The Ramanujan Journal

, Volume 1, Issue 4, pp 431–448

# Algebraic Dilogarithm Identities

• Basil Gordon
• Richard J. Mcintosh
Article

## 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 ck are integers.

dilogarithm basic hypergeometric series q-series

## Preview

### References

1. 1.
N.H. Abel, Oeuvres Complètes, 2, 189-192.Google Scholar
2. 2.
L. Euler, “Institutiones calculi integralis,” 1(1768), 110-113.Google Scholar
3. 3.
G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990.Google Scholar
4. 4.
B. Gordon and R.J. McIntosh, “Eighth order mock theta functions,” (to appear).Google Scholar
5. 5.
A.N. Kirillov, “Dilogarithm identities,” Preprint hep-th/9408113.Google Scholar
6. 6.
J. Landen, Mathematical memoirs, 1(1780), 112.Google Scholar
7. 7.
L. Lewin, “The dilogarithm in algebraic fields,” J. Austral. Soc. Ser. A 33(1982), 302-330.Google Scholar
8. 8.
---(Ed.), Structural Properties of Polylogarithms, “Mathematical Surveys and Monographs,” 37 (American Mathematical Society, Providence, 1991).Google Scholar
9. 9.
J.H. Loxton, “Special values of the dilogarithm function,” Acta Arith. 43(1984), 155-166.Google Scholar
10. 10.
R.J. McIntosh, “Some asymptotic formulae for q-hypergeometric series,” J. London Math. Soc. (2) 51(1995) 120-136.Google Scholar
11. 11.
---, “Some asymptotic formulae for q-shifted factorials,” (in preparation).Google Scholar
12. 12.
L.J. Rogers, “On function sum theorems connected with the series ∑1 xn / n2,” Proc. London Math. Soc. 4(1907), 169-189.Google Scholar
13. 13.
G.N. Watson, “A note on Spence's logarithmic transcendent,” Quart. J. Math., Oxford Ser. 8(1937), 39-42.Google Scholar
14. 14.
D. Zagier, “Special values and functional equations of polylogarithms,” Appendix A of [8].Google Scholar