SUMS OF SERIES OF ROGERS DILOGARITHM FUNCTIONS

ABDOLHOSSEIN HOORFAR AND FENG QI

Abstract. Some sums of series of Rogers dilogarithm functions are established by Abel’s functional equation.

1. Introduction

The dilogarithm is defined [2, p.102] by the series

Li_{2}(x) = ∑_{n=1}^{∞} \frac{x^{n}}{n^{2}}  (1)

for −1 ≤ x ≤ 1. The Rogers dilogarithm function L_{R}(x) is defined in [8] and [13, p. 287] for 0 ≤ x ≤ 1 by

L_{R}(x) = \begin{cases}Li_{2}(x) + \frac{1}{2} ln x ln(1 − x), & 0 < x < 1, \\ 0, & x = 0, \\ \frac{π^{2}}{6} , & x = 1. \end{cases}  (2)

The function L_{R}(x) satisfies the concise identity

L_{R}(x) + L_{R}(1 − x) = \frac{π^{2}}{6}  (3)

for 0 ≤ x ≤ 1, see [7, pp. 110–113], and Abel’s functional equation

L_{R}(x) + L_{R}(y) = L_{R}(xy) + L_{R}(\frac{x(1 − y)}{1 − xy}) + L_{R}(\frac{y(1 − x)}{1 − xy})  (4)

for 0 < x, y < 1, see [1, pp. 189–192] and [5]. The duplication formula for L_{R}(x) follows from Abel’s functional equation (4) and is given for 0 ≤ x ≤ 1 by

L_{R}(x) = \frac{1}{2}L_{R}(x^{2}) + L_{R}(\frac{x}{1 + x}).  (5)

Date: This paper was finalized on 8 February 2007.

2000 Mathematics Subject Classification. Primary 33E20, 11B65, 33D15, 34E05; Secondary 34E05, 41A60.

Key words and phrases. Rogers dilogarithm function, Abel’s functional equation, series, sum, dilogarithm function, identity.

This paper was typeset using AMS-L^{A}TEX.

2 A. HOORFAR AND F. QI

The function L_{R}(x) satisfies also the following identities:

L_{R}(\frac{1}{2}) = \frac{π^{2}}{12} , L_{R}(ρ) = \frac{π^{2}}{10} , L_{R}(ρ^{2}) = L_{R}(1 − ρ) = \frac{π^{2}}{15} ,  (6)

where ρ = \frac{\sqrt{5} −1}{2}, and has the nice infinite series

∑_{n=2}^{∞} L_{R}(\frac{1}{n^{2}} ) = \frac{π^{2}}{6}  (7)

obtained in [13, p. 298] and [14].

It is remarked that the formulas from (1) to (7) can be looked up at [18, 19]. For more information on its history, properties, identities, generalizations, ap-

plications and recent developments of the dilogarithms and Rogers dilogarithm functions, please refer to [1, pp. 189–192], [2, pp.102–107], [4, pp. 323–326], [7, pp. 110–113], [3, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22] and the references therein.

The main aim of this paper is to generalize the series (7).

Our main results are the following four theorems.

Theorem 1. For p, q ∈ ℕ and α ≥ 0,

∑_{n=0}^{∞} L_{R}(\frac{pq}{(n + p + α)(n + q + α)})

= ∑_{n=0}^{q−1} L_{R}(\frac{p}{n + p + α}) + ∑_{n=0}^{p−1} L_{R}(\frac{q}{n + q + α}). (8)

Remark 1. The series (7) is a special case of (8) for p = q = α = 1.

Theorem 2. For p, q ∈ ℕ and 0 < θ, β < 1,

∑_{n=0}^{∞} L_{R}(\frac{β(1 − θ^{p})(1 − θ^{q})θ^{n}}{(1 − βθ^{n+p})(1 − βθ^{n+q})})

= ∑_{n=0}^{q−1} L_{R}(\frac{β(1 − θ^{p})θ^{n}}{1 − βθ^{n+p}} ) + ∑_{n=0}^{p−1} L_{R}(\frac{1 − θ^{q}}{1 − βθ^{n+q}} ) − pL_{R}(1 − θ^{q}). (9)

