DI Dr. Axel Riese

Research Area

Algorithmic q-Theory

Software

Bibasic Telescope

A Mathematica Implementation of a Generalization of Gosper's Algorithm to Bibasic Hypergeometric Summation

This package is part of the RISCErgoSum bundle. pqTelescope is a Mathematica implementation of a generalization of Gosper’s algorithm to indefinite bibasic hypergeometric summation. The package has been developed by Axel Riese, a former member of the RISC Combinatorics group. ...

Authors: Axel Riese

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MultiSum

A Mathematica Package for Proving Hypergeometric Multi-Sum Identities

This package is part of the RISCErgoSum bundle. MultiSum is a Mathematica package for proving hypergeometric multi-sum identities. It uses an efficient generalization of Sister Celine’s technique to find a homogeneous polynomial recurrence relation for the sum. The package has ...

Authors: Axel Riese, Kurt Wegschaider

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Omega

A Mathematica Implementation of Partition Analysis

Omega is a Mathematica implementation of MacMahon’s Partition Analysis carried out by Axel Riese, a Postdoc of the RISC Combinatorics group. It has been developed together with George E. Andrews and Peter Paule within the frame of a project initiated ...

Authors: Axel Riese

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qMultiSum

A Mathematica Package for Proving q-Hypergeometric Multi-Sum Identities

This package is part of the RISCErgoSum bundle. qMultiSum is a Mathematica package for proving q-hypergeometric multiple summation identities. The package has been developed by Axel Riese, a former member of the RISC Combinatorics group. ...

Authors: Axel Riese

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qZeil

A Mathematica Implementation of q-Analogues of Gosper's and Zeilberger's Algorithm

This package is part of the RISCErgoSum bundle. qZeil is a Mathematica implementation of q-analogues of Gosper’s and Zeilberger’s algorithm for proving and finding indefinite and definite q-hypergeometric summation identities. The package has been developed by Axel Riese, a former ...

Authors: Axel Riese

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RatDiff

A Mathematica Implementation of Mark van Hoeij's Algorithm for Finding Rational Solutions of Linear Difference Equations

RatDiff is a Mathematica implementation of Mark van Hoeij's algorithm for finding rational solutions of linear difference equations. The package has been developed by Axel Riese, a Postdoc of the RISC Combinatorics group during a stay at the University of ...

Authors: Axel Riese

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Publications

2006

[Riese]

Hypergeometric Summation Algorithms for High Order Finite Elements

A. Becirovic, P. Paule, V. Pillwein, A. Riese, C. Schneider, J. Schoeberl

Computing 78(3), pp. 235-249. 2006. ISSN 0010-485X. Preliminary version available. [doi] [pdf] [ps]

[bib]

@article{RISC2949,
author = {A. Becirovic and P. Paule and V. Pillwein and A. Riese and C. Schneider and J. Schoeberl},
title = {{Hypergeometric Summation Algorithms for High Order Finite Elements}},
language = {english},
journal = {Computing},
volume = {78},
number = {3},
pages = {235--249},
isbn_issn = {ISSN 0010-485X},
year = {2006},
note = {Preliminary version available},
refereed = {yes},
length = {15},
url = {https://doi.org/10.1007/s00607-006-0179-x}
}

2004

[Riese]

Computer Proofs of Matrix Product Identities

P.J. Larcombe, A. Riese, B. Zimmermann

J. Algebra Appl. 3, pp. 105-109. 2004. ISSN 0219-4988. [pdf]

[bib]

@article{RISC428,
author = {P.J. Larcombe and A. Riese and B. Zimmermann},
title = {{Computer Proofs of Matrix Product Identities}},
language = {english},
journal = {J. Algebra Appl.},
volume = {3},
pages = {105--109},
isbn_issn = {ISSN 0219-4988},
year = {2004},
refereed = {yes},
length = {5}
}

[Riese]

MacMahon's Partition Analysis XI: Hexagonal Plane Partitions

G. E. Andrews, P. Paule, A. Riese

J. Kepler University Linz. Technical report no. 2004-4, March 2004. SFB-Report. [pdf]

[bib]

@techreport{RISC430,
author = {G. E. Andrews and P. Paule and A. Riese},
title = {{MacMahon's Partition Analysis XI: Hexagonal Plane Partitions}},
language = {english},
number = {2004-4},
year = {2004},
month = {March},
institution = {J. Kepler University Linz},
length = {10},
type = {SFB-Report}
}

[Riese]

MacMahon's Partition Analysis X: Plane Partitions with Diagonals.

