A study of elliptic gamma function and allies. (English) Zbl 1440.11063

Summary: We study analytic and arithmetic properties of the elliptic gamma function \[ \prod _{m,n=0}^\infty \frac{1-x^{-1}q^{m+1}p^{n+1}}{1-xq^mp^n}, \quad |q|,|p|<1,\] in the regime \(p=q\), in particular, its connection with the elliptic dilogarithm and a formula of S. J. Bloch [Higher regulators, algebraic \(K\)-theory, and zeta functions of elliptic curves. Providence, RI: American Mathematical Society (AMS) (2000; Zbl 0958.19001)]. We further extend the results to more general products by linking them to non-holomorphic Eisenstein series and, via some formulae of D. Zagier [Math. Ann. 286, No. 1–3, 613–624 (1990; Zbl 0698.33001)], to elliptic polylogarithms.


11F27 Theta series; Weil representation; theta correspondences
11G55 Polylogarithms and relations with \(K\)-theory
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[1] Bloch, S.J.: Higher regulators, algebraic \[KK\]-theory, and zeta functions of elliptic curves. In: Lecture Notes (UC Irvine, 1977); CRM Monograph Series, vol. 11. American Mathematical Society, Providence (2000)
[2] Duke, W., Imamo \[\bar{\text{g}}\] g¯lu, Ö.: On a formula of Bloch. Funct. Approx. 37(1), 109-117 (2007) · Zbl 1213.11141
[3] Felder, G., Varchenko, A.: The elliptic gamma function and \[\text{ SL }(3,{\mathbb{Z}})<imes {\mathbb{Z}}^3\] SL(3,Z)⋉Z3. Adv. Math. 156(1), 44-76 (2000) · Zbl 1038.11029
[4] Felder, G.; Varchenko, A.; Fuchs, J. (ed.); Mickelsson, J. (ed.); Rozenblioum, G. (ed.); Stolin, A. (ed.); Westerberg, A. (ed.), Multiplication formulae for the elliptic gamma function, No. 391, 69-73 (2005), Providence
[5] Ruijsenaars, S.: First order analytic difference equations and integrable quantum systems. J. Math. Phys. 38, 1069-1146 (1997) · Zbl 0877.39002
[6] Stienstra, J.; Yui, N. (ed.); Yau, S-T (ed.); Lewis, JD (ed.), Mahler measure variations, Eisenstein series and instanton expansions, No. 38, 139-150 (2006), Providence · Zbl 1118.11047
[7] Zagier, D.: The Bloch-Wigner-Ramakrishnan polylogarithm function. Math. Ann. 286, 613-624 (1990) · Zbl 0698.33001
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