# Fast computer algebra for special functions [START]

### Project Lead

### Project Duration

01/04/2010 - 31/03/2016## Publications

### 2016

### Workshop on Symbolic Combinatorics and Algorithmic Differential Algebra

#### Manuel Kauers, Peter Paule, Greg Reid

ACM Communications in Computer Algebra 50(Issue 1), pp. 27-34. March 2016. 1932-2240. [pdf]@

author = {Manuel Kauers and Peter Paule and Greg Reid},

title = {{Workshop on Symbolic Combinatorics and Algorithmic Differential Algebra}},

language = {english},

journal = {ACM Communications in Computer Algebra},

volume = {50},

number = {Issue 1},

pages = {27--34},

isbn_issn = {1932-2240},

year = {2016},

month = {March},

refereed = {no},

length = {8}

}

**article**{RISC5284,author = {Manuel Kauers and Peter Paule and Greg Reid},

title = {{Workshop on Symbolic Combinatorics and Algorithmic Differential Algebra}},

language = {english},

journal = {ACM Communications in Computer Algebra},

volume = {50},

number = {Issue 1},

pages = {27--34},

isbn_issn = {1932-2240},

year = {2016},

month = {March},

refereed = {no},

length = {8}

}

### 2014

### Radicals of Ore Polynomials

#### Maximilian Jaroschek

In: Proceedings of EACA 2014, , pp. -. 2014. to appear. [pdf]@

author = {Maximilian Jaroschek},

title = {{Radicals of Ore Polynomials}},

booktitle = {{Proceedings of EACA 2014}},

language = {english},

pages = {--},

isbn_issn = {?},

year = {2014},

note = {to appear},

editor = {?},

refereed = {yes},

length = {4}

}

**inproceedings**{RISC4979,author = {Maximilian Jaroschek},

title = {{Radicals of Ore Polynomials}},

booktitle = {{Proceedings of EACA 2014}},

language = {english},

pages = {--},

isbn_issn = {?},

year = {2014},

note = {to appear},

editor = {?},

refereed = {yes},

length = {4}

}

### Ore Polynomials in Sage

#### Manuel Kauers, Maximilian Jaroschek, Fredrik Johansson

In: Computer Algebra and Polynomials, Jaime Gutierrez, Josef Schicho, Martin Weimann (ed.), Lecture Notes in Computer Science , pp. ?-?. 2014. tba. [pdf] [ps]@

author = {Manuel Kauers and Maximilian Jaroschek and Fredrik Johansson},

title = {{Ore Polynomials in Sage}},

booktitle = {{Computer Algebra and Polynomials}},

language = {english},

abstract = {We present a Sage implementation of Ore algebras. The main features for the mostcommon instances include basic arithmetic and actions; GCRD and LCLM; D-finiteclosure properties; natural transformations between related algebras; guessing;desingularization; solvers for polynomials, rational functions and (generalized)power series. This paper is a tutorial on how to use the package.},

series = {Lecture Notes in Computer Science},

pages = {?--?},

isbn_issn = {tba},

year = {2014},

editor = {Jaime Gutierrez and Josef Schicho and Martin Weimann},

refereed = {yes},

length = {17}

}

**inproceedings**{RISC4944,author = {Manuel Kauers and Maximilian Jaroschek and Fredrik Johansson},

title = {{Ore Polynomials in Sage}},

booktitle = {{Computer Algebra and Polynomials}},

language = {english},

abstract = {We present a Sage implementation of Ore algebras. The main features for the mostcommon instances include basic arithmetic and actions; GCRD and LCLM; D-finiteclosure properties; natural transformations between related algebras; guessing;desingularization; solvers for polynomials, rational functions and (generalized)power series. This paper is a tutorial on how to use the package.},

