Abstract not found

We formulate general conjectures about the relationship between the A-model connection on the cohomology of a d-dimensional Calabi-Yau complete intersection V of r hypersurfaces V1,...,Vr in a toric variety PΣ and the system of differential operators annihilating the special generalized hypergeometric function Φ0 depending on the fan Σ. In this context, the mirror symmetry phenomenon can be interpreted as the twofold characterization of the series Φ0. First, Φ0 is defined by intersection numbers of rational curves in PΣ with the hypersurfaces Vi and their toric degenerations. Second, Φ0 is the power expansion near a boundary point of the moduli space of the monodromy invariant period of the homolomorphic differential d-form on an another Calabi-Yau d-fold V ′ called the mirror of V. Using this generalized hypergeometric series, we propose a general construction for mirrors V ′ of V and canonical q-coordinates on the moduli spaces of Calabi-Yau manifolds. 1

Besides an elementary introduction to q-identities and basic hypergeometric series, a newly developed Mathematica implementation of a q-analogue of Zeilberger's fast algorithm for proving terminating q-hypergeometric identities together with its theoretical background is described. To illustrate the usage of the package and its range of applicability, non-trivial examples are presented as well as additional features like the computation of companion and dual identities.

Abstract. Historically, the polylogarithm has attracted specialists and nonspecialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including an intriguing conjecture of Don Zagier. 1.

. We compute the inverse of a specific infinite-dimensional matrix, thus unifying a number of previous matrix inversions. Our inversion theorem is applied to derive a number of summation formulas of hypergeometric type. 1. Introduction. Let F = (f nk ) n;k2Z (Zdenotes the set of integers) be an infinitedimensional lower-triangular matrix; i.e. f nk = 0 unless n k. The matrix (f \Gamma1 kl ) k;l2Z is the inverse matrix of F if and only if X nkl f nk f \Gamma1 kl = ffi nl for all n; l 2 Z. Such matrix inversions are very important in many fields of combinatorics and special functions. For example, when dealing with combinatorial sums, application of the so-called "inverse relations" (see (4.1) and (4.2)), which base on matrix inversion, helps to simplify problems, or yields new identities. Riordan dedicated two chapters of his book [21] to inverse relations and its applications. Riordans inverse relations were classified and given a unified method of proof by Egorychev [7, ch.3]...

Introduction Binomial series and q-binomial series, such as 1 X k=0 ` M k '` N R \Gamma k ' ; respectively 1 X k=0 q (M \Gammak)(R\Gammak) M k q N R \Gamma k q ; where the q-binomial coefficient is defined by n k q = (1 \Gamma q n )(1 \Gamma q n\Gamma1 ) \Delta \Delta \Delta (1 \Gamma q n\Gammak+1 ) (1 \Gamma q k )(1 \Gamma q k\Gamma1 ) \Delta \D

Several characteristic parameters of randomly grown quadtrees of any dimension are analyzed. Additive parameters have expectations whose generating functions are expressible in terms of generalized hypergeometric functions. A complex asymptotic process based on singularity analysis and integral representations akin to Mellin transforms leads to explicit values for various structure constants related to path length, retrieval costs, and storage occupation.

Abstract. We compute the number of rhombus tilings of a hexagon with sides N, M, N, N, M, N, which contain a fixed rhombus on the symmetry axis that cuts through the sides of length M. 1.

We prove a constant term conjecture of Robbins and Zeilberger (J. Cambin.

. We deal with the unweighted and weighted enumerations of lozenge tilings of a hexagon with side lengths a; b + m; c; a + m; b; c + m, where an equilateral triangle of side length m has been removed from the center. We give closed formulas for the plain enumeration and for a certain (\Gamma1)-enumeration of these lozenge tilings. In the case that a = b = c, we also provide closed formulas for certain weighted enumerations of those lozenge tilings that are cyclically symmetric. For m = 0, the latter formulas specialize to statements about weighted enumerations of cyclically symmetric plane partitions. One such specialization gives a proof of a conjecture of Stembridge on a certain weighted count of cyclically symmetric plane partitions. The tools employed in our proofs are nonstandard applications of the theory of nonintersecting lattice paths and determinant evaluations. In particular, we evaluate the determinants det 0i;jn\Gamma1 \Gamma !ffi ij + \Gamma m+i+j j \Delta\Delta , w...