Theorem 3. For p, q ∈ ℕ, 0 < β ≤ 1 and 0 < θ < 1,

∑_{n=0}^{∞} L_{R}(\frac{β(1 − θ^{p})(1 − θ^{q})θ^{n}}{(1 + βθ^{n})(1 + βθ^{n+p+q})})

= ∑_{n=0}^{q−1} L_{R}(\frac{θ^{p}(1 + βθ^{n})}{1 + βθ^{n+p}}) + ∑_{n=0}^{p−1} L_{R}(\frac{β(1 − θ^{q})θ^{n}}{1 + βθ^{n}} ) − qL_{R}(θ^{p}). (10)

SUMS OF SERIES OF ROGERS DILOGARITHM FUNCTIONS 3

Theorem 4. For r > 1,

∑_{n=0}^{∞} \frac{1}{2^{n}} L_{R}(\frac{1}{r^{2^{n}} + 1}) = L_{R}(\frac{1}{r}).  (11)

As straightforward consequences of above theorems, some sums of series of special Rogers dilogarithm functions are deduced as follows. Corollary 1. Let t > 0 and ϕ = \frac{\sqrt{5} +1}{2}, then the following identities are valid:

∑_{n=2}^{∞} L_{R}(\frac{2}{n(n + 1)}) = \frac{π^{2}}{4} ,  (12)
∑_{n=0}^{∞} \frac{1}{2^{n}} L_{R}(\frac{1}{2^{2^{n}} + 1}) = \frac{π^{2}}{12} ,  (13)
∑_{n=1}^{∞} L_{R}(\frac{2}{n^{2} + \sqrt{5} n + 1}) = \frac{π^{2}}{6} + L_{R}(3 − \sqrt{5} ),  (14)
∑_{n=1}^{∞} L_{R} (\frac{2}{(n + \sqrt{2} )(n + 1 + \sqrt{2} )}) = \frac{π^{2}}{6} + L_{R}(\frac{1}{2 + \sqrt{2}} ),  (15)
∑_{n=1}^{∞} L_{R} (\frac{4}{(2n − 1 + \sqrt{5} )^{2}} ) = \frac{π^{2}}{5} ,  (16)
∑_{n=2}^{∞}(−1)^{n}L_{R}(\frac{4}{n^{2}} ) = \frac{π^{2}}{3} − 2L_{R}(\frac{2}{3}),  (17)
∑_{n=1}^{∞} L_{R}(\frac{2^{n}}{(2^{n+1} − 1)^{2}} ) = \frac{π^{2}}{12} ,  (18)
∑_{n=1}^{∞} L_{R}(\frac{ϕ^{n−2}}{(ϕ^{n+1} − 1)^{2}} ) = \frac{π^{2}}{10} ,  (19)
∑_{n=1}^{∞} L_{R}(\frac{2^{n−1}}{(2^{n−1} + 1)(2^{n+1} + 1)}) = \frac{3}{2}L_{R}(\frac{1}{4}),  (20)
∑_{n=1}^{∞} L_{R}(\frac{2^{2}3^{n−1}}{(3^{n−1} + 1)(3^{n+1} + 1)}) = \frac{π^{2}}{12} ,  (21)
∑_{n=2}^{∞} L_{R}(\frac{sinh^{2} t}{sinh^{2}(nt)}) = L_{R}(e^{−2t}),  (22)

and

∑_{n=1}^{∞} L_{R}(\frac{sinh^{2} t}{cosh[(n − 1)t] cosh[(n + 1)t]})

= L_{R}(\frac{e^{−t}}{cosh t}) + L_{R}(e^{−t} sinh t) − L_{R}(e^{−2t}). (23)

4 A. HOORFAR AND F. QI

2. Proofs of theorems and corollary

Proof of Theorem 1. Let

x_{n} = \frac{p}{n + p + α} and y_{n} = \frac{q}{n + q + α}

for n = 0, 1, 2, . . .. It is clear that

\frac{x_{n}(1 − y_{n})}{1 − x_{n}y_{n}} = \frac{p}{(n + q) + p + α} = x_{n+q},

and

\frac{y_{n}(1 − x_{n})}{1 − x_{n}y_{n}} = \frac{q}{(n + p) + q + α} = y_{n+p}.

