A notion of additive dilogarithm for a field k is introduced, based on the K-theory and higher Chow groups of the affine line relative to 2(0). Analogues of the K2-regulator, the polylogarithm Lie algebra, and the `-adic realization of the dilogarithm motive are discussed. The higher Chow groups of 0-cycles in this theory are identified with the Kähler differential forms Ωk. It is hoped that th...

LiI(x1, . . . , xn) = AI(∞;x1, . . . , xk) are called multiple polylogarithms [2]: they generalize the classical polylogarithm Lin(x1) = A(n)(∞;x1). It is immediate from the defining equations (1) and (2) that the sums SI can be written in terms of the AI . To state the relation precisely, let C(n) be the set of compositions of n, i.e., ordered sequences (i1, . . . , ik) of positive integers wi...

Let T be the triangle with vertices (1,0), (0,1), (1,1). We study certain integrals over T , one of which was computed by Euler. We give expressions for them both as a linear combination of multiple zeta values, and as a polynomial in single zeta values. We obtain asymptotic expansions of the integrals, and of sums of certain multiple zeta values with constant weight. We also give related expre...

The specializations of the motivic elliptic polylogarithm on the universal elliptic curve to the modular curve are referred to as Eisenstein classes. In this paper, we prove that the syntomic realization of the Eisenstein classes restricted to the ordinary locus of the modular curve may be expressed using p-adic Eisenstein series of negative weight, which are p-adic modular forms defined using ...


 In this paper, the generalized Laguerre-Apostol-Frobenius-type poly-Genocchi polyno- mials of higher order with parameters a, b and c are defined using concept polylogarithm, Laguerre, Apostol Frobenius polynomials. These polynomials possess numerous properties including recurrence relations, explicit formulas certain differential identity. Moreover, some connections these poly- nomials ...

The N = ∞ vector O(N) model is a solvable, interacting field theory in three dimensions (D). In a recent paper with A. Chubukov and J. Ye [1], we have computed a universal number, c̃, characterizing the size dependence of the free energy at the conformally-invariant critical point of this theory. The result [1] for c̃ can be expressed in terms of polylogarithms. Here, we use non-trivial polylogar...

Let N be a positive integer. In this paper we shall study the special values of multiple polylogarithms atNth roots of unity, called multiple polylogarithm values (MPVs) of levelN . These objects are generalizations of multiple zeta values and alternating Euler sums, which was studied by Euler, and more recently, many mathematicians and theoretical physicists.. Our primary goal in this paper is...