# Symbolic Integration and Special Functions [P20162-N18]

### Project Lead

### Project Duration

01/01/2008 - 31/12/2011## Partners

### The Austrian Science Fund (FWF)

## Publications

### 2017

[Koutschan]

### Computing the number of realizations of a Laman graph

#### Jose Capco, Georg Grasegger, Matteo Gallet, Christoph Koutschan, Niels Lubbes, Josef Schicho

In: Electronic Notes in Discrete Mathematics (Proceedings of Eurocomb 2017), Vadim Lozin (ed.), Proceedings of The European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB'17)61, pp. 207-213. 2017. ISSN 1571-0653. [url]@

author = {Jose Capco and Georg Grasegger and Matteo Gallet and Christoph Koutschan and Niels Lubbes and Josef Schicho},

title = {{Computing the number of realizations of a Laman graph}},

booktitle = {{Electronic Notes in Discrete Mathematics (Proceedings of Eurocomb 2017)}},

language = {english},

abstract = {Laman graphs model planar frameworks which are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. In a recent paper we provide a recursion formula for this number of realizations using ideas from algebraic and tropical geometry. Here, we present a concise summary of this result focusing on the main ideas and the combinatorial point of view.},

volume = {61},

pages = {207--213},

isbn_issn = {ISSN 1571-0653},

year = {2017},

editor = {Vadim Lozin},

refereed = {yes},

keywords = {Laman graph; minimally rigid graph; tropical geometry; euclidean embedding; graph realization},

length = {7},

conferencename = {The European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB'17)},

url = {http://www.koutschan.de/data/laman/}

}

**inproceedings**{RISC5478,author = {Jose Capco and Georg Grasegger and Matteo Gallet and Christoph Koutschan and Niels Lubbes and Josef Schicho},

title = {{Computing the number of realizations of a Laman graph}},

booktitle = {{Electronic Notes in Discrete Mathematics (Proceedings of Eurocomb 2017)}},

language = {english},

abstract = {Laman graphs model planar frameworks which are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. In a recent paper we provide a recursion formula for this number of realizations using ideas from algebraic and tropical geometry. Here, we present a concise summary of this result focusing on the main ideas and the combinatorial point of view.},

volume = {61},

pages = {207--213},

isbn_issn = {ISSN 1571-0653},

year = {2017},

editor = {Vadim Lozin},

refereed = {yes},

keywords = {Laman graph; minimally rigid graph; tropical geometry; euclidean embedding; graph realization},

length = {7},

conferencename = {The European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB'17)},

url = {http://www.koutschan.de/data/laman/}

}

### 2015

[Koutschan]

### Integral D-Finite Functions

#### Manuel Kauers, Christoph Koutschan

arxiv. Technical report no. 1501.03691, 2015. [pdf]@

author = {Manuel Kauers and Christoph Koutschan},

title = {{Integral D-Finite Functions}},

language = {english},

abstract = { We propose a differential analog of the notion of integral closure ofalgebraic function fields. We present an algorithm for computing theintegralclosure of the algebra defined by a linear differential operator. Ouralgorithmis a direct analog of van Hoeij's algorithm for computing integral bases ofalgebraic function fields.},

number = {1501.03691},

year = {2015},

institution = {arxiv},

length = {8}

}

**techreport**{RISC5101,author = {Manuel Kauers and Christoph Koutschan},

title = {{Integral D-Finite Functions}},

language = {english},

abstract = { We propose a differential analog of the notion of integral closure ofalgebraic function fields. We present an algorithm for computing theintegralclosure of the algebra defined by a linear differential operator. Ouralgorithmis a direct analog of van Hoeij's algorithm for computing integral bases ofalgebraic function fields.},

number = {1501.03691},

year = {2015},

institution = {arxiv},

length = {8}

}

### 2014

[Koutschan]

### Relativistic Coulomb Integrals and Zeilberger's Holonomic Systems Approach II

#### Christoph Koutschan, Peter Paule, Sergei K. Suslov

In: Algebraic and Algorithmic Aspects of Differential and Integral Operators, Moulay Barkatou and Thomas Cluzeau and Georg Regensburger and Markus Rosenkranz (ed.), Lecture Notes in Computer Science 8372, pp. 135-145. 2014. Springer, Berlin Heidelberg, ISBN 978-3-642-54478-1. [pdf]@

author = {Christoph Koutschan and Peter Paule and Sergei K. Suslov},

title = {{Relativistic Coulomb Integrals and Zeilberger's Holonomic Systems Approach II}},

booktitle = {{Algebraic and Algorithmic Aspects of Differential and Integral Operators}},

language = {english},

abstract = {We derive the recurrence relations for relativistic Coulomb integrals directly from the integral representations with the help of computer algebra methods. In order to manage the computational complexity of this problem, we employ holonomic closure properties in a sophisticated way.},

series = {Lecture Notes in Computer Science},

volume = {8372},

pages = {135--145},

publisher = {Springer},

address = {Berlin Heidelberg},

isbn_issn = {ISBN 978-3-642-54478-1},

year = {2014},

editor = {Moulay Barkatou and Thomas Cluzeau and Georg Regensburger and Markus Rosenkranz},