G. E. Andrews, P. Paule, A. Riese

Southeast Asian Journal Mathematics and Mathematical Sciences 3, pp. 3-14. 2004. ISSN 0972-7752. [pdf]

[bib]

@article{RISC2473,
author = {G. E. Andrews and P. Paule and A. Riese},
title = {{MacMahon's Partition Analysis X: Plane Partitions with Diagonals.}},
language = {english},
journal = {Southeast Asian Journal Mathematics and Mathematical Sciences},
volume = {3},
pages = {3--14},
isbn_issn = {ISSN 0972-7752},
year = {2004},
refereed = {yes},
length = {12}
}

2003

[Riese]

qMultiSum -- A Package for Proving q-Hypergeometric Multiple Summation Identities

A. Riese

J. Symbolic Comput. 35, pp. 349-376. 2003. ISSN 0747-7171. [pdf]

[bib]

@article{RISC427,
author = {A. Riese},
title = {{qMultiSum --- A Package for Proving q-Hypergeometric Multiple Summation Identities}},
language = {english},
journal = {J. Symbolic Comput.},
volume = {35},
pages = {349--376},
isbn_issn = {ISSN 0747-7171},
year = {2003},
refereed = {yes},
length = {28}
}

2002

[Riese]

A Computer Proof of a Polynomial Identity Implying a Partition Theorem of Göllnitz

A. Berkovich, A. Riese

Adv. in Appl. Math. 28, pp. 1-16. 2002. ISSN 0196-8858. [pdf]

[bib]

@article{RISC216,
author = {A. Berkovich and A. Riese},
title = {{A Computer Proof of a Polynomial Identity Implying a Partition Theorem of Göllnitz}},
language = {english},
journal = {Adv. in Appl. Math.},
volume = {28},
pages = {1--16},
isbn_issn = {ISSN 0196-8858},
year = {2002},
refereed = {yes},
length = {16}
}

[Riese]

A Computer Proof of a Series Evaluation in Terms of Harmonic Numbers

R. Lyons, P. Paule, A. Riese

Appl. Algebra Engrg. Comm. Comput. 13, pp. 327-333. 2002. ISSN 0938-1279. [pdf]

[bib]

@article{RISC220,
author = {R. Lyons and P. Paule and A. Riese},
title = {{A Computer Proof of a Series Evaluation in Terms of Harmonic Numbers}},
language = {english},
journal = {Appl. Algebra Engrg. Comm. Comput.},
volume = {13},
pages = {327--333},
isbn_issn = {ISSN 0938-1279},
year = {2002},
refereed = {yes},
length = {7}
}

2001

[Riese]

MacMahon's Partition Analysis III: The Omega Package

G.E. Andrews, P. Paule, A. Riese

European J. Combin. 22, pp. 887-904. 2001. ISSN 0195-6698. [pdf]

[bib]

@article{RISC154,
author = {G.E. Andrews and P. Paule and A. Riese},
title = {{MacMahon's Partition Analysis III: The Omega Package}},
language = {english},
journal = {European J. Combin.},
volume = {22},
pages = {887--904},
isbn_issn = {ISSN 0195-6698},
year = {2001},
refereed = {yes},
length = {18}
}

[Riese]

MacMahon's Partition Analysis VI: A New Reduction Algorithm

G.E. Andrews, P. Paule, A. Riese

Ann. Comb. 5, pp. 251-270. 2001. ISSN 0218-0006. [pdf]

[bib]

@article{RISC156,
author = {G.E. Andrews and P. Paule and A. Riese},
title = {{MacMahon's Partition Analysis VI: A New Reduction Algorithm}},
language = {english},
journal = {Ann. Comb.},
volume = {5},
pages = {251--270},
isbn_issn = {ISSN 0218-0006},
year = {2001},
refereed = {yes},
length = {20}
}

[Riese]

MacMahon's Partition Analysis VII: Constrained Compositions

G.E. Andrews, P. Paule, A. Riese

In: q-Series with Applications to Combinatorics, Number Theory, and Physics, B.C. Berndt and K. Ono (ed.), Contemp. Math. 291, pp. 11-27. 2001. Amer. Math. Soc., ISBN 0-8218-2746-4. [pdf]

[bib]

@inproceedings{RISC159,
author = {G.E. Andrews and P. Paule and A. Riese},
title = {{MacMahon's Partition Analysis VII: Constrained Compositions}},
booktitle = {{q-Series with Applications to Combinatorics, Number Theory, and Physics}},
language = {english},
series = {Contemp. Math.},
volume = {291},
pages = {11--27},
publisher = {Amer. Math. Soc.},
isbn_issn = {ISBN 0-8218-2746-4},
year = {2001},
editor = {B.C. Berndt and K. Ono},
refereed = {yes},
length = {17}
}

[Riese]

MacMahon's Partition Analysis VIII: Plane Partition Diamonds

G.E. Andrews, P. Paule, A. Riese

Adv. in Appl. Math. 27, pp. 231-242. 2001. ISSN 0196-8858. [pdf]

[bib]

@article{RISC161,
author = {G.E. Andrews and P. Paule and A. Riese},
title = {{MacMahon's Partition Analysis VIII: Plane Partition Diamonds}},
language = {english},
journal = {Adv. in Appl. Math.},
volume = {27},
pages = {231--242},
isbn_issn = {ISSN 0196-8858},
year = {2001},
refereed = {yes},
length = {12}
}

[Riese]

