7 research outputs found
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
Hypergeometric-type sequences
We introduce hypergeometric-type sequences. They are linear combinations of interlaced hypergeometric sequences (of arbitrary interlacements). We prove that they form a subring of the ring of holonomic sequences. An interesting family of sequences in this class are those defined by trigonometric functions with linear arguments in the index and π, such as Chebyshev polynomials, sin2 (n π/4) · cos (n π/6))n , and compositions like (sin (cos(nπ/3)π))n . We describe an algorithm that computes a hypergeometric-type normal form of a given holonomic nth term whenever it exists. Our implementation enables us to generate several identities for terms defined via trigonometric function
Trading Order for Degree in Creative Telescoping
We analyze the differential equations produced by the method of creative
telescoping applied to a hyperexponential term in two variables. We show that
equations of low order have high degree, and that higher order equations have
lower degree. More precisely, we derive degree bounding formulas which allow to
estimate the degree of the output equations from creative telescoping as a
function of the order. As an application, we show how the knowledge of these
formulas can be used to improve, at least in principle, the performance of
creative telescoping implementations, and we deduce bounds on the asymptotic
complexity of creative telescoping for hyperexponential terms
A Non-Holonomic Systems Approach to Special Function Identities
We extend Zeilberger’s approach to special function identities to cases that are not holonomic. The method of creative telescoping is thus applied to definite sums or integrals involving Stirling or Bernoulli numbers, incomplete Gamma function or polylogarithms, which are not covered by the holonomic framework. The basic idea is to take into account the dimension of appropriate ideals in Ore algebras. This unifies several earlier extensions and provides algorithms for summation and integration in classes that had not been accessible to computer algebra before