Taking x = x_{n} and y = y_{n} in (4) leads to

L_{R}(x_{n}) + L_{R}(y_{n}) = L_{R}(\frac{pq}{(n + p + α)(n + q + α)}) + L_{R}(x_{n+q}) + L_{R}(y_{n+p})

for n = 0, 1, 2, . . .. Summing up on both sides of above equality for n from 0 to N ≥ max{p, q} gives

∑_{n=0}^{q−1} L_{R}(x_{n}) + ∑_{n=0}^{p−1} L_{R}(y_{n}) = ∑_{n=0}^{N} L_{R}(\frac{pq}{(n + p + α)(n + q + α)})

+ ∑_{n=N+1−q}^{N} L_{R}(x_{n+q}) + ∑_{n=N+1−p}^{N} L_{R}(y_{n+p}).

Letting N → ∞ yields

lim_{N→∞} ∑_{n=N+1−q}^{N} L_{R}(x_{n+q}) = lim_{N→∞} ∑_{n=N+1−p}^{N} L_{R}(y_{n+p}) = 0.

The proof of Theorem 1 is complete. □ Proof of Theorem 2. Now let us consider the sequences

x_{n} = \frac{β(1 − θ^{p})θ^{n}}{1 − βθ^{n+p}} and y_{n} = \frac{1 − θ^{q}}{1 − βθ^{n+q}}

for n = 0, 1, 2, . . .. It is obvious that 0 < x_{n} < 1 and 0 < y_{n} < 1. Straightforward computation gives

\frac{x_{n}(1 − y_{n})}{1 − x_{n}y_{n}} = \frac{β(1 − θ^{p})θ^{n+q}}{1 − βθ^{(n+q)+p}} = x_{n+q}

and

\frac{y_{n}(1 − x_{n})}{1 − x_{n}y_{n}} = \frac{1 − θ^{q}}{1 − βθ^{(n+p)+q}} = y_{n+p}.

SUMS OF SERIES OF ROGERS DILOGARITHM FUNCTIONS 5

Using identity (4) again gives

L_{R}(x_{n}) + L_{R}(y_{n}) = L_{R}(x_{n}y_{n}) + L_{R}(x_{n+q}) + L_{R}(y_{n+p}).

Summing up for n from 0 to N ≥ max{p, q} leads to

∑_{n=0}^{q−1} L_{R}(x_{n}) + ∑_{n=0}^{p−1} L_{R}(y_{n}) = ∑_{n=0}^{N} L_{R}(x_{n}y_{n})

+ ∑_{n=N+1−q}^{N} L_{R}(x_{n+q}) + ∑_{n=N+1−p}^{N} L_{R}(y_{n+p}).

Since lim_{n→∞} L_{R}(x_{n}) = 0 and lim_{n→∞} L_{R}(y_{n}) = L_{R}(1 − θ^{q}), if taking N → ∞ in above identity, then formula (9) follows. The proof of Theorem 3 is finished. □ Proof of Theorem 3. Let

x_{n} = \frac{θ^{p}(1 + βθ^{n})}{1 + βθ^{n+p}} and y_{n} = \frac{β(1 − θ^{q})θ^{n}}{1 + βθ^{n}}

for n = 0, 1, 2, . . .. It is apparent that 0 < x_{n}, y_{n} < 1. Direct calculation reveals

x_{n}y_{n} = \frac{β(1 − θ^{q})θ^{n+p}}{1 + βθ^{n+p}} = y_{n+p}

and

\frac{x_{n}(1 − y_{n})}{1 − x_{n}y_{n}} = \frac{θ^{p}(1 + βθ^{n+q})}{1 + βθ^{(n+q)+p}} = x_{n+q}

with

\frac{y_{n}(1 − x_{n})}{1 − x_{n}y_{n}} = \frac{β(1 − θ^{p})(1 − θ^{q})θ^{n}}{(1 + θ^{n})(1 + θ^{n+p+q})} ≜ z_{n}.

From identity (4), it follows that

L_{R}(x_{n}) + L_{R}(y_{n}) = L_{R}(y_{n+p}) + L_{R}(x_{n+q}) + L_{R}(z_{n}).

