Manuel Kauers
Order-Degree Curves for Hypergeometric Creative Telescoping.
Wed, 01/11/2012 - 15:01 — SomNambulAuthors: Manuel Kauers, Shaoshi Chen
Subjects: Symbolic Computation
AbstractCreative telescoping applied to a bivariate proper hypergeometric term
produces linear recurrence operators with polynomial coefficients, called
telescopers. We provide bounds for the degrees of the polynomials appearing in
these operators. Our bounds are expressed as curves in the (r,d)-plane which
assign to every order r a bound on the degree d of the telescopers. These
curves are hyperbolas, which reflect the phenomenon that higher order
telescopers tend to have lower degree, and vice versa.Telescopers for Rational and Algebraic Functions via Residues.
Wed, 01/11/2012 - 15:01 — SomNambulAuthors: Manuel Kauers, Shaoshi Chen, Michael Singer
Subjects: Symbolic Computation
AbstractWe show that the problem of constructing telescopers for functions of m
variables is equivalent to the problem of constructing telescopers for
algebraic functions of m -1 variables and present a new algorithm to construct
telescopers for algebraic functions of two variables. These considerations are
based on analyzing the residues of the input. According to experiments, the
resulting algorithm for rational functions of three variables is faster than
known algorithms, at least in some examples of combinatorial interest.A Proof of George Andrews' and David Robbins' $q$-TSPP Conjecture.
Wed, 02/24/2010 - 16:01 — SomNambulAuthors: Doron Zeilberger, Manuel Kauers, Christoph Koutschan
Subjects: Combinatorics
AbstractThe conjecture that the orbit-counting generating function for totally
symmetric plane partitions can be written as an explicit product-formula, has
been stated independently by George Andrews and David Robbins around 1983. We
present a proof of this long-standing conjecture.The complete Generating Function for Gessel Walks is Algebraic.
Fri, 09/11/2009 - 19:12 — SomNambulAuthors: Alin Bostan, Manuel Kauers
Subjects: Combinatorics
AbstractGessel walks are lattice walks in the quarter plane $\set N^2$ which start at
the origin $(0,0)\in\set N^2$ and consist only of steps chosen from the set
$\{\leftarrow,\swarrow,\nearrow,\to\}$. We prove that if $g(n;i,j)$ denotes the
number of Gessel walks of length $n$ which end at the point $(i,j)\in\set N^2$,
then the trivariate generating series $G(t;x,y)=\sum_{n,i,j\geq 0} g(n;i,j)x^i
y^j t^n$ is an algebraic function.The complete Generating Function for Gessel Walks is Algebraic.
Fri, 09/11/2009 - 19:12 — SomNambulAuthors: Alin Bostan, Manuel Kauers
Subjects: Combinatorics
AbstractGessel walks are lattice walks in the quarter plane $\set N^2$ which start at
the origin $(0,0)\in\set N^2$ and consist only of steps chosen from the set
$\{\leftarrow,\swarrow,\nearrow,\to\}$. We prove that if $g(n;i,j)$ denotes the
number of Gessel walks of length $n$ which end at the point $(i,j)\in\set N^2$,
then the trivariate generating series $G(t;x,y)=\sum_{n,i,j\geq 0} g(n;i,j)x^i
y^j t^n$ is an algebraic function.
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