# Computer Algebra for Nested Sums and Products [FWF SFB F050-09]

### Project Lead

### Project Duration

01/03/2013 - 28/02/2021### Project URL

Go to Website## Partners

### The Austrian Science Fund (FWF)

## Software

### qFunctions

#### The qFunctions package is a Mathematica package for q-series and partition theory applications.

The qFunctions package by Jakob Ablinger and Ali K. Uncu is a Mathematica package for q-series and partition theory applications. This package includes both experimental and symbolic tools. The experimental set of elements includes guessers for q-shift equations and recurrences ...

Authors: Jakob Ablinger, Ali Uncu

More## Publications

### 2019

[Ablinger]

### Discovering and Proving Infinite Pochhammer Sum Identities

#### J. Ablinger

Experimental Mathematics, pp. 1-15. 2019. Taylor & Francis, 10.1080/10586458.2019.1627254. [url]@

author = {J. Ablinger},

title = {{Discovering and Proving Infinite Pochhammer Sum Identities}},

language = {english},

journal = {Experimental Mathematics},

pages = {1--15},

publisher = {Taylor & Francis},

isbn_issn = {?},

year = {2019},

note = {10.1080/10586458.2019.1627254},

refereed = {yes},

length = {15},

url = {https://doi.org/10.1080/10586458.2019.1627254}

}

**article**{RISC5896,author = {J. Ablinger},

title = {{Discovering and Proving Infinite Pochhammer Sum Identities}},

language = {english},

journal = {Experimental Mathematics},

pages = {1--15},

publisher = {Taylor & Francis},

isbn_issn = {?},

year = {2019},

note = {10.1080/10586458.2019.1627254},

refereed = {yes},

length = {15},

url = {https://doi.org/10.1080/10586458.2019.1627254}

}

[Ablinger]

### Proving two conjectural series for $\zeta(7)$ and discovering more series for $\zeta(7)$.

#### J. Ablinger

arXiv. Technical report, 2019. [url]@

author = {J. Ablinger},

title = {{Proving two conjectural series for $\zeta(7)$ and discovering more series for $\zeta(7)$.}},

language = {english},

year = {2019},

institution = {arXiv},

length = {5},

url = {https://arxiv.org/pdf/1908.06631.pdf}

}

**techreport**{RISC5968,author = {J. Ablinger},

title = {{Proving two conjectural series for $\zeta(7)$ and discovering more series for $\zeta(7)$.}},

language = {english},

year = {2019},

institution = {arXiv},

length = {5},

url = {https://arxiv.org/pdf/1908.06631.pdf}

}

[Berkovich]

### Refined q-Trinomial Coefficients and Two Infinite Hierarchies of q-Series Identities

#### Ali Kemal Uncu, Alexander Berkovich

ArXiv e-prints (accepted), pp. 1-10. 2019. N/A. [url]@

author = {Ali Kemal Uncu and Alexander Berkovich},

title = {{Refined q-Trinomial Coefficients and Two Infinite Hierarchies of q-Series Identities }},

language = {english},

abstract = {We will prove an identity involving refined q-trinomial coefficients. We then extend this identity to two infinite families of doubly bounded polynomial identities using transformation properties of the refined q-trinomials in an iterative fashion in the spirit of Bailey chains. One of these two hierarchies contains an identity which is equivalent to Capparelli's first Partition Theorem. },

journal = {ArXiv e-prints (accepted)},

pages = {1--10},

isbn_issn = {N/A},

year = {2019},

refereed = {yes},

length = {10},

url = {https://arxiv.org/abs/1810.12048}

}

**article**{RISC5801,author = {Ali Kemal Uncu and Alexander Berkovich},

title = {{Refined q-Trinomial Coefficients and Two Infinite Hierarchies of q-Series Identities }},

language = {english},

abstract = {We will prove an identity involving refined q-trinomial coefficients. We then extend this identity to two infinite families of doubly bounded polynomial identities using transformation properties of the refined q-trinomials in an iterative fashion in the spirit of Bailey chains. One of these two hierarchies contains an identity which is equivalent to Capparelli's first Partition Theorem. },

journal = {ArXiv e-prints (accepted)},

pages = {1--10},

isbn_issn = {N/A},

year = {2019},

refereed = {yes},

length = {10},

url = {https://arxiv.org/abs/1810.12048}

}

[Schneider]