Therefore, for N ≥ max{p, q},

∑_{n=0}^{q−1} L_{R}(x_{n}) + ∑_{n=0}^{p−1} L_{R}(y_{n}) = ∑_{n=0}^{N} L_{R}(z_{n})

+ ∑_{n=N+1−q}^{N} L_{R}(x_{n+q}) + ∑_{n=N+1−p}^{N} L_{R}(y_{n+p}).

Since lim_{n→∞} L_{R}(x_{n}) = L_{R}(θ^{p}) and lim_{n→∞} L_{R}(y_{n}) = 0, then formula (10) is deduced by taking N → ∞. Theorem 3 is proved. □

6 A. HOORFAR AND F. QI

Proof of Theorem 4. Applying (5) to x = x_{n} = \frac{1}{r^{2^{n}} +1} for n = 0, 1, 2, … gives

L_{R}(x_{n}) = \frac{1}{2}L_{R}(x_{n+1}) + L_{R}(\frac{1}{r^{2^{n}} + 1})

and

\frac{1}{2^{n}} L_{R}(x_{n}) = \frac{1}{2^{n+1}} L_{R}(x_{n+1}) + \frac{1}{2^{n}} L_{R}(\frac{1}{r^{2^{n}} + 1})

for n = 0, 1, 2, . . .. Summing up for n from 0 to ∞ yields

∑_{n=0}^{∞} \frac{1}{2^{n}} L_{R}(\frac{1}{r^{2^{n}} + 1}) = L_{R}(x_{0}) = L_{R}(\frac{1}{r}).

The proof of Theorem 4 is complete. □

Proof of Corollary 1. Taking p = 2, q = 1 and α = 1, \frac{\sqrt{5} −1}{2}, \sqrt{2} in (8) and simplifying by employing (3) and (6) leads to the identities (12), (14) and (15) respectively.

Identity (13) is a direct consequence of (11) for r = 2.

Letting p = q = 1 and α = \frac{\sqrt{5} −1}{2} in (8) yields (16).

It is easy to see that

∑_{n=2}^{∞}(−1)^{n}L_{R}(\frac{4}{n^{2}} ) = ∑_{n=1}^{∞} L_{R}(\frac{1}{n^{2}} ) − ∑_{n=1}^{∞} L_{R}(\frac{1}{(n + 1/2)^{2}} ).

Combining this with (8) for p = q = 1 and α = \frac{1}{2} leads to (17).

Identities (18) and (19) are special cases of (9) for p = q = 1, β = θ = \frac{1}{2} and β = θ = \frac{1}{ϕ} = \frac{\sqrt{5} −1}{2}, respectively.

Applying p = q = β = 1 and θ = \frac{1}{2} in (10) gives

∑_{n=1}^{∞} L_{R}(\frac{2^{n−1}}{(2^{n−1} + 1)(2^{n+1} + 1)}) = L_{R}(\frac{2}{3}) + L_{R}(\frac{1}{4}) − L_{R}(\frac{1}{2}).

Taking x = \frac{1}{2} in identity (5) yields

L_{R}(\frac{2}{3}) − L_{R}(\frac{1}{2}) = \frac{1}{2}L_{R}(\frac{1}{4}).

Thus, identity (20) is obtained.

Identity (21) is a direct consequence of (10) for p = q = β = 1 and θ = \frac{1}{3}.

Taking θ = e^{−2t} and β = e^{−2b} in (9) and (10) and simplifying gives

∑_{n=0}^{∞} L_{R}(\frac{sinh(pt) sinh(qt)}{sinh((n + p)t + b) sinh((n + q)t + b)}) = ∑_{n=0}^{q−1} L_{R}(\frac{e^{−(nt+b)} sinh(pt)}{sinh((n + p)t + b)})

+ ∑_{n=0}^{p−1} L_{R}(\frac{e^{(nt+b)} sinh(qt)}{sinh((n + q)t + b)}) − pL_{R}(1 − e^{2qt}) (24)

for t > 0 and b > 0 and

∑_{n=0}^{∞} L_{R}(\frac{sinh(pt) sinh(qt)}{cosh(nt + b) cosh((n + p + q)t + b)}) = ∑_{n=0}^{q−1} L_{R}(\frac{e^{−pt} cosh(nt + b)}{cosh((n + p)t + b)})

