PROOF OF IRA GESSEL’S LATTICE PATH CONJECTURE (2009)
Manuel Kauers, Doron Zeilberger
Abstract. We present a computer-aided, yet fully rigorous, proof of Ira Gessel’s tantalizingly simply-stated conjecture that the number of ways of walking 2n steps in the region x + y ≥ 0, y ≥...
Manuel Kauers, Christoph Koutschan, Doron Zeilberger
In the historic conference Combinatoire Énumérative [6] wonderfully organized by Gilbert Labelle and Pierre Leroux there were many stimulating lectures, including a very interesting one by Pierre...
The complete Generating Function for Gessel Walks is Algebraic (2009)
Gessel 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...
A Non-Holonomic Systems Approach to Special Function Identities (2009)
Chyzak, Frédéric, Kauers, Manuel, Salvy, Bruno
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...
A Non-Holonomic Systems Approach to Special Function Identities (2009)
Chyzak, Frédéric, Kauers, Manuel, Salvy, Bruno
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...
A Non-Holonomic Systems Approach to Special Function Identities (2009)
Chyzak, Frédéric, Kauers, Manuel, Salvy, Bruno
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...
Shift Equivalence of P-finite Sequences (2008)
We present an algorithm which decides the shift equivalence problem for P-finite sequences. A sequence is called P-finite if it satisfies a homogeneous linear recurrence equation with polynomial...
EXPERIMENTS WITH A POSITIVITY PRESERVING OPERATOR (2008)
Manuel Kauers, Doron Zeilberger
Abstract. We consider some multivariate rational functions which have (or are conjectured to have) only positive coefficients in their series expansion. We consider an operator that preserves...
Automatic Classification of Restricted Lattice Walks (2008)
We propose an experimental mathematics approach leading to the computer-driven discovery of various structural properties of general counting functions coming from enumeration of walks.
ABSTRACT Application of Unspecified Sequences in Symbolic Summation (2008)
We consider symbolic sums which contain subexpressions representing unspecified sequences. Existing symbolic summation technology is extended to sums of this kind. We show how this can be applied in...
Kauers, Manuel, Koutschan, Christoph, Zeilberger, Doron
In the historic conference Combinatoire Enumerative[LL] wonderfully organized by Gilbert Labelle and Pierre Leroux there were many stimulating lectures, including a very interesting one by Pierre...
Proof of Ira Gessel's Lattice Path Conjecture (2008)
Kauers, Manuel, Koutschan, Christoph, Zeilberger, Doron
We present a computer-aided, yet fully rigorous, proof of Ira Gessel's tantalizingly simply-stated conjecture that the number of ways of walking $2n$ steps in the region $x+y \geq 0, y \geq 0$ of the...
The Quasi-Holonomic Ansatz and Restricted Lattice Walks (2008)
Kauers, Manuel, Zeilberger, Doron
The great enumerator Germain Kreweras empirically discovered this intriguing fact, and then needed lots of pages[K], and lots of human ingenuity, to prove it. Other great enumerators, for example,...
Automated Proofs for Some Stirling Number Identities (2008)
Manuel Kauers, Carsten Schneider
We present computer-generated proofs for some summation identities for (q-)Stirling and (q-)Eulerian numbers that were obtained by combining a recent summation algorithm for Stirling number...
Shift Equivalence of P-finite Sequences (2008)
We present an algorithm which decides the shift equivalence problem for Pfinite sequences. A sequence is called P-finite if it satisfies a homogeneous linear recurrence equation with polynomial...
recurrent sequences, zero equivalence (2008)
We introduce the class of nested polynomially recurrent sequences which includes a large number of sequences that are of combinatorial interest. We present an algorithm for deciding zero equivalence...
Comments: The great enumerator Germain Kreweras empirically discovered this intriguing fact, (2008)
Manuel Kauers, Doron Zeilberger, For Example, Found Other
and then needed lots of pages[K], and lots of human ingenuity, to prove it. Other great enumerators,
A computer proof of Turán’s inequality (2008)
Abstract. We show how Turán’s inequality Pn(x) 2 −Pn−1(x)Pn+1(x) ≥ 0 for Legendre polynomials and related inequalities can be proven by means of a computer procedure. The use of this...
AUTOMATED PROOFS FOR SOME STIRLING NUMBER IDENTITIES (2008)
Manuel Kauers, Carsten Schneider
Abstract. We present computer-generated proofs of some summation identities for (q-)Stirling and (q-)Eulerian numbers that were obtained by combining a recent summation algorithm for Stirling number...