series = {Lecture Notes in Computer Science},

pages = {?--?},

isbn_issn = {tba},

year = {2014},

editor = {Jaime Gutierrez and Josef Schicho and Martin Weimann},

refereed = {yes},

length = {17}

}

### Fast and rigorous computation of special functions to high precision

#### F. Johansson

RISC. PhD Thesis. 2014. [pdf]@

author = {F. Johansson},

title = {{Fast and rigorous computation of special functions to high precision}},

language = {english},

year = {2014},

translation = {0},

school = {RISC},

length = {0}

}

**phdthesis**{RISC4972,author = {F. Johansson},

title = {{Fast and rigorous computation of special functions to high precision}},

language = {english},

year = {2014},

translation = {0},

school = {RISC},

length = {0}

}

### Using functional equations to enumerate 1324-avoiding permutations

#### Fredrik Johansson, Brian Nakamura

Advances in Applied Mathematics 56(0), pp. 20 - 34. 2014. ISSN 0196-8858. [url]@

author = {Fredrik Johansson and Brian Nakamura},

title = {{Using functional equations to enumerate 1324-avoiding permutations}},

language = {english},

journal = {Advances in Applied Mathematics},

volume = {56},

number = {0},

pages = {20 -- 34},

isbn_issn = {ISSN 0196-8858},

year = {2014},

refereed = {yes},

keywords = {Enumeration algorithm},

length = {15},

url = {http://www.sciencedirect.com/science/article/pii/S0196885814000256}

}

**article**{RISC4987,author = {Fredrik Johansson and Brian Nakamura},

title = {{Using functional equations to enumerate 1324-avoiding permutations}},

language = {english},

journal = {Advances in Applied Mathematics},

volume = {56},

number = {0},

pages = {20 -- 34},

isbn_issn = {ISSN 0196-8858},

year = {2014},

refereed = {yes},

keywords = {Enumeration algorithm},

length = {15},

url = {http://www.sciencedirect.com/science/article/pii/S0196885814000256}

}

### Hypercontractive inequalities via SOS, and the Frankl-R\"odl graph

#### Manuel Kauers, Ryan ODonnell, Li-Yang Tan, Yuan Zhou

In: Proceedings of SODA'14, tba (ed.), pp. ?-?. 2014. tba. [pdf]@

author = {Manuel Kauers and Ryan ODonnell and Li-Yang Tan and Yuan Zhou},

title = {{Hypercontractive inequalities via SOS, and the Frankl-R\"odl graph}},

booktitle = {{Proceedings of SODA'14}},

language = {english},

pages = {?--?},

isbn_issn = {tba},

year = {2014},

editor = {tba},

refereed = {yes},

length = {15}

}

**inproceedings**{RISC4829,author = {Manuel Kauers and Ryan ODonnell and Li-Yang Tan and Yuan Zhou},

title = {{Hypercontractive inequalities via SOS, and the Frankl-R\"odl graph}},

booktitle = {{Proceedings of SODA'14}},

language = {english},

pages = {?--?},

isbn_issn = {tba},

year = {2014},

editor = {tba},

refereed = {yes},

length = {15}

}

### On the length of integers in telescopers for proper hypergeometric terms

#### Manuel Kauers, Lily Yen

Journal of Symbolic Computation, pp. ?-?. 2014. ISSN 0747-7171. to appear. [pdf] [ps]@

author = {Manuel Kauers and Lily Yen},

title = {{On the length of integers in telescopers for proper hypergeometric terms}},

language = {english},

journal = {Journal of Symbolic Computation},

pages = {?--?},

isbn_issn = {ISSN 0747-7171},

year = {2014},

note = {to appear},

refereed = {yes},

length = {15}

}

**article**{RISC4955,author = {Manuel Kauers and Lily Yen},

title = {{On the length of integers in telescopers for proper hypergeometric terms}},

language = {english},

journal = {Journal of Symbolic Computation},

pages = {?--?},

isbn_issn = {ISSN 0747-7171},

year = {2014},

note = {to appear},

refereed = {yes},

length = {15}

}

### Computer Algebra

#### Manuel Kauers

In: Handbook of Combinatorics, Miklos Bona (ed.), pp. ?-?. 2014. Taylor and Francis, tba.@

