Symbolic Summation/Integration and Algebraic Relations [SSIAR]

@article{RISC3945,
author = {Alin Bostan and Manuel Kauers},
title = {{The Complete Generating Function for Gessel Walks is Algebraic}},
language = {english},
abstract = { 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 $\{\leftarrow,\penalty0\swarrow,\penalty0\nearrow,\penalty0\rightarrow\}$. 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 $\displaystyle\smash{ G(t;x,y)=\sum_{n,i,j\geq 0} g(n;i,j)x^i y^j t^n } $ is an algebraic function.},
journal = {Proceedings of the AMS},
volume = {138},
number = {9},
pages = {3063--3078},
isbn_issn = {ISSN 0002-9939},
year = {2010},
month = {September},
refereed = {yes},
length = {16}
}

@article{RISC3437,
author = {Manuel Kauers and Christoph Koutschan},
title = {{A Mathematica Package for q-Holonomic Sequences and Power Series}},
language = {english},
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 GeneratingFunctions. It provides commands for addition, multiplication, and substitution of these objects, for converting between various representations ($q$-differential equations, $q$-recurrence equations, $q$-shift equations), for computing sequence terms and power series coefficients, and for guessing recurrence equations given initial terms of a sequence.},
journal = {The Ramanujan Journal},
volume = {19},
number = {2},
pages = {137--150},
publisher = {Springer},
isbn_issn = {ISSN 1382-4090},
year = {2009},
refereed = {yes},
length = {14}
}

@article{RISC3801,
author = {Manuel Kauers and Christoph Koutschan and Doron Zeilberger},
title = {{A Proof of George Andrews' and Dave Robbins' q-TSPP-Conjecture (modulo a finite amount of routine calculations)}},
language = {english},
journal = {The personal Journal of Ekhad and Zeilberger},
pages = {1--8},
isbn_issn = { },
year = {2009},
month = {January},
refereed = {no},
length = {8},
url = {http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/qtspp.html}
}

@article{RISC3837,
author = {Manuel Kauers and Christoph Koutschan and Doron Zeilberger},
title = {{Proof of Ira Gessel's Lattice Path Conjecture}},
language = {english},
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 $x+y \geq 0, y \geq 0$ of the square-lattice with unit steps in the east, west, north, and south directions, that start and end at the origin, equals $16^n\frac{(5/6)_n(1/2)_n}{(5/3)_n(2)_n}$ .},
journal = {Proceedings of the National Academy of Sciences},
volume = {106},
number = {28},
pages = {11502--11505},
isbn_issn = {ISSN 0027-8424},
year = {2009},
month = {July},
refereed = {yes},
length = {4}
}

@article{RISC3846,
author = {J. Bluemlein and M. Kauers and S. Klein and C. Schneider},
title = {{Determining the closed forms of the {$O(a_s^3)$} anomalous dimensions and {W}ilson coefficients from {M}ellin moments by means of computer algebra}},
language = {english},
journal = {Comput. Phys. Comm. },
volume = {180},
pages = {2143--2165},
isbn_issn = {ISSN 0010-4655},
year = {2009},
note = {arXiv:0902.4091 [hep-ph]},
refereed = {yes},
length = {23},
url = {https://www.doi.org/10.1016/j.cpc.2009.06.020}
}

@article{RISC3368,
author = {Manuel Kauers},
title = {{Solving Difference Equations whose Coefficients are not Transcendental}},
language = {english},
abstract = {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 is presented. Also a procedure for solving linear difference equations with such sequences as coefficients is proposed. The methods are illustrated on some problems arising in the literature on special functions and combinatorial sequences.},
journal = {Theoretical Computer Science},
volume = {401},
number = {1-3},
pages = {217--227},
isbn_issn = {ISSN 0304-3975},
year = {2008},
month = {July},
refereed = {yes},
length = {11}
}

@article{RISC3370,
author = {Manuel Kauers and Doron Zeilberger},
title = {{The Quasi-Holonomic Ansatz and Restricted Lattice Walks}},
language = {english},
journal = {Journal of Difference Equations and Applications},
volume = {14},
number = {10--11},
pages = {1119--1126},
isbn_issn = {ISSN 1023-6198},
year = {2008},
note = {to appear},
refereed = {yes},
length = {10}
}

@article{RISC3394,
author = {Manuel Kauers and Doron Zeilberger},
title = {{Experiments with a Positivity Preserving Operator}},
language = {english},
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 positivity of series coefficients, and apply the inverse of this operator to the rational functions. We obtain new rational functions which seem to have only positive coefficients, whose positivity would imply positivity of the original series, and which, in a certain sense, cannot be improved any further.},
journal = {Experimental Mathematics},
volume = {17},
number = {3},
pages = {341--345},
isbn_issn = {ISSN 1058-6458},
year = {2008},
refereed = {yes},
length = {5}
}

@article{RISC3778,
author = {Manuel Kauers},
title = {{Fast Solvers for Dense Linear Systems}},
language = {english},
abstract = {It appears that large scale calculations in particle physicsoften require to solve systems of linear equations with rational numbercoefficients exactly. If classical Gaussian elimination is applied to a\emph{dense} system, the time needed to solve such a system grows exponentiallyin the size of the system. In this tutorial paper, we present a standardtechnique from computer algebra that avoids this exponential growth:homomorphic images. Using this technique, big dense linear systemscan be solved in a much more reasonable time than using Gaussianelimination over the rationals.},
journal = {Nuclear Physics B (Proc. Suppl.)},
volume = {183},
pages = {245--250},
isbn_issn = {ISSN 0550-3213},
year = {2008},
refereed = {yes},
length = {6}
}

@article{RISC3366,
author = {Manuel Kauers and Carsten Schneider},
title = {{Automated Proofs for Some Stirling Number Identities}},
language = {english},
abstract = {We present computer-generated proofs for some summation identities for($q$-)Stir-ling and ($q$-)Eulerian numbers that were obtained by combining arecent summation algorithm for Stirling number identities with arecurrence solver for difference fields.},
journal = {The Electronic Journal of Combinatorics},
volume = {15},
number = {1},
pages = {1--7},
isbn_issn = {ISSN 1077-8926},
year = {2008},
note = {R2},
refereed = {yes},
length = {7},
url = {https://doi.org/10.37236/726 }
}

@article{RISC3341,
author = {Manuel Kauers and Burkhard Zimmermann},
title = {{Computing the Algebraic Relations of C-finite Sequences and Multisequences}},
language = {english},
abstract = {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 these generators makes it possible to use Groebner bases methods for carrying out certain basic operations in the ring of such sequences effectively. In particular, one can answer the question whether a given sequence can be represented in terms of other given sequences. },
journal = {Journal of Symbolic Computation},
volume = {43},
number = {11},
pages = {787--803},
isbn_issn = {ISSN 0747-7171.},
year = {2008},
refereed = {yes},
length = {25}
}