refereed = {yes},

length = {11}

}

**incollection**{RISC4847,author = {Christoph Koutschan and Peter Paule and Sergei K. Suslov},

title = {{Relativistic Coulomb Integrals and Zeilberger's Holonomic Systems Approach II}},

booktitle = {{Algebraic and Algorithmic Aspects of Differential and Integral Operators}},

language = {english},

abstract = {We derive the recurrence relations for relativistic Coulomb integrals directly from the integral representations with the help of computer algebra methods. In order to manage the computational complexity of this problem, we employ holonomic closure properties in a sophisticated way.},

series = {Lecture Notes in Computer Science},

volume = {8372},

pages = {135--145},

publisher = {Springer},

address = {Berlin Heidelberg},

isbn_issn = {ISBN 978-3-642-54478-1},

year = {2014},

editor = {Moulay Barkatou and Thomas Cluzeau and Georg Regensburger and Markus Rosenkranz},

refereed = {yes},

length = {11}

}

[Koutschan]

### A Generalized Apagodu-Zeilberger Algorithm

#### Shaoshi Chen, Manuel Kauers, Christoph Koutschan

In: Proceedings of ISSAC 2014, Katsusuke Nabeshima (ed.), pp. 107-114. 2014. ISBN 978-1-4503-2501-1. [pdf]@

author = {Shaoshi Chen and Manuel Kauers and Christoph Koutschan},

title = {{A Generalized Apagodu-Zeilberger Algorithm}},

booktitle = {{Proceedings of ISSAC 2014}},

language = {english},

pages = {107--114},

isbn_issn = {ISBN 978-1-4503-2501-1},

year = {2014},

editor = {Katsusuke Nabeshima},

refereed = {yes},

length = {8}

}

**inproceedings**{RISC5034,author = {Shaoshi Chen and Manuel Kauers and Christoph Koutschan},

title = {{A Generalized Apagodu-Zeilberger Algorithm}},

booktitle = {{Proceedings of ISSAC 2014}},

language = {english},

pages = {107--114},

isbn_issn = {ISBN 978-1-4503-2501-1},

year = {2014},

editor = {Katsusuke Nabeshima},

refereed = {yes},

length = {8}

}

### 2013

[Koutschan]

### Advanced Computer Algebra for Determinants

#### Christoph Koutschan, Thotsaporn Thanatipanonda

Annals of Combinatorics 17(3), pp. 509-523. 2013. ISSN 0218-0006. [url] [pdf]@

author = {Christoph Koutschan and Thotsaporn Thanatipanonda},

title = {{Advanced Computer Algebra for Determinants}},

language = {english},

abstract = {We prove three conjectures concerning the evaluation of determinants, which are related to the counting of plane partitions and rhombus tilings. One of them was posed by George Andrews in 1980, the other two were by Guoce Xin and Christian Krattenthaler. Our proofs employ computer algebra methods, namely, the holonomic ansatz proposed by Doron Zeilberger and variations thereof. These variations make Zeilberger's original approach even more powerful and allow for addressing a wider variety of determinants. Finally, we present, as a challenge problem, a conjecture about a closed-form evaluation of Andrews's determinant.},

journal = {Annals of Combinatorics},

volume = {17},

number = {3},

pages = {509--523},

isbn_issn = {ISSN 0218-0006},

year = {2013},

refereed = {yes},

length = {15},

url = {http://www.risc.jku.at/people/ckoutsch/det/}

}

**article**{RISC4531,author = {Christoph Koutschan and Thotsaporn Thanatipanonda},

title = {{Advanced Computer Algebra for Determinants}},

language = {english},

abstract = {We prove three conjectures concerning the evaluation of determinants, which are related to the counting of plane partitions and rhombus tilings. One of them was posed by George Andrews in 1980, the other two were by Guoce Xin and Christian Krattenthaler. Our proofs employ computer algebra methods, namely, the holonomic ansatz proposed by Doron Zeilberger and variations thereof. These variations make Zeilberger's original approach even more powerful and allow for addressing a wider variety of determinants. Finally, we present, as a challenge problem, a conjecture about a closed-form evaluation of Andrews's determinant.},

journal = {Annals of Combinatorics},

volume = {17},

number = {3},

pages = {509--523},

isbn_issn = {ISSN 0218-0006},

year = {2013},

refereed = {yes},

length = {15},

url = {http://www.risc.jku.at/people/ckoutsch/det/}

}

[Koutschan]

### Harmonic interpolation based on Radon projections along the sides of regular polygons

#### Irina Georgieva, Clemens Hofreither, Christoph Koutschan, Veronika Pillwein, Thotsaporn Thanatipanonda

Central European Journal of Mathematics 11(4), pp. 609-620. 2013. ISSN 1895-1074. [pdf]@

author = {Irina Georgieva and Clemens Hofreither and Christoph Koutschan and Veronika Pillwein and Thotsaporn Thanatipanonda},

title = {{Harmonic interpolation based on Radon projections along the sides of regular polygons}},

language = {english},

abstract = {Given information about a harmonic function in two variables, consisting of a finitenumber of values of its Radon projections, i.e., integrals along some chords of the unitcircle, we study the problem of interpolating these data by a harmonic polynomial.With the help of symbolic summation techniques we show that this interpolationproblem has a unique solution in the case when the chords form a regular polygon.Numerical experiments for this and more general cases are presented.},