MacMahon's Partition Analysis IX: k-Gon Partitions

G.E. Andrews, P. Paule, A. Riese

Bull. Austral. Math. Soc. 64, pp. 321-329. 2001. ISSN 0004-9727. [pdf]

[bib]

@article{RISC163,
author = {G.E. Andrews and P. Paule and A. Riese},
title = {{MacMahon's Partition Analysis IX: k-Gon Partitions}},
language = {english},
journal = {Bull. Austral. Math. Soc.},
volume = {64},
pages = {321--329},
isbn_issn = {ISSN 0004-9727},
year = {2001},
refereed = {yes},
length = {19}
}

[Riese]

MacMahon's Partition Analysis V: Bijections, Recursions, and Magic Squares

G.E. Andrews, P. Paule, A. Riese, V. Strehl

In: Algebraic Combinatorics and Applications, A. Betten and others (ed.), pp. 1-39. 2001. Springer, ISBN 3-5404-1110-0. [pdf]

[bib]

@inproceedings{RISC164,
author = {G.E. Andrews and P. Paule and A. Riese and V. Strehl},
title = {{MacMahon's Partition Analysis V: Bijections, Recursions, and Magic Squares}},
booktitle = {{Algebraic Combinatorics and Applications}},
language = {english},
pages = {1--39},
publisher = {Springer},
isbn_issn = {ISBN 3-5404-1110-0},
year = {2001},
editor = {A. Betten and others},
refereed = {yes},
length = {39}
}

[Riese]

Fine-Tuning Zeilberger's Algorithm - The Methods of Automatic Filtering and Creative Substituting

A. Riese

In: Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics, F.G. Garvan and M.E.H. Ismail (ed.), Developments in Mathematics 4, pp. 243-254. 2001. Kluwer, ISBN 1-4020-0101-0. [pdf]

[bib]

@inproceedings{RISC167,
author = {A. Riese},
title = {{Fine-Tuning Zeilberger's Algorithm - The Methods of Automatic Filtering and Creative Substituting}},
booktitle = {{Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics}},
language = {english},
series = {Developments in Mathematics},
volume = {4},
pages = {243--254},
publisher = {Kluwer},
isbn_issn = {ISBN 1-4020-0101-0},
year = {2001},
editor = {F.G. Garvan and M.E.H. Ismail},
refereed = {yes},
length = {12}
}

1997

[Riese]

A Mathematica q-Analogue of Zeilberger's Algorithm Based on an Algebraically Motivated Approach to q-Hypergeometric Telescoping

P. Paule, A. Riese

In: Special Functions, q-Series and Related Topics, M.E.H. Ismail and M. Rahman (ed.), Fields Inst. Commun. 14, pp. 179-210. 1997. Amer. Math. Soc., ISBN 0-8218-0524-X. [pdf]

[bib]

@incollection{RISC117,
author = {P. Paule and A. Riese},
title = {{A Mathematica q-Analogue of Zeilberger's Algorithm Based on an Algebraically Motivated Approach to q-Hypergeometric Telescoping}},
booktitle = {{Special Functions, q-Series and Related Topics}},
language = {english},
series = {Fields Inst. Commun.},
volume = {14},
pages = {179--210},
publisher = {Amer. Math. Soc.},
isbn_issn = {ISBN 0-8218-0524-X},
year = {1997},
editor = {M.E.H. Ismail and M. Rahman},
refereed = {yes},
length = {32}
}

[Riese]

Contributions to Symbolic q-Hypergeometric Summation

A. Riese

RISC, J. Kepler University Linz. PhD Thesis. 1997. [pdf]

[bib]

@phdthesis{RISC118,
author = {A. Riese},
title = {{Contributions to Symbolic q-Hypergeometric Summation}},
language = {english},
year = {1997},
translation = {0},
school = {RISC, J. Kepler University Linz},
length = {95}
}

1996

[Riese]

A Generalization of Gosper's Algorithm to Bibasic Hypergeometric Summation

A. Riese

Electron. J. Combin. 3(#R19), pp. 1-16. 1996. ISSN 1077-8926. [pdf]

[bib]

@article{RISC210,
author = {A. Riese},
title = {{A Generalization of Gosper's Algorithm to Bibasic Hypergeometric Summation}},
language = {english},
journal = {Electron. J. Combin.},
volume = {3},
number = {#R19},
pages = {1--16},
isbn_issn = {ISSN 1077-8926},
year = {1996},
refereed = {yes},
length = {16}
}

1995

[Riese]

A Mathematica q-Analogue of Zeilberger's Algorithm for Proving q-Hypergeometric Identities

A. Riese

RISC, J. Kepler University Linz. Diploma Thesis. 1995. [pdf]

[bib]

@mastersthesis{RISC114,
author = {A. Riese},
title = {{A Mathematica q-Analogue of Zeilberger's Algorithm for Proving q-Hypergeometric Identities}},
language = {english},
year = {1995},
translation = {0},
school = {RISC, J. Kepler University Linz},
length = {76}
}