author = {Manuel Kauers},

title = {{Computer Algebra}},

booktitle = {{Handbook of Combinatorics}},

language = {english},

pages = {?--?},

publisher = {Taylor and Francis},

isbn_issn = {tba},

year = {2014},

editor = {Miklos Bona},

refereed = {yes},

length = {59}

}

**incollection**{RISC4956,author = {Manuel Kauers},

title = {{Computer Algebra}},

booktitle = {{Handbook of Combinatorics}},

language = {english},

pages = {?--?},

publisher = {Taylor and Francis},

isbn_issn = {tba},

year = {2014},

editor = {Miklos Bona},

refereed = {yes},

length = {59}

}

### Bounds for D-Finite Closure Properties

#### Manuel Kauers

In: Proceedings of ISSAC 2014, Katsusuke Nabeshima (ed.), pp. 288-295. 2014. isbn 978-1-4503-2501-1/14/07. [pdf]@

author = {Manuel Kauers},

title = {{Bounds for D-Finite Closure Properties}},

booktitle = {{Proceedings of ISSAC 2014}},

language = {english},

pages = {288--295},

isbn_issn = {isbn 978-1-4503-2501-1/14/07},

year = {2014},

editor = {Katsusuke Nabeshima},

refereed = {yes},

length = {8}

}

**inproceedings**{RISC4989,author = {Manuel Kauers},

title = {{Bounds for D-Finite Closure Properties}},

booktitle = {{Proceedings of ISSAC 2014}},

language = {english},

pages = {288--295},

isbn_issn = {isbn 978-1-4503-2501-1/14/07},

year = {2014},

editor = {Katsusuke Nabeshima},

refereed = {yes},

length = {8}

}

### On 3-dimensional lattice walks confined to the positive octant

#### Alin Bostan, Mireille Bousquet-Melou, Manuel Kauers, Stephen Melczer

Annals of Combinatorics, pp. ??-??. 2014. ISSN 0218-0006. to appear. [pdf]@

author = {Alin Bostan and Mireille Bousquet-Melou and Manuel Kauers and Stephen Melczer},

title = {{ On 3-dimensional lattice walks confined to the positive octant}},

language = {english},

journal = {Annals of Combinatorics},

pages = {??--??},

isbn_issn = {ISSN 0218-0006},

year = {2014},

note = {to appear},

refereed = {yes},

length = {36}

}

**article**{RISC5082,author = {Alin Bostan and Mireille Bousquet-Melou and Manuel Kauers and Stephen Melczer},

title = {{ On 3-dimensional lattice walks confined to the positive octant}},

language = {english},

journal = {Annals of Combinatorics},

pages = {??--??},

isbn_issn = {ISSN 0218-0006},

year = {2014},

note = {to appear},

refereed = {yes},

length = {36}

}

### 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}

}

### 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

### Formal Laurent Series in Several Variables

#### Ainhoa Aparicio Monforte, Manuel Kauers

Expositiones Mathematicae 31(4), pp. 350-367. 2013. ISSN 0723-0869. [pdf]@

author = {Ainhoa Aparicio Monforte and Manuel Kauers},

title = {{Formal Laurent Series in Several Variables}},

language = {english},

abstract = { We explain the construction of fields of formal infinite series in several variables, generalizing the classical notion of formal Laurent series in one variable. Our discussion addresses the field operations for these series (addition, multiplication, and division), the composition, and includes an implicit function theorem.},

journal = {Expositiones Mathematicae},

volume = {31},

number = {4},

pages = {350--367},

isbn_issn = {ISSN 0723-0869},

year = {2013},

refereed = {yes},

length = {24}

}

**article**{RISC4600,author = {Ainhoa Aparicio Monforte and Manuel Kauers},

title = {{Formal Laurent Series in Several Variables}},

language = {english},

abstract = { We explain the construction of fields of formal infinite series in several variables, generalizing the classical notion of formal Laurent series in one variable. Our discussion addresses the field operations for these series (addition, multiplication, and division), the composition, and includes an implicit function theorem.},