journal = {Central European Journal of Mathematics},

volume = {11},

number = {4},

pages = {609--620},

isbn_issn = {ISSN 1895-1074},

year = {2013},

refereed = {yes},

length = {12}

}

**article**{RISC4655,author = {Irina Georgieva and Clemens Hofreither and Christoph Koutschan and Veronika Pillwein and Thotsaporn Thanatipanonda},

title = {{Harmonic interpolation based on Radon projections along the sides of regular polygons}},

language = {english},

abstract = {Given information about a harmonic function in two variables, consisting of a finitenumber of values of its Radon projections, i.e., integrals along some chords of the unitcircle, we study the problem of interpolating these data by a harmonic polynomial.With the help of symbolic summation techniques we show that this interpolationproblem has a unique solution in the case when the chords form a regular polygon.Numerical experiments for this and more general cases are presented.},

journal = {Central European Journal of Mathematics},

volume = {11},

number = {4},

pages = {609--620},

isbn_issn = {ISSN 1895-1074},

year = {2013},

refereed = {yes},

length = {12}

}

[Zimmermann]

### Computer-Assisted Proofs of Some Identities for Bessel Functions of Fractional Order

#### S. Gerhold, M. Kauers, C. Koutschan, P. Paule, C. Schneider, B. Zimmermann

In: Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, C. Schneider, J. Bluemlein (ed.), Texts and Monographs in Symbolic Computation , pp. 75-96. 2013. Springer, ISBN-13: 978-3709116159. arXiv:1305.4818 [cs.SC]. [url]@

author = {S. Gerhold and M. Kauers and C. Koutschan and P. Paule and C. Schneider and B. Zimmermann},

title = {{Computer-Assisted Proofs of Some Identities for Bessel Functions of Fractional Order}},

booktitle = {{Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions}},

language = {english},

series = {Texts and Monographs in Symbolic Computation},

pages = {75--96},

publisher = {Springer},

isbn_issn = {ISBN-13: 978-3709116159},

year = {2013},

note = {arXiv:1305.4818 [cs.SC]},

editor = {C. Schneider and J. Bluemlein},

refereed = {no},

length = {22},

url = {http://arxiv.org/abs/1305.4818}

}

**incollection**{RISC4721,author = {S. Gerhold and M. Kauers and C. Koutschan and P. Paule and C. Schneider and B. Zimmermann},

title = {{Computer-Assisted Proofs of Some Identities for Bessel Functions of Fractional Order}},

booktitle = {{Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions}},

language = {english},

series = {Texts and Monographs in Symbolic Computation},

pages = {75--96},

publisher = {Springer},

isbn_issn = {ISBN-13: 978-3709116159},

year = {2013},

note = {arXiv:1305.4818 [cs.SC]},

editor = {C. Schneider and J. Bluemlein},

refereed = {no},

length = {22},

url = {http://arxiv.org/abs/1305.4818}

}

### 2012

[Koutschan]

### Computer Algebra meets Finite Elements: an Efficient Implementation for Maxwell's Equations

#### Christoph Koutschan, Christoph Lehrenfeld, Joachim Schoeberl

In: Numerical and Symbolic Scientific Computing: Progress and Prospects, Ulrich Langer, Peter Paule (ed.), Texts and Monographs in Symbolic Computation 1, pp. 105-121. 2012. Springer, Wien, ISBN 978-3-7091-0793-5. [pdf]@

author = {Christoph Koutschan and Christoph Lehrenfeld and Joachim Schoeberl},

title = {{Computer Algebra meets Finite Elements: an Efficient Implementation for Maxwell's Equations}},

booktitle = {{Numerical and Symbolic Scientific Computing: Progress and Prospects}},

language = {english},

abstract = {We consider the numerical discretization of the time-domain Maxwell'sequations with an energy-conserving discontinuous Galerkin finiteelement formulation. This particular formulation allows for higherorder approximations of the electric and magnetic field. Specialemphasis is placed on an efficient implementation which is achieved bytaking advantage of recurrence properties and the tensor-productstructure of the chosen shape functions. These recurrences have beenderived symbolically with computer algebra methods reminiscent of theholonomic systems approach.},

series = {Texts and Monographs in Symbolic Computation},

volume = {1},

pages = {105--121},

publisher = {Springer},

address = {Wien},

isbn_issn = {ISBN 978-3-7091-0793-5},

year = {2012},

editor = {Ulrich Langer and Peter Paule},

refereed = {yes},

length = {18}

}

**incollection**{RISC4252,author = {Christoph Koutschan and Christoph Lehrenfeld and Joachim Schoeberl},

title = {{Computer Algebra meets Finite Elements: an Efficient Implementation for Maxwell's Equations}},

booktitle = {{Numerical and Symbolic Scientific Computing: Progress and Prospects}},

language = {english},

abstract = {We consider the numerical discretization of the time-domain Maxwell'sequations with an energy-conserving discontinuous Galerkin finiteelement formulation. This particular formulation allows for higherorder approximations of the electric and magnetic field. Specialemphasis is placed on an efficient implementation which is achieved bytaking advantage of recurrence properties and the tensor-productstructure of the chosen shape functions. These recurrences have beenderived symbolically with computer algebra methods reminiscent of theholonomic systems approach.},

series = {Texts and Monographs in Symbolic Computation},

volume = {1},

pages = {105--121},

publisher = {Springer},

address = {Wien},

isbn_issn = {ISBN 978-3-7091-0793-5},

year = {2012},

editor = {Ulrich Langer and Peter Paule},

refereed = {yes},

length = {18}

}

[Koutschan]

