IDENTITIES FOR SUMS OF A q-ANALOGUE OF POLYLOGARITHM FUNCTIONS

We present several identities for sums of q-polylogarithm functions. Our motivation for these are the relations between the q-zeta function (see [3, 4, 12] and references therein), q-polylogarithm functions (see below) and the quantum group SUq(2). More precisely, the left regular representation of the quantum group SUq(2) is the coordinate ring A(SUq(2)) represented as a left Uq(sl(2))-module. It is well known that it has the homogeneous decomposition A(SUq(2)) = ⊕k≥1Ak, where Ak corresponds to the irreducible representation of Uq(sl(2)) with spin (k − 1)/2 and dimAk = k . Let C = q k+qk−2 (q−q−1)2 + ef be the Casimir operator of Uq(sl(2)) that acts on Ak as a scalar operator, that is, C|Ak = an = ( q−q q−q−1 2 . Thus one can introduce a spectral zeta-function associated to the quantum group SUq(2) by Z(s, SUq(2)) = (1 + q ) ∑