Computing Limits of Sequences (2008)
Abstract. This poster presents a simple approach for computing lim-its of certain ΠΣ-expressions using a discrete analogue of l’Hospital’s rule. While details concerning ter-mination and the...
A MATHEMATICA PACKAGE FOR q-HOLONOMIC SEQUENCES AND POWER SERIES (2008)
Manuel Kauers, Christoph Koutschan
Abstract. We describe a Mathematica package for dealing with q-holonomic sequences and power series. The package is intended as a q-analogue of the Maple package gfun and the Mathematica package...
An Interface between Mathematica and Singular (2008)
The purpose of this package is to provide an easy way to access some of the functionality provided by the special purpose computer algebra system Singular from within a Mathematica session or...
Comments: The great enumerator Germain Kreweras empirically discovered this intriguing fact, (2008)
Manuel Kauers, Doron Zeilberger, For Example, Found Other
and then needed lots of pages[K], and lots of human ingenuity, to prove it. Other great enumerators,
General Terms Algorithms (2008)
An extension of Karr’s summation algorithm is presented by which symbolic sums involving radical expressions can be simplified. We discuss the construction of appropriate difference fields as well...
Abstract Indefinite Summation with Unspecified Summands (2008)
Manuel Kauers, Carsten Schneider
We provide a new algorithm for indefinite nested summation which is applicable to summands involving unspecified sequences x(n). More than that, we show how to extend Karr’s algorithm to a general...
Computer Algebra for Special Function Inequalities (2008)
Abstract. Recent computer proofs for some special function inequalities are presented. The algorithmic ideas underlying these computer proofs are described, and the conceptual difference to existing...
A computer proof of Turán’s inequality (2008)
Abstract. We show how Turán’s inequality Pn(x) 2 −Pn−1(x)Pn+1(x) ≥ 0 for Legendre polynomials and related inequalities can be proven by means of a computer procedure. The use of this...
SumCracker: A package for manipulating symbolic sums and related objects (2008)
We describe a new software package, named SumCracker, for proving and finding identities involving symbolic sums and related objects. SumCracker is applicable to a wide range of expressions for many...
Computer Algebra for Special Function Inequalities (2008)
Abstract. Recent computer proofs for some special function inequalities are presented. The algorithmic ideas underlying these computer proofs are described, and the conceptual difference to existing...
ABSTRACT Application of Unspecified Sequences in Symbolic Summation (2008)
We consider symbolic sums which contain subexpressions representing unspecified sequences. Existing symbolic summation technology is extended to sums of this kind. We show how this can be applied in...
A MATHEMATICA PACKAGE FOR q-HOLONOMIC SEQUENCES AND POWER SERIES (2008)
Manuel Kauers, Christoph Koutschan
Abstract. We describe a Mathematica package for dealing with q-holonomic sequences and power series. The package is intended as a q-analogue of the Maple package gfun and the Mathematica package...
Experiments with a Positivity Preserving Operator (2007)
Kauers, Manuel, Zeilberger, Doron
We consider some multivariate rational functions which have (or are conjectured to have) only positive coefficients in their series expansion. We consider an operator that preserves positivity of...
Computer algebra and power series with positive coefficients (2007)
Abstract. We consider the question whether all the coefficients in the series expansions of some specific rational functions are positive, and we demonstrate how computer algebra can help answering...
Summation Algorithms for Stirling Number Identities (2007)
We consider a class of sequences defined by triangular recurrence equations. This class contains Stirling numbers and Eulerian numbers of both kinds, and hypergeometric multiples of those. We give a...
Automated Proofs for Some Stirling Number Identities (2007)
Manuel Kauers, Carsten Schneider
We present computer-generated proofs for some summation identities for (q-)Stirling and (q-)Eulerian numbers that were obtained by combining a recent summation algorithm for Stirling number...
Computer algebra and power series with positive coefficients (2007)
Abstract. We consider the question whether all the coefficients in the series expansions of some specific rational functions are positive, and we demonstrate how computer algebra can help answering...
Summation Algorithms for Stirling Number Identities (2007)
We consider a class of sequences defined by triangular recurrence equations. This class contains Stirling numbers and Eulerian numbers of both kinds, and hypergeometric multiples of those. We give a...
Computer algebra and power series with positive coefficients (2007)
Abstract. We consider the question whether all the coefficients in the series expansions of some specific rational functions are positive, and we demonstrate how computer algebra can help answering...
Computer algebra and power series with positive coefficients (2007)
Abstract. We consider the question whether all the coefficients in the series expansions of some specific rational functions are positive, and we demonstrate how computer algebra can help answering...