### The iterated integrals of ln(1+x^n)

#### Tewodros Amdeberhan, Christoph Koutschan, Victor H. Moll, Eric S. Rowland

International Journal of Number Theory 8(1), pp. 71-94. 2012. ISSN 1793-0421. [pdf]@

author = {Tewodros Amdeberhan and Christoph Koutschan and Victor H. Moll and Eric S. Rowland},

title = {{The iterated integrals of ln(1+x^n)}},

language = {english},

abstract = {For a polynomial P, we consider the sequence of iterated integralsof ln P(x). This sequence is expressed in terms of the zeros of P(x). In thespecial case of ln(1+x^2 ), arithmetic properties of certain coefficients arisingare described. Similar observations are made for ln(1+x^3).},

journal = {International Journal of Number Theory},

volume = {8},

number = {1},

pages = {71--94},

isbn_issn = {ISSN 1793-0421},

year = {2012},

refereed = {yes},

length = {24}

}

**article**{RISC4306,author = {Tewodros Amdeberhan and Christoph Koutschan and Victor H. Moll and Eric S. Rowland},

title = {{The iterated integrals of ln(1+x^n)}},

language = {english},

abstract = {For a polynomial P, we consider the sequence of iterated integralsof ln P(x). This sequence is expressed in terms of the zeros of P(x). In thespecial case of ln(1+x^2 ), arithmetic properties of certain coefficients arisingare described. Similar observations are made for ln(1+x^3).},

journal = {International Journal of Number Theory},

volume = {8},

number = {1},

pages = {71--94},

isbn_issn = {ISSN 1793-0421},

year = {2012},

refereed = {yes},

length = {24}

}

[Koutschan]

### The non-commutative A-polynomial of (-2, 3, n) pretzel knots

#### Stavros Garoufalidis, Christoph Koutschan

Experimental Mathematics 21(3), pp. 241-251. 2012. ISSN 1058-6458. [url] [pdf]@

author = {Stavros Garoufalidis and Christoph Koutschan},

title = {{The non-commutative A-polynomial of (-2,3,n) pretzel knots}},

language = {english},

abstract = {We study q-holonomic sequences that arise as the colored Jones polynomial of knots in 3-space. The minimal-order recurrence for such a sequence is called the (non-commutative) A-polynomial of a knot. Using the method of guessing, we obtain this polynomial explicitly for the K_p=(-2,3,3+2p) pretzel knots for p=-5, ..., 5. This is a particularly interesting family since the pairs (K_p, -K_{-p}) are geometrically similar (in particular, scissors congruent) with similar character varieties. Our computation of thenon-commutative A-polynomial (a) complements the computation of the A-polynomial of the pretzel knots done by the first author and Mattman, (b) supports the AJ Conjecture for knots with reducible A-polynomial and (c) numerically computes the Kashaev invariant of pretzel knots in linear time. In a later publication, we will use the numerical computation of the Kashaev invariantto numerically verify the Volume Conjecture for the above mentioned pretzel knots.},

journal = {Experimental Mathematics},

volume = {21},

number = {3},

pages = {241--251},

isbn_issn = {ISSN 1058-6458},

year = {2012},

refereed = {yes},

length = {11},

url = {http://www.risc.jku.at/people/ckoutsch/pretzel/}

}

**article**{RISC4408,author = {Stavros Garoufalidis and Christoph Koutschan},

title = {{The non-commutative A-polynomial of (-2,3,n) pretzel knots}},

language = {english},

abstract = {We study q-holonomic sequences that arise as the colored Jones polynomial of knots in 3-space. The minimal-order recurrence for such a sequence is called the (non-commutative) A-polynomial of a knot. Using the method of guessing, we obtain this polynomial explicitly for the K_p=(-2,3,3+2p) pretzel knots for p=-5, ..., 5. This is a particularly interesting family since the pairs (K_p, -K_{-p}) are geometrically similar (in particular, scissors congruent) with similar character varieties. Our computation of thenon-commutative A-polynomial (a) complements the computation of the A-polynomial of the pretzel knots done by the first author and Mattman, (b) supports the AJ Conjecture for knots with reducible A-polynomial and (c) numerically computes the Kashaev invariant of pretzel knots in linear time. In a later publication, we will use the numerical computation of the Kashaev invariantto numerically verify the Volume Conjecture for the above mentioned pretzel knots.},

journal = {Experimental Mathematics},

volume = {21},

number = {3},

pages = {241--251},

isbn_issn = {ISSN 1058-6458},

year = {2012},

refereed = {yes},

length = {11},

url = {http://www.risc.jku.at/people/ckoutsch/pretzel/}

}

[Koutschan]

