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. We construct integral bases for the SO(3)-TQFT-modules of surfaces in genus one and two at roots of unity of prime order and show that the corresponding mapping class group representations preserve a unimodular Hermitian form over a ring of algebraic integers. For higher genus surfaces the Hermitian form sometimes must be non-unimodular. In one such case, genus 3 and p = 5, we still give an explicit basis. 1.

Abstract. We compute the number of rhombus tilings of a hexagon with side lengths a,b,c,a,b,c which contain the central rhombus and the number of rhombus tilings of a hexagon with side lengths a,b,c,a,b,c which contain the ‘almost central ’ rhombus above the centre.

1.1. Partitions, generalized Vandermonde determinants and representation theory. 5 1.2. The Frobenius character formulae. 8

Abstract. A determinant evaluation is proven, a special case of which establishes a conjecture of Bombieri, Hunt, and van der Poorten (Experimental Math. 4 (1995), 87–96) that arose in the study of Thue’s method of approximating algebraic numbers. 1. Introduction. In their study [2] of Thue’s method of approximating an algebraic number, Bombieri, Hunt, and van der Poorten conjectured two determinant evaluations, one of which can be restated as follows. Conjecture (Bombieri, Hunt, van der Poorten [2, next-to-last paragraph]). Let b, c be nonnegative integers, c ≤ b, and let ∆(b, c) be the determinant of the

In this article hypergeometric identities (or transformations) for p+1 F p - series and for Kampe de Feriet series of unit arguments are derived systematically from known transformations of hypergeometric series and products of hypergeometric series, respectively, using the beta integral method in an automated manner, based on the Mathematica package HYP. As a result we obtain some known and some identities which seem to not have been recorded before in literature.

From numerical experiments, D. E. Knuth conjectured that 0 ! D n+4 ! D n for a combinatorial sequence (D n ) defined as the difference D n = R n \Gamma L n of two definite hypergeometric sums. The conjecture implies an identity of type L n = bR n c, involving the floor function.

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A polynomial method for the enumeration of plane partitions and alternating sign matrices

Abstract. In 1879, Thomae discussed the relations between two generic hypergeometric 3F2-series with argument 1. It is well-known since then that there are 120 such relations (including the trivial ones which come from permutations of the parameters of the hypergeometric series). More recently, Rhin and Viola asked the following question (in a different, but equivalent language of integrals): If there exists a linear dependence relation over Q between two convergent 3F2-series with argument 1, with integral parameters, and whose values are irrational numbers, is this relation a specialisation of one of the 120 Thomae relations? A few years later, Sato answered this question in the negative, by giving six examples of relations which cannot be explained by Thomae’s relations. We show that Sato’s counter-examples can be naturally embedded into two families of infinitely many 3F2-relations, both parametrised by three independent parameters. Moreover, we find two more infinite families of the same nature. The families, which do not seem to have been recorded before, come from certain 3F2-transformation formulae and contiguous relations. We also explain in detail the relationship between the integrals of Rhin and Viola and 3F2-series.