### Evaluation of binomial double sums involving absolute values

#### C. Krattenthaler, C. Schneider

In: Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra, in Honour of Peter Paule on his 60th Birthday, V. Pillwein, C. Schneider (ed.), Texts and Monographs in Symbolic Computuation, in press , pp. 1-36. 2019. Springer, arXiv:1607.05314 [math.CO]. [url]@

author = {C. Krattenthaler and C. Schneider},

title = {{Evaluation of binomial double sums involving absolute values}},

booktitle = {{Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra, in Honour of Peter Paule on his 60th Birthday}},

language = {english},

series = {Texts and Monographs in Symbolic Computuation, in press},

pages = {1--36},

publisher = {Springer},

isbn_issn = {?},

year = {2019},

note = {arXiv:1607.05314 [math.CO]},

editor = {V. Pillwein and C. Schneider},

refereed = {yes},

length = {36},

url = {https://arxiv.org/abs/1607.05314}

}

**incollection**{RISC5970,author = {C. Krattenthaler and C. Schneider},

title = {{Evaluation of binomial double sums involving absolute values}},

booktitle = {{Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra, in Honour of Peter Paule on his 60th Birthday}},

language = {english},

series = {Texts and Monographs in Symbolic Computuation, in press},

pages = {1--36},

publisher = {Springer},

isbn_issn = {?},

year = {2019},

note = {arXiv:1607.05314 [math.CO]},

editor = {V. Pillwein and C. Schneider},

refereed = {yes},

length = {36},

url = {https://arxiv.org/abs/1607.05314}

}

[Schneider]

### Three loop QCD corrections to heavy quark form factors

#### J. Ablinger, J. Blümlein, P. Marquard, N. Rana, C. Schneider

In: Proceedings of ACAT 2019, to appear, pp. -. 2019. arXiv:1905.03728 [hep-ph]. [url]@

author = {J. Ablinger and J. Blümlein and P. Marquard and N. Rana and C. Schneider},

title = {{Three loop QCD corrections to heavy quark form factors}},

booktitle = {{Proceedings of ACAT 2019}},

language = {english},

volume = {to appear},

pages = {--},

isbn_issn = {?},

year = {2019},

note = {arXiv:1905.03728 [hep-ph]},

editor = {?},

refereed = {no},

length = {9},

url = {https://arxiv.org/abs/1905.03728}

}

**inproceedings**{RISC6004,author = {J. Ablinger and J. Blümlein and P. Marquard and N. Rana and C. Schneider},

title = {{Three loop QCD corrections to heavy quark form factors}},

booktitle = {{Proceedings of ACAT 2019}},

language = {english},

volume = {to appear},

pages = {--},

isbn_issn = {?},

year = {2019},

note = {arXiv:1905.03728 [hep-ph]},

editor = {?},

refereed = {no},

length = {9},

url = {https://arxiv.org/abs/1905.03728}

}

[Uncu]

### A Polynomial Identity Implying Schur's Partition Theorem

#### Ali Kemal Uncu

ArXiv e-prints (submitted), pp. 1-11. 2019. N/A. [url]@

author = {Ali Kemal Uncu},

title = {{A Polynomial Identity Implying Schur's Partition Theorem }},

language = {english},

abstract = {We propose and prove a new polynomial identity that implies Schur's partition theorem. We give combinatorial interpretations of some of our expressions in the spirit of Kurşungöz. We also present some related polynomial and q-series identities. },

journal = {ArXiv e-prints (submitted)},

pages = {1--11},

isbn_issn = {N/A},

year = {2019},

refereed = {yes},

length = {11},

url = {https://arxiv.org/abs/1903.01157}

}

**article**{RISC5898,author = {Ali Kemal Uncu},

title = {{A Polynomial Identity Implying Schur's Partition Theorem }},

language = {english},

abstract = {We propose and prove a new polynomial identity that implies Schur's partition theorem. We give combinatorial interpretations of some of our expressions in the spirit of Kurşungöz. We also present some related polynomial and q-series identities. },

journal = {ArXiv e-prints (submitted)},

pages = {1--11},

isbn_issn = {N/A},

year = {2019},

refereed = {yes},

length = {11},

url = {https://arxiv.org/abs/1903.01157}

}

[Uncu]

### qFunctions - A Mathematica package for q-series and partition theory applications

#### J. Ablinger, A. Uncu

arxiv. Technical report, 2019. [url]@

author = {J. Ablinger and A. Uncu},

title = {{qFunctions -- A Mathematica package for q-series and partition theory applications}},

language = {english},

year = {2019},

institution = {arxiv},

length = {17},

url = {https://arxiv.org/pdf/1910.12410.pdf}

}

**techreport**{RISC6005,author = {J. Ablinger and A. Uncu},

title = {{qFunctions -- A Mathematica package for q-series and partition theory applications}},

language = {english},

year = {2019},

institution = {arxiv},

length = {17},

url = {https://arxiv.org/pdf/1910.12410.pdf}

}

### 2018

[Ablinger]

