On multiple $L$ -values
Journal of the Mathematical Society of Japan
We formulate and prove reguralized double shu¿e and derivation relations for multiple L-values. A description of principal part of a multiple L-function is also given. Introduction. In the present paper, we study the regularized double shu¿e and the derivation relations of the multiple L-values and give some applications. A fair amount of work related to the multiple L-values has already been done, e.g., A. Goncharov [G1], [G2], G. Racinet [R] and the references therein. In particular, the
... arization stu¤ is also treated in a series of works of Goncharov and Racinet. Our approach here, which largely follows the setup and method given in [IKZ] for multiple zeta values, is less abstract and more directly aimed at obtaining relations among multiple L-values. In particular, a generalization of the derivation relation of multiple zeta values, which as shown in [IKZ] is in a sense equivalent to the regularized double shu¿e relation, is established by using the regularization and the method developed in [IKZ]. In §1 we present some basic definitions and algebraic setup introduced by M. Ho¤man [H] which is suitable for our study. In §2, after the discussion on the finite double shu¿e relation (Proposition 2.1), we give the regularized double shu¿e relations (Theorem 2.3, Theorem 2.4). The derivation relation (Theorem 3.1) is formulated and proved in §3. The final §4 is devoted to a couple of applications of the results and ideas developed in the previous sections. Of them, the principal part of a certain multiple Lfunction is determined (Theorem 4.1) in terms of the polynomials defined algebraically to describe the regularization procedure.