Automated Proofs for Some Stirling Number Identities (2007)
Manuel Kauers, Carsten Schneider
We present computer-generated proofs for some summation identities for (q-)Stirling and (q-)Eulerian numbers that were obtained by combining a recent summation algorithm for Stirling number...
RISC-Linz Report Series No. 07-06 (2007)
Manuel Kauers, Manfred Kerber, Robert Miner, Wolfgang Windsteiger (eds, B. Buchberger, R. Hemmecke, ...
series of conferences dedicated to the integration of computer algebra systems (CAS) and automated deduction systems (ADS). MKM 2007 was the Sixth International Conference on Mathematical Knowledge...
A computer proof of Moll’s log-concavity conjecture (2006)
Manuel Kauers, Peter Paule, Communicated Jim Haglund
Abstract. In a study on quartic integrals, Moll met a specialized family of Jacobi polynomials. He conjectured that the corresponding coefficient sequences are log-concave. In this paper we settle...
A computer proof of Moll’s log-concavity conjecture (2006)
In a study on quartic integrals, Moll met a specialized family of Jacobi polynomials. He conjectured that the corresponding coefficient sequences are log-concave. In this paper we settle Moll’s...
Computing the algebraic relations of C-finite sequences and multisequences (2006)
Manuel Kauers, Burkhard Zimmermann
We present an algorithm for computing generators for the ideal of algebraic relations among sequences which are given by homogeneous linear recurrence equations with constant coefficients. Knowing...
SumCracker: A package for manipulating symbolic sums and related objects (2006)
We describe a new software package, named SumCracker, for proving and finding identities involving symbolic sums and related objects. SumCracker is applicable to a wide range of expressions for many...
Shift Equivalence of P-finite Sequences (2006)
We present an algorithm which decides the shift equivalence problem for Pfinite sequences. A sequence is called P-finite if it satisfies a homogeneous linear recurrence equation with polynomial...
Computing the algebraic relations of C-finite sequences and multisequences (2006)
Manuel Kauers, Burkhard Zimmermann
We present an algorithm for computing generators for the ideal of algebraic relations among sequences which are given by homogeneous linear recurrence equations with constant coefficients. Knowing...
Indefinite summation with unspecified summands (2006)
Manuel Kauers, Carsten Schneider
We provide a new algorithm for indefinite nested summation which is applicable to summands involving unspecified sequences x(n). More than that, we show how to extend Karr’s algorithm to a general...
Solving difference equations whose coefficients are not transcendental (2005)
We consider a large class of sequences which are defined by systems of (possibly nonlinear) difference equations. A procedure for recursively enumerating the algebraic dependencies of such sequences...
Solving difference equations whose coefficients are not transcendental (2005)
We consider a large class of sequences which are defined by systems of (possibly nonlinear) difference equations. A procedure for recursively enumerating the algebraic dependencies of such sequences...
A procedure for proving special function inequalities involving a discrete parameter (2005)
We define a class of special function inequalities that contains many classical examples, such as the Cauchy-Schwarz inequality, and introduce a proving procedure based on induction and Cylindrical...
Computer proofs for polynomial identities in arbitrary many variables (2004)
Categories and Subject Descriptors I.1.2 [Computing Methodologies]: Symbolic and Algebraic Manipulation--Algorithms
An Algorithm for Deciding Zero Equivalence of Nested Polynomially Recurrent Sequences (2003)
We introduce the class of nested polynomially recurrent sequences which includes a large number of sequences that are of combinatorial interest. We present an algorithm for deciding zero equivalence...
Computing Limits of Sequences (2003)
Manuel Kauers, Johannes Kepler
So far, it seems that more emphasis was laid on the treatment of continuous functions, and although some mathematical fundaments are already available [1, 2], algorithmic approaches for the discrete...
An Algorithm for Deciding Zero Equivalence of Nested Polynomially Recurrent Sequences (2003)
We introduce the class of nested polynomially recurrent sequences which includes a large number of sequences that are of combinatorial interest. We present an algorithm for deciding zero equivalence...
Solving parametric linear systems: an experiment with constraint algebraic programming (2002)
Clemens Ballarin, Manuel Kauers
Algorithms in computer algebra are usually designed for a fixed set of domains. For example, algorithms over the domain of polynomials are not applicable to parameters because the inherent assumption...
Solving Parametric Linear Systems: an Experiment with Constraint Algebraic Programming (2002)
Clemens Ballarin, Manuel Kauers
Algorithms in computer algebra are usually designed for a fixed set of domains. For example, algorithms over the domain of polynomials are not applicable to parameters because the inherent assumption...