### An Improved Method to Compute the Inverse Mellin Transform of Holonomic Sequences

#### J. Ablinger

In: Proceedings of "Loops and Legs in Quantum Field Theory - LL 2018, J. Blümlein and P. Marquard (ed.), PoS(LL2018) , pp. 1-10. 2018. ISSN 1824-8039. [url]@

author = {J. Ablinger},

title = {{An Improved Method to Compute the Inverse Mellin Transform of Holonomic Sequences}},

booktitle = {{Proceedings of "Loops and Legs in Quantum Field Theory - LL 2018}},

language = {english},

series = {PoS(LL2018)},

pages = {1--10},

isbn_issn = {ISSN 1824-8039},

year = {2018},

editor = {J. Blümlein and P. Marquard},

refereed = {yes},

length = {10},

url = {https://pos.sissa.it/303/063/pdf}

}

**inproceedings**{RISC5789,author = {J. Ablinger},

title = {{An Improved Method to Compute the Inverse Mellin Transform of Holonomic Sequences}},

booktitle = {{Proceedings of "Loops and Legs in Quantum Field Theory - LL 2018}},

language = {english},

series = {PoS(LL2018)},

pages = {1--10},

isbn_issn = {ISSN 1824-8039},

year = {2018},

editor = {J. Blümlein and P. Marquard},

refereed = {yes},

length = {10},

url = {https://pos.sissa.it/303/063/pdf}

}

[Berkovich]

### Elementary Polynomial Identities Involving q-Trinomial Coefficients

#### Ali Kemal Uncu, Alexander Berkovich

ArXiv e-prints (submitted), pp. -. 2018. N/A. [url]@

author = {Ali Kemal Uncu and Alexander Berkovich},

title = {{Elementary Polynomial Identities Involving q-Trinomial Coefficients }},

language = {english},

journal = {ArXiv e-prints (submitted)},

pages = {--},

isbn_issn = {N/A},

year = {2018},

refereed = {yes},

length = {0},

url = {https://arxiv.org/abs/1810.06497}

}

**article**{RISC5791,author = {Ali Kemal Uncu and Alexander Berkovich},

title = {{Elementary Polynomial Identities Involving q-Trinomial Coefficients }},

language = {english},

journal = {ArXiv e-prints (submitted)},

pages = {--},

isbn_issn = {N/A},

year = {2018},

refereed = {yes},

length = {0},

url = {https://arxiv.org/abs/1810.06497}

}

[Jiu]

### The Method of Brackets in Experimental Mathematics

#### Ivan Gonzalez, Karen Kohl, Lin Jiu, and Victor H. Moll

In: Frontiers in Orthogonal Polynomials and q-Series, Xin Li, Zuhair Nashed (ed.), pp. -. 2018. World Scientific Publishing, 978-981-3228-87-0. [url]@

author = {Ivan Gonzalez and Karen Kohl and Lin Jiu and and Victor H. Moll},

title = {{The Method of Brackets in Experimental Mathematics}},

booktitle = {{Frontiers in Orthogonal Polynomials and q-Series}},

language = {english},

pages = {--},

publisher = {World Scientific Publishing},

isbn_issn = {978-981-3228-87-0},

year = {2018},

editor = {Xin Li and Zuhair Nashed},

refereed = {no},

length = {0},

url = {http://www.worldscientific.com/worldscibooks/10.1142/10677}

}

**incollection**{RISC5497,author = {Ivan Gonzalez and Karen Kohl and Lin Jiu and and Victor H. Moll},

title = {{The Method of Brackets in Experimental Mathematics}},

booktitle = {{Frontiers in Orthogonal Polynomials and q-Series}},

language = {english},

pages = {--},

publisher = {World Scientific Publishing},

isbn_issn = {978-981-3228-87-0},

year = {2018},

editor = {Xin Li and Zuhair Nashed},

refereed = {no},

length = {0},

url = {http://www.worldscientific.com/worldscibooks/10.1142/10677}

}

[Middeke]

### Towards a Direct Method for Finding Hypergeometric Solutions of Linear First Order Recurrence Systems

#### J. Middeke, C. Schneider

To appear in ACM Communications in Computer Algebra, pp. -. 2018. ISSN 1932-2240. Extended abstract of the poster presentation at 43st International Symposium on Symbolic and Algebraic Computation (ISSAC'18). [pdf] [pdf]@

author = {J. Middeke and C.~Schneider},

title = {{Towards a Direct Method for Finding Hypergeometric Solutions of Linear First Order Recurrence Systems}},

language = {english},

journal = {To appear in ACM Communications in Computer Algebra},