journal = {Expositiones Mathematicae},

volume = {31},

number = {4},

pages = {350--367},

isbn_issn = {ISSN 0723-0869},

year = {2013},

refereed = {yes},

length = {24}

}

### Desingularization Explains Order-Degree Curves for Ore Operators

#### Shaoshi Chen, Maximilian Jaroschek, Manuel Kauers, Michael F. Singer

In: Proceedings of ISSAC'13, Manuel Kauers (ed.), pp. 157-164. 2013. isbn 978-1-4503-2059-7/13/06. [pdf] [ps]@

author = {Shaoshi Chen and Maximilian Jaroschek and Manuel Kauers and Michael F. Singer},

title = {{Desingularization Explains Order-Degree Curves for Ore Operators}},

booktitle = {{Proceedings of ISSAC'13}},

language = {english},

abstract = { Desingularization is the problem of finding a left multiple of a given Ore operator in which some factor of the leading coefficient of the original operator is removed. An order-degree curve for a given Ore operator is a curve in the $(r,d)$-plane such that for all points $(r,d)$ above this curve, there exists a left multiple of order~$r$ and degree~$d$ of the given operator. We give a new proof of a desingularization result by Abramov and van Hoeij for the shift case, and show how desingularization implies order-degree curves which are extremely accurate in examples. },

pages = {157--164},

isbn_issn = {isbn 978-1-4503-2059-7/13/06},

year = {2013},

editor = {Manuel Kauers},

refereed = {yes},

length = {8}

}

**inproceedings**{RISC4706,author = {Shaoshi Chen and Maximilian Jaroschek and Manuel Kauers and Michael F. Singer},

title = {{Desingularization Explains Order-Degree Curves for Ore Operators}},

booktitle = {{Proceedings of ISSAC'13}},

language = {english},

abstract = { Desingularization is the problem of finding a left multiple of a given Ore operator in which some factor of the leading coefficient of the original operator is removed. An order-degree curve for a given Ore operator is a curve in the $(r,d)$-plane such that for all points $(r,d)$ above this curve, there exists a left multiple of order~$r$ and degree~$d$ of the given operator. We give a new proof of a desingularization result by Abramov and van Hoeij for the shift case, and show how desingularization implies order-degree curves which are extremely accurate in examples. },

pages = {157--164},

isbn_issn = {isbn 978-1-4503-2059-7/13/06},

year = {2013},

editor = {Manuel Kauers},

refereed = {yes},

length = {8}

}

### Improved Polynomial Remainder Sequences for Ore Polynomials

#### Maximilian Jaroschek

Journal of Symbolic Computation 58, pp. 64-76. 2013. ISSN 0747-7171. [url]@

author = {Maximilian Jaroschek},

title = {{Improved Polynomial Remainder Sequences for Ore Polynomials}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {58},

pages = {64--76},

isbn_issn = {ISSN 0747-7171},

year = {2013},

refereed = {yes},

length = {14},

url = {http://www.sciencedirect.com/science/article/pii/S0747717113000849}

}

**article**{RISC4726,author = {Maximilian Jaroschek},

title = {{Improved Polynomial Remainder Sequences for Ore Polynomials}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {58},

pages = {64--76},

isbn_issn = {ISSN 0747-7171},

year = {2013},

refereed = {yes},

length = {14},

url = {http://www.sciencedirect.com/science/article/pii/S0747717113000849}

}

### Removable Singularities of Ore Operators

#### Maximilian Jaroschek

RISC. PhD Thesis. November 2013. [pdf]@

author = {Maximilian Jaroschek},

title = {{Removable Singularities of Ore Operators}},

language = {english},

year = {2013},

month = {November},

translation = {0},

school = {RISC},

length = {115}

}

**phdthesis**{RISC4848,author = {Maximilian Jaroschek},

title = {{Removable Singularities of Ore Operators}},

language = {english},

year = {2013},

month = {November},

translation = {0},

school = {RISC},

length = {115}

}

### Finding Hyperexponential Solutions of Linear ODEs by Numerical Evaluation

#### Fredrik Johansson, Manuel Kauers, Marc Mezzarobba

In: Proceedings of ISSAC'13, Manuel Kauers (ed.), pp. 211-218. 2013. isbn 978-1-4503-2059-7/13/06. [pdf] [ps]@

