2+1 terms which simplifies to Euler's identity for m=1 and Abel's functional equation for m=2 (Gordon and McIntosh 1997). For m=3, it is equivalent to

L(a)+L(b)+L(c)-L(u)-L(v) =L(abc)+L(ac/u)+L(bc/v)-L(av/u)-L(bu/v),

(32)

with

av(1-bc)+bu(1-ac) = uv(1-ab)

(33)

v(1-a)+u(1-b) = 1-abc

(34)

(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 M(3,t), M(4,5), M(5,6) and M(6,7) 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. https://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 sum_1^(infty)x^n/n^2." 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.

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Rogers L-Function

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Weisstein, Eric W. "Rogers L-Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RogersL-Function.html

Rogers L-Function

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