Abstract
The Rogers L-function \(L(x) = \sum\limits_{n = 1}^\infty {\frac{{x^n }} {{n^2 }} + \frac{1} {2}\log x} \log (1 - x) \)satisfies the functional equation \(L(x) + L(y) = L(xy) + L\left( {\frac{{x(1 - y)}} {{1 - xy}}} \right) + L\left( {\frac{{y(1 - x)}} {{1 - xy}}} \right) \).From this we derive several other such equations, including Euler's identity L(x)+L(1-x)=L(1) and various identities arising from summation and transformation formulas for basic hypergeometric series. We also obtain some new equations of the form \(\sum\limits_{k = 0}^n {c_k L(\theta ^k ) = 0} \) where θ is algebraic and the c k are integers.
Similar content being viewed by others
References
N.H. Abel, Oeuvres Complètes, 2, 189-192.
L. Euler, “Institutiones calculi integralis,” 1(1768), 110-113.
G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990.
B. Gordon and R.J. McIntosh, “Eighth order mock theta functions,” (to appear).
A.N. Kirillov, “Dilogarithm identities,” Preprint hep-th/9408113.
J. Landen, Mathematical memoirs, 1(1780), 112.
L. Lewin, “The dilogarithm in algebraic fields,” J. Austral. Soc. Ser. A 33(1982), 302-330.
---(Ed.), Structural Properties of Polylogarithms, “Mathematical Surveys and Monographs,” 37 (American Mathematical Society, Providence, 1991).
J.H. Loxton, “Special values of the dilogarithm function,” Acta Arith. 43(1984), 155-166.
R.J. McIntosh, “Some asymptotic formulae for q-hypergeometric series,” J. London Math. Soc. (2) 51(1995) 120-136.
---, “Some asymptotic formulae for q-shifted factorials,” (in preparation).
L.J. Rogers, “On function sum theorems connected with the series ∑ ∞1 xn / n2,” Proc. London Math. Soc. 4(1907), 169-189.
G.N. Watson, “A note on Spence's logarithmic transcendent,” Quart. J. Math., Oxford Ser. 8(1937), 39-42.
D. Zagier, “Special values and functional equations of polylogarithms,” Appendix A of [8].
Rights and permissions
About this article
Cite this article
Gordon, B., Mcintosh, R.J. Algebraic Dilogarithm Identities. The Ramanujan Journal 1, 431–448 (1997). https://doi.org/10.1023/A:1009709927327
Issue Date:
DOI: https://doi.org/10.1023/A:1009709927327