+ ∑_{n=0}^{p−1} L_{R}(\frac{e^{−(n+q)t−b} sinh(qt)}{cosh(nt + b)}) − qL_{R}(e^{−2t}) (25)

for t > 0 and b ≥ 0. Identities (22) and (23) are special cases of (24) and (25) for p = q = 1, b = t and b = 0, respectively. □

References

[1] N. H. Abel (Ed. L. Sylow and S. Lie), Oeuvres Completes, Vol. 2, Johnson Reprint Corp., New York, 1988.

[2] G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.

[3] J. Baddoura, Integration in finite terms with elementary functions and dilogarithms, J. Symbolic Comput. 41 (2006), no.8, 909–942.

[4] B. C. Berndt, Romanujan’s Notebooks, Part IV, Springer-Verlag, New York, 1994.

[5] A. G. Bytsko, 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 (1999), 8045–8058.

[6] F. Chapoton, Functional identities for the Rogers dilogarithm associated to cluster Y -systems, Bull. London Math. Soc. 37 (2005), no. 5, 755–760.

[7] L. Euler, Institutiones calculi integralis, Vol. 1, Basel, Switzerland: Birkhäuser, 1768.

[8] B. Gordon and R. J. McIntosh, Algebraic dilogarithm identities, Ramanujan J. 1 (1997), no. 4, 431–448.

[9] M. Hassani, Approximation of the dilogarithm function, RGMIA Res. Rep. Coll. 8 (2005),

no. 4, Art. 18. Available onine at http://rgmia.vu.edu.au/v8n4.html.

[10] A. N. Kirillov, Dilogarithm identities, Progr. Theor. Phys. Suppl. 118 (1995), 61–142.

[11] L. Lewin, Dilogarithms and Associated Functions, Maodnald, London, 1958.

[12] L. Lewin, Polylogarithms and Associated Functions, North-Holland, New York, 1981.

[13] L. Lewin, Structural Properties of Polylogarithms, Amer. Math. Soc., Providence, 1991.

[14] L. Lewin, The dilogarithm in algebraic fields, J. Austral. Math. Soc. (Ser. A) 33 (1982), 302–330.

[15] S. Myung, A bilinear form of dilogarithm and motivic regulator map, Adv. Math. 199 (2006), no. 2, 331–355.

[16] G. Rhin and C. Viola, The permutation group method for the dilogarithm, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), no. 3, 389–437.

[17] L. J. Rogers, On function sum theorems connected with the series ∑_{1}^{∞} x^{n}/n^{2}, Proc. London Math. Soc. 4 (1907), 169–189.

[18] E. W. Weisstein, Dilogarithm, From MathWorld—A Wolfram Web Resource. http://

    mathworld.wolfram.com/Dilogarithm.html.

[19] E. W. Weisstein, Rogers L-Function, From MathWorld—A Wolfram Web Resource. http:

    //mathworld.wolfram.com/RogersL-Function.html.

[20] E. W. Weisstein, Polylogarithm, From MathWorld—A Wolfram Web Resource. http://

    mathworld.wolfram.com/Polylogarithm.html.

[21] D. Zagier, The dilogarithm function in geometry and number theory, Number theory and related topics (Bombay, 1988), 231–249, Tata Inst. Fund. Res. Stud. Math., 12, Tata Inst. Fund. Res., Bombay, 1989.

[22] W. Zudilin, Approximations to q-logarithms and q-dilogarithms, with applications to q-zeta values, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 322 (2005), Trudy po Teorii Chisel, 107–124, 253–254.

(A. Hoorfar) Department of Irrigation Engineering, College of Agriculture, Tehran University, Karaj, 31587-77871, Iran

   E-mail address: hoorfar@ut.ac.ir

(F. Qi) Research Institute of Mathematical Inequality Theory, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China

   E-mail address: qifeng618@gmail.com, qifeng618@hotmail.com, qifeng618@msn.com, qifeng618@qq.com,
qifeng@hpu.edu.cn, fengqi618@member.ams.org
   URL: http://rgmia.vu.edu.au/qi.html