159,602 research outputs found

    New Multiple Harmonic Sum Identities

    Full text link
    We consider a special class of binomial sums involving harmonic numbers and we prove three identities by using the elementary method of the partial fraction decomposition. Some applications to infinite series and congruences are given

    The Many Faces of a Character

    Full text link
    We prove an identity between three infinite families of polynomials which are defined in terms of `bosonic', `fermionic', and `one-dimensional configuration' sums. In the limit where the polynomials become infinite series, they give different-looking expressions for the characters of the two integrable representations of the affine su(2)su(2) algebra at level one. We conjecture yet another fermionic sum representation for the polynomials which is constructed directly from the Bethe-Ansatz solution of the Heisenberg spin chain.Comment: 14/9 pages in harvmac, Tel-Aviv preprint TAUP 2125-9

    Calculating Three Loop Ladder and V-Topologies for Massive Operator Matrix Elements by Computer Algebra

    Full text link
    Three loop ladder and VV-topology diagrams contributing to the massive operator matrix element AQgA_{Qg} are calculated. The corresponding objects can all be expressed in terms of nested sums and recurrences depending on the Mellin variable NN and the dimensional parameter ε\varepsilon. Given these representations, the desired Laurent series expansions in ε\varepsilon can be obtained with the help of our computer algebra toolbox. Here we rely on generalized hypergeometric functions and Mellin-Barnes representations, on difference ring algorithms for symbolic summation, on an optimized version of the multivariate Almkvist-Zeilberger algorithm for symbolic integration, and on new methods to calculate Laurent series solutions of coupled systems of differential equations. The solutions can be computed for general coefficient matrices directly for any basis also performing the expansion in the dimensional parameter in case it is expressible in terms of indefinite nested product-sum expressions. This structural result is based on new results of our difference ring theory. In the cases discussed we deal with iterative sum- and integral-solutions over general alphabets. The final results are expressed in terms of special sums, forming quasi-shuffle algebras, such as nested harmonic sums, generalized harmonic sums, and nested binomially weighted (cyclotomic) sums. Analytic continuations to complex values of NN are possible through the recursion relations obeyed by these quantities and their analytic asymptotic expansions. The latter lead to a host of new constants beyond the multiple zeta values, the infinite generalized harmonic and cyclotomic sums in the case of VV-topologies.Comment: 110 pages Latex, 4 Figure
    • …
    corecore