### Zeilberger's Holonomic Ansatz for Pfaffians

#### Masao Ishikawa, Christoph Koutschan

In: Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation (ISSAC), Joris van der Hoeven, Mark van Hoeij (ed.), pp. 227-233. 2012. ACM, ISBN 978-1-4503-1269. [url] [pdf]@

author = {Masao Ishikawa and Christoph Koutschan},

title = {{Zeilberger's Holonomic Ansatz for Pfaffians}},

booktitle = {{Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation (ISSAC)}},

language = {english},

abstract = {A variation of Zeilberger's holonomic ansatz for symbolic determinant evaluations is proposed which is tailored to deal with Pfaffians. The method is also applicable to determinants of skew-symmetric matrices, for which the original approach does not work. As Zeilberger's approach is based on the Laplace expansion (cofactor expansion) of the determinant, we derive our approach from the cofactor expansion of the Pfaffian. To demonstrate the power of our method, we prove, using computer algebra algorithms, some conjectures proposed in the paper "Pfaffian decomposition and a Pfaffian analogue of q-Catalan Hankel determinants" by Ishikawa, Tagawa, and Zeng. A minor summation formula related to partitions and Motzkin paths follows as a corollary.},

pages = {227--233},

publisher = {ACM},

isbn_issn = {ISBN 978-1-4503-1269},

year = {2012},

editor = {Joris van der Hoeven and Mark van Hoeij},

refereed = {yes},

length = {7},

url = {http://www.risc.jku.at/people/ckoutsch/pfaffians/}

}

**inproceedings**{RISC4512,author = {Masao Ishikawa and Christoph Koutschan},

title = {{Zeilberger's Holonomic Ansatz for Pfaffians}},

booktitle = {{Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation (ISSAC)}},

language = {english},

abstract = {A variation of Zeilberger's holonomic ansatz for symbolic determinant evaluations is proposed which is tailored to deal with Pfaffians. The method is also applicable to determinants of skew-symmetric matrices, for which the original approach does not work. As Zeilberger's approach is based on the Laplace expansion (cofactor expansion) of the determinant, we derive our approach from the cofactor expansion of the Pfaffian. To demonstrate the power of our method, we prove, using computer algebra algorithms, some conjectures proposed in the paper "Pfaffian decomposition and a Pfaffian analogue of q-Catalan Hankel determinants" by Ishikawa, Tagawa, and Zeng. A minor summation formula related to partitions and Motzkin paths follows as a corollary.},

pages = {227--233},

publisher = {ACM},

isbn_issn = {ISBN 978-1-4503-1269},

year = {2012},

editor = {Joris van der Hoeven and Mark van Hoeij},

refereed = {yes},

length = {7},

url = {http://www.risc.jku.at/people/ckoutsch/pfaffians/}

}

[Koutschan]

### Twisting q-holonomic sequences by complex roots of unity

#### Stavros Garoufalidis, Christoph Koutschan

In: Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation (ISSAC), Joris van der Hoeven, Mark van Hoeij (ed.), pp. 179-186. 2012. ACM, ISBN 978-1-4503-1269. [url] [pdf]@

author = {Stavros Garoufalidis and Christoph Koutschan},

title = {{Twisting q-holonomic sequences by complex roots of unity}},

booktitle = {{Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation (ISSAC)}},

language = {english},

abstract = {A sequence f_n(q) is q-holonomic if it satisfies a nontrivial linear recurrence with coefficients polynomials in q and q^n. Our main theorems state that q-holonomicity is preserved under twisting, i.e., replacing q by w*q where w is a complex root of unity, and under the substitution q -> q^alpha where alpha is a rational number. Our proofs are constructive, work in the multivariate setting of \partial-finite sequences and are implemented in the Mathematica package HolonomicFunctions. Our results are illustrated by twisting natural q-holonomic sequences which appear in quantum topology, namely the colored Jones polynomial of pretzel knots and twist knots. The recurrence of the twisted colored Jones polynomial can be used to compute the asymptotics of the Kashaev invariant of a knot at an arbitrary complex root of unity.},

pages = {179--186},

publisher = {ACM},

isbn_issn = {ISBN 978-1-4503-1269},

year = {2012},

editor = {Joris van der Hoeven and Mark van Hoeij},

refereed = {yes},

length = {8},

url = {http://www.math.gatech.edu/~stavros/publications/twisting.qholonomic.data/}

}

**inproceedings**{RISC4513,author = {Stavros Garoufalidis and Christoph Koutschan},

title = {{Twisting q-holonomic sequences by complex roots of unity}},

booktitle = {{Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation (ISSAC)}},

language = {english},

abstract = {A sequence f_n(q) is q-holonomic if it satisfies a nontrivial linear recurrence with coefficients polynomials in q and q^n. Our main theorems state that q-holonomicity is preserved under twisting, i.e., replacing q by w*q where w is a complex root of unity, and under the substitution q -> q^alpha where alpha is a rational number. Our proofs are constructive, work in the multivariate setting of \partial-finite sequences and are implemented in the Mathematica package HolonomicFunctions. Our results are illustrated by twisting natural q-holonomic sequences which appear in quantum topology, namely the colored Jones polynomial of pretzel knots and twist knots. The recurrence of the twisted colored Jones polynomial can be used to compute the asymptotics of the Kashaev invariant of a knot at an arbitrary complex root of unity.},

pages = {179--186},

publisher = {ACM},

isbn_issn = {ISBN 978-1-4503-1269},

year = {2012},

editor = {Joris van der Hoeven and Mark van Hoeij},

refereed = {yes},

length = {8},

url = {http://www.math.gatech.edu/~stavros/publications/twisting.qholonomic.data/}

