q-Analogues of the Barnes multiple zeta functions

In this paper, we introduce $q$-analogues of the Barnes multiple zeta functions. We show that these functions can be extended meromorphically to the whole plane, and moreover, tend to the Barnes multiple zeta functions when $q\uparrow 1$ for all complex numbers.Comment: 13 page

Checkerboard style Schur multiple zeta values and odd single zeta values

We give explicit formulas for the recently introduced Schur multiple zeta values, which generalize multiple zeta(-star) values and which assign to a Young tableaux a real number. In this note we consider Young tableaux of various shapes, filled with alternating entries like a Checkerboard. In particular we obtain new sum representation for odd single zeta values in terms of these Schur multiple zeta values. As a special case we show that some Schur multiple zeta values of Checkerboard style, filled with 1 and 3, are given by determinants of matrices with odd single zeta values as entries.Comment: 21 pages. Added Corollary 3.7 and the case (a,b)=(1,2) in Section

On Schur multiple zeta functions: A combinatoric generalization of multiple zeta functions

We introduce Schur multiple zeta functions which interpolate both the multiple zeta and multiple zeta-star functions of the Euler-Zagier type combinatorially. We first study their basic properties including a region of absolute convergence and the case where all variables are the same. Then, under an assumption on variables, some determinant formulas coming from theory of Schur functions such as the Jacobi-Trudi, Giambelli and dual Cauchy formula are established with the help of Macdonald's ninth variation of Schur functions. Moreover, we investigate the quasi-symmetric functions corresponding to the Schur multiple zeta functions. We obtain the similar results as above for them and, furthermore, describe the images of them by the antipode of the Hopf algebra of quasi-symmetric functions explicitly. Finally, we establish iterated integral representations of the Schur multiple zeta values of ribbon type, which yield a duality for them in some cases.Comment: 42 pages, 2 figure

Abductive Proof Procedure with Adjusting Derivations for General Logic Programs

In this paper, we formulate a new integrity constraint in correlation with 3-valued stable models in an abduction framework based on general logic programs. Under the constraint, not every ground atom or its negation is a logical consequence of the theory and an expected abductive explanation, but some atom may be unspecified as a logical consequence by an adjustment. As a reflection of the integrity constraint with an adjustment, we augment an adjusting derivation to Eshghi and Kowalski abductive proof procedure, in which such an unspecified atom can be dealt with