author = {Fredrik Johansson and Manuel Kauers and Marc Mezzarobba},

title = {{Finding Hyperexponential Solutions of Linear ODEs by Numerical Evaluation}},

booktitle = {{Proceedings of ISSAC'13}},

language = {english},

abstract = { We present a new algorithm for computing hyperexponential solutions of linear ordinary differential equations with polynomial coefficients. The algorithm relies on interpreting formal series solutions at the singular points as analytic functions and evaluating them numerically at some common ordinary point. The numerical data is used to determine a small number of combinations of the formal series that may give rise to hyperexponential solutions. },

pages = {211--218},

isbn_issn = {isbn 978-1-4503-2059-7/13/06},

year = {2013},

editor = {Manuel Kauers},

refereed = {yes},

length = {8}

}

**inproceedings**{RISC4707,author = {Fredrik Johansson and Manuel Kauers and Marc Mezzarobba},

title = {{Finding Hyperexponential Solutions of Linear ODEs by Numerical Evaluation}},

booktitle = {{Proceedings of ISSAC'13}},

language = {english},

abstract = { We present a new algorithm for computing hyperexponential solutions of linear ordinary differential equations with polynomial coefficients. The algorithm relies on interpreting formal series solutions at the singular points as analytic functions and evaluating them numerically at some common ordinary point. The numerical data is used to determine a small number of combinations of the formal series that may give rise to hyperexponential solutions. },

pages = {211--218},

isbn_issn = {isbn 978-1-4503-2059-7/13/06},

year = {2013},

editor = {Manuel Kauers},

refereed = {yes},

length = {8}

}

### The Holonomic Toolbox

#### Manuel Kauers

In: Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, Johannes Bluemlein and Carsten Schneider (ed.), pp. ??-??. 2013. Springer, tba. to appear. [pdf]@

author = {Manuel Kauers},

title = {{The Holonomic Toolbox}},

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

language = {english},

abstract = {This is an overview over standard techniques for holonomic functions,written for readers who are new to the subject. We state the definition for holonomyin a couple of different ways, including some concrete special cases as well asa more abstract and more general version. We give a collection of standard examplesand state several fundamental properties of holonomic objects. Two techniqueswhich are most useful in applications are explained in some more detail: closureproperties, which can be used to prove identities among holonomic functions, andguessing, which can be used to generate plausible conjectures for equations satisfiedby a given function.},

pages = {??--??},

publisher = {Springer},

isbn_issn = {tba},

year = {2013},

note = {to appear},

editor = {Johannes Bluemlein and Carsten Schneider},

refereed = {yes},

length = {25}

}

**incollection**{RISC4710,author = {Manuel Kauers},

title = {{The Holonomic Toolbox}},

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

language = {english},

abstract = {This is an overview over standard techniques for holonomic functions,written for readers who are new to the subject. We state the definition for holonomyin a couple of different ways, including some concrete special cases as well asa more abstract and more general version. We give a collection of standard examplesand state several fundamental properties of holonomic objects. Two techniqueswhich are most useful in applications are explained in some more detail: closureproperties, which can be used to prove identities among holonomic functions, andguessing, which can be used to generate plausible conjectures for equations satisfiedby a given function.},

pages = {??--??},

publisher = {Springer},

isbn_issn = {tba},

year = {2013},

note = {to appear},

editor = {Johannes Bluemlein and Carsten Schneider},

refereed = {yes},

length = {25}

}

### 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}

}

### 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}

}