}

[Koutschan]

### Third order integrability conditions for homogeneous potentials of degree -1

#### Thierry Combot, Christoph Koutschan

Journal of Mathematical Physics 53(8), pp. 082704-. 2012. ISSN 0022-2488. [pdf]@

author = {Thierry Combot and Christoph Koutschan},

title = {{Third order integrability conditions for homogeneous potentials of degree -1}},

language = {english},

abstract = {We prove an integrability criterion of order 3 for a homogeneous potential of degree -1 in the plane. Still, this criterion depends on some integer and it is impossible to apply it directly except for families of potentials whose eigenvalues are bounded. To address this issue, we use holonomic and asymptotic computations with error control of this criterion and apply it to the potential of the form V(r, theta) = 1/r * h(exp(i*theta)) with h in C[z], deg(h) <= 3. We then find all meromorphically integrable potentials of this form.},

journal = {Journal of Mathematical Physics},

volume = {53},

number = {8},

pages = {082704--},

isbn_issn = {ISSN 0022-2488},

year = {2012},

refereed = {yes},

length = {26}

}

**article**{RISC4572,author = {Thierry Combot and Christoph Koutschan},

title = {{Third order integrability conditions for homogeneous potentials of degree -1}},

language = {english},

abstract = {We prove an integrability criterion of order 3 for a homogeneous potential of degree -1 in the plane. Still, this criterion depends on some integer and it is impossible to apply it directly except for families of potentials whose eigenvalues are bounded. To address this issue, we use holonomic and asymptotic computations with error control of this criterion and apply it to the potential of the form V(r, theta) = 1/r * h(exp(i*theta)) with h in C[z], deg(h) <= 3. We then find all meromorphically integrable potentials of this form.},

journal = {Journal of Mathematical Physics},

volume = {53},

number = {8},

pages = {082704--},

isbn_issn = {ISSN 0022-2488},

year = {2012},

refereed = {yes},

length = {26}

}

[Pillwein]

### Fast summation techniques for sparse shape functions in tetrahedral hp-FEM

#### S. Beuchler, V. Pillwein and S. Zaglmayr

In: Domain Decomposition Methods in Science and Engineering XX, , pp. 537-544. 2012. to appear. [pdf]@

author = {S. Beuchler and V. Pillwein and S. Zaglmayr},

title = {{Fast summation techniques for sparse shape functions in tetrahedral hp-FEM}},

booktitle = {{Domain Decomposition Methods in Science and Engineering XX}},

language = {english},

pages = {537--544},

isbn_issn = {?},

year = {2012},

note = {to appear},

editor = {?},

refereed = {yes},

length = {8}

}

**inproceedings**{RISC3283,author = {S. Beuchler and V. Pillwein and S. Zaglmayr},

title = {{Fast summation techniques for sparse shape functions in tetrahedral hp-FEM}},

booktitle = {{Domain Decomposition Methods in Science and Engineering XX}},

language = {english},

pages = {537--544},

isbn_issn = {?},

year = {2012},

note = {to appear},

editor = {?},

refereed = {yes},

length = {8}

}

[Pillwein]

### Sparsity optimized high order finite element functions for H(div) on simplices

#### S. Beuchler, V. Pillwein and S. Zaglmayr

Numerische Mathematik 122(2), pp. 197-225. 2012. [pdf]@

author = {S. Beuchler and V. Pillwein and S. Zaglmayr},

title = {{Sparsity optimized high order finite element functions for H(div) on simplices}},

language = {english},

journal = {Numerische Mathematik},

volume = {122},

number = {2},

pages = {197--225},

isbn_issn = {?},

year = {2012},

refereed = {yes},

length = {29}

}

**article**{RISC4524,author = {S. Beuchler and V. Pillwein and S. Zaglmayr},

title = {{Sparsity optimized high order finite element functions for H(div) on simplices}},

language = {english},

journal = {Numerische Mathematik},

volume = {122},

number = {2},

pages = {197--225},

isbn_issn = {?},

year = {2012},

refereed = {yes},

length = {29}

}

### 2011

[Koutschan]

### The integrals in Gradshteyn and Ryzhik. Part 18: Some automatic proofs

#### Christoph Koutschan, Victor H. Moll

SCIENTIA Series A: Mathematical Sciences 20, pp. 93-111. 2011. Universidad Tecnica Federico Santa Maria, Valparaiso, Chile, ISSN 0716-8446. [pdf]@

author = {Christoph Koutschan and Victor H. Moll},

title = {{The integrals in Gradshteyn and Ryzhik. Part 18: Some automatic proofs}},

language = {english},

journal = {SCIENTIA Series A: Mathematical Sciences},

volume = {20},

pages = {93--111},

address = {Universidad Tecnica Federico Santa Maria, Valparaiso, Chile},

isbn_issn = {ISSN 0716-8446},

year = {2011},

refereed = {yes},

length = {19}

}

**article**{RISC4099,author = {Christoph Koutschan and Victor H. Moll},

title = {{The integrals in Gradshteyn and Ryzhik. Part 18: Some automatic proofs}},

language = {english},

journal = {SCIENTIA Series A: Mathematical Sciences},

volume = {20},

pages = {93--111},

address = {Universidad Tecnica Federico Santa Maria, Valparaiso, Chile},

isbn_issn = {ISSN 0716-8446},

year = {2011},

refereed = {yes},

length = {19}

}

[Koutschan]

