92 research outputs found
Efficient Algorithms for Mixed Creative Telescoping
Creative telescoping is a powerful computer algebra paradigm -initiated by
Doron Zeilberger in the 90's- for dealing with definite integrals and sums with
parameters. We address the mixed continuous-discrete case, and focus on the
integration of bivariate hypergeometric-hyperexponential terms. We design a new
creative telescoping algorithm operating on this class of inputs, based on a
Hermite-like reduction procedure. The new algorithm has two nice features: it
is efficient and it delivers, for a suitable representation of the input, a
minimal-order telescoper. Its analysis reveals tight bounds on the sizes of the
telescoper it produces.Comment: To be published in the proceedings of ISSAC'1
Explicit formula for the generating series of diagonal 3D rook paths
Let denote the number of ways in which a chess rook can move from a
corner cell to the opposite corner cell of an
three-dimensional chessboard, assuming that the piece moves closer to the goal
cell at each step. We describe the computer-driven \emph{discovery and proof}
of the fact that the generating series admits
the following explicit expression in terms of a Gaussian hypergeometric
function: G(x) = 1 + 6 \cdot \int_0^x \frac{\,\pFq21{1/3}{2/3}{2} {\frac{27
w(2-3w)}{(1-4w)^3}}}{(1-4w)(1-64w)} \, dw.Comment: To appear in "S\'eminaire Lotharingien de Combinatoire
The C-polynomial of a knot
In an earlier paper the first author defined a non-commutative A-polynomial
for knots in 3-space, using the colored Jones function. The idea is that the
colored Jones function of a knot satisfies a non-trivial linear q-difference
equation. Said differently, the colored Jones function of a knot is annihilated
by a non-zero ideal of the Weyl algebra which is generalted (after
localization) by the non-commutative A-polynomial of a knot.
In that paper, it was conjectured that this polynomial (which has to do with
representations of the quantum group U_q(SL_2)) specializes at q=1 to the
better known A-polynomial of a knot, which has to do with genuine SL_2(C)
representations of the knot complement.
Computing the non-commutative A-polynomial of a knot is a difficult task
which so far has been achieved for the two simplest knots. In the present
paper, we introduce the C-polynomial of a knot, along with its non-commutative
version, and give an explicit computation for all twist knots. In a forthcoming
paper, we will use this information to compute the non-commutative A-polynomial
of twist knots. Finally, we formulate a number of conjectures relating the A,
the C-polynomial and the Alexander polynomial, all confirmed for the class of
twist knots.Comment: This is the version published by Algebraic & Geometric Topology on 11
October 200
Bounds for D-finite closure properties
We provide bounds on the size of operators obtained by algorithms for
executing D-finite closure properties. For operators of small order, we give
bounds on the degree and on the height (bit-size). For higher order operators,
we give degree bounds that are parameterized with respect to the order and
reflect the phenomenon that higher order operators may have lower degrees
(order-degree curves)
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