### The sl3 Jones polynomial of the trefoil: a case study of q-holonomic recursions

#### Stavros Garoufalidis, Christoph Koutschan

Advances in Applied Mathematics 47(4), pp. 829-839. 2011. ISSN 0196-8858. [pdf]@

author = {Stavros Garoufalidis and Christoph Koutschan},

title = {{The sl3 Jones polynomial of the trefoil: a case study of q-holonomic recursions}},

language = {english},

abstract = {The sl3 colored Jones polynomial of the trefoil knot is a q-holonomicsequence of two variables with natural origin, namely quantum topology.The paper presents an explicit set of generators for the annihilator ideal of this q-holonomic sequence as a case study.On the one hand, our results are new and useful to quantum topology: this is the first example of a rank 2 Lie algebra computation concerning the colored Jones polynomial of a knot. On the other hand, thiswork illustrates the applicability and computational power of the employed computer algebra methods.},

journal = {Advances in Applied Mathematics},

volume = {47},

number = {4},

pages = {829--839},

isbn_issn = {ISSN 0196-8858},

year = {2011},

refereed = {yes},

length = {11}

}

**article**{RISC4340,author = {Stavros Garoufalidis and Christoph Koutschan},

title = {{The sl3 Jones polynomial of the trefoil: a case study of q-holonomic recursions}},

language = {english},

abstract = {The sl3 colored Jones polynomial of the trefoil knot is a q-holonomicsequence of two variables with natural origin, namely quantum topology.The paper presents an explicit set of generators for the annihilator ideal of this q-holonomic sequence as a case study.On the one hand, our results are new and useful to quantum topology: this is the first example of a rank 2 Lie algebra computation concerning the colored Jones polynomial of a knot. On the other hand, thiswork illustrates the applicability and computational power of the employed computer algebra methods.},

journal = {Advances in Applied Mathematics},

volume = {47},

number = {4},

pages = {829--839},

isbn_issn = {ISSN 0196-8858},

year = {2011},

refereed = {yes},

length = {11}

}

[Koutschan]

### The 1958 Pekeris-Accad-WEIZAC Ground-Breaking Collaboration that computed Ground States of Two-Electron Atoms (and its 2010 Redux)

#### Christoph Koutschan, Doron Zeilberger

The Mathematical Intelligencer 33(2), pp. 52-57. 2011. ISSN 0343-6993. [url] [pdf]@

author = {Christoph Koutschan and Doron Zeilberger},

title = {{The 1958 Pekeris-Accad-WEIZAC Ground-Breaking Collaboration that computed Ground States of Two-Electron Atoms (and its 2010 Redux)}},

language = {english},

abstract = {In order to appreciate how good we as mathematicians and scientists have ittoday, with extremely fast hardware and lots and lots of memory, as well aswith readily available high-level software, both for numeric and symboliccomputation, it may be a good idea to go back to the early days of electroniccomputers and carefully examine, as a case study, a problem that was considereda huge challenge back then, and compare notes. We chose C.L. Pekeris' 1958seminal work on the ground state energies of two-electron atoms. Allcomputations will be done ab initio with today's software and hardware, with aspecial emphasis on the symbolic computations which in 1958 had to be made byhand, and which nowadays can be automated and generalized.},

journal = {The Mathematical Intelligencer},

volume = {33},

number = {2},

pages = {52--57},

isbn_issn = {ISSN 0343-6993},

year = {2011},

refereed = {yes},

length = {6},

url = {http://www.risc.jku.at/people/ckoutsch/pekeris/}

}

**article**{RISC4356,author = {Christoph Koutschan and Doron Zeilberger},

title = {{The 1958 Pekeris-Accad-WEIZAC Ground-Breaking Collaboration that computed Ground States of Two-Electron Atoms (and its 2010 Redux)}},

language = {english},

abstract = {In order to appreciate how good we as mathematicians and scientists have ittoday, with extremely fast hardware and lots and lots of memory, as well aswith readily available high-level software, both for numeric and symboliccomputation, it may be a good idea to go back to the early days of electroniccomputers and carefully examine, as a case study, a problem that was considereda huge challenge back then, and compare notes. We chose C.L. Pekeris' 1958seminal work on the ground state energies of two-electron atoms. Allcomputations will be done ab initio with today's software and hardware, with aspecial emphasis on the symbolic computations which in 1958 had to be made byhand, and which nowadays can be automated and generalized.},

journal = {The Mathematical Intelligencer},

volume = {33},

number = {2},

pages = {52--57},

isbn_issn = {ISSN 0343-6993},

year = {2011},

refereed = {yes},

length = {6},

url = {http://www.risc.jku.at/people/ckoutsch/pekeris/}

}

[Koutschan]

### On Two-generated Non-commutative Algebras Subject to the Affine Relation

#### Viktor Levandovskyy, Christoph Koutschan, Oleksandr Motsak

In: Proceedings of CASC 2011, Vladimir Gerdt, Wolfram Koepf, Ernst W. Mayr, Evgenii Vorozhtsov (ed.), Lecture Notes in Computer Science 6885, pp. 309-320. 2011. Springer, ISBN 978-3-642-23567-2. [pdf]@

author = {Viktor Levandovskyy and Christoph Koutschan and Oleksandr Motsak},

title = {{On Two-generated Non-commutative Algebras Subject to the Affine Relation}},

booktitle = {{Proceedings of CASC 2011}},

language = {english},

abstract = {We consider algebras over a field K, generated by two variables x and y subject to the single relation yx = qxy + ax + by + c for q in K^* and a, b, c in K. We prove, that among such algebras there are precisely five isomorphism classes. The representatives of these classes, which are ubiquitous operator algebras, are called model algebras. We derive explicit multiplication formulas for y^m x^n in terms of standard monomials x^i y^j for many algebras of the considered type. Such formulas are used in e.g. establishing formulas of binomial type and in an implementation of non-commutative multiplication in a computer algebra system. By using the formulas we also study centers and ring-theoretic properties of the non-commutative model algebras.},

series = {Lecture Notes in Computer Science},

volume = {6885},

pages = {309--320},

publisher = {Springer},

isbn_issn = {ISBN 978-3-642-23567-2},

year = {2011},

editor = {Vladimir Gerdt and Wolfram Koepf and Ernst W. Mayr and Evgenii Vorozhtsov},

refereed = {yes},

length = {12}

}

**inproceedings**{RISC4374,author = {Viktor Levandovskyy and Christoph Koutschan and Oleksandr Motsak},

title = {{On Two-generated Non-commutative Algebras Subject to the Affine Relation}},

booktitle = {{Proceedings of CASC 2011}},

language = {english},

abstract = {We consider algebras over a field K, generated by two variables x and y subject to the single relation yx = qxy + ax + by + c for q in K^* and a, b, c in K. We prove, that among such algebras there are precisely five isomorphism classes. The representatives of these classes, which are ubiquitous operator algebras, are called model algebras. We derive explicit multiplication formulas for y^m x^n in terms of standard monomials x^i y^j for many algebras of the considered type. Such formulas are used in e.g. establishing formulas of binomial type and in an implementation of non-commutative multiplication in a computer algebra system. By using the formulas we also study centers and ring-theoretic properties of the non-commutative model algebras.},

series = {Lecture Notes in Computer Science},

volume = {6885},

pages = {309--320},

publisher = {Springer},

isbn_issn = {ISBN 978-3-642-23567-2},

year = {2011},

editor = {Vladimir Gerdt and Wolfram Koepf and Ernst W. Mayr and Evgenii Vorozhtsov},

refereed = {yes},

length = {12}

}

[Koutschan]

### On Kahan's Rules for Determining Branch Cuts

#### Frederic Chyzak, James H. Davenport, Christoph Koutschan, Bruno Salvy

In: Proceedings of the 13th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), Dongming Wang et al. (ed.), pp. 47-51. 2011. IEEE Computer Society: Conference Publishing Services (CPS), ISBN 978-0-7695-4630-8. [pdf]@

author = {Frederic Chyzak and James H. Davenport and Christoph Koutschan and Bruno Salvy},

title = {{On Kahan's Rules for Determining Branch Cuts}},

booktitle = {{Proceedings of the 13th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC)}},

language = {english},

abstract = {In computer algebra there are different ways of approaching the mathematical concept of functions, one of which is by defining them as solutions of differential equations. We compare different such approaches and discuss the occurring problems. The main focus is on the question of determining possible branch cuts. We explore the extent to which the treatment of branch cuts can be rendered (more) algorithmic, by adapting Kahan's rules to the differential equation setting.},

pages = {47--51},

publisher = {IEEE Computer Society: Conference Publishing Services (CPS)},

isbn_issn = {ISBN 978-0-7695-4630-8},

year = {2011},

editor = {Dongming Wang et al.},

refereed = {yes},

length = {5}

}

**inproceedings**{RISC4375,author = {Frederic Chyzak and James H. Davenport and Christoph Koutschan and Bruno Salvy},

title = {{On Kahan's Rules for Determining Branch Cuts}},

booktitle = {{Proceedings of the 13th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC)}},

language = {english},

abstract = {In computer algebra there are different ways of approaching the mathematical concept of functions, one of which is by defining them as solutions of differential equations. We compare different such approaches and discuss the occurring problems. The main focus is on the question of determining possible branch cuts. We explore the extent to which the treatment of branch cuts can be rendered (more) algorithmic, by adapting Kahan's rules to the differential equation setting.},

pages = {47--51},

publisher = {IEEE Computer Society: Conference Publishing Services (CPS)},

isbn_issn = {ISBN 978-0-7695-4630-8},

year = {2011},

editor = {Dongming Wang et al.},

refereed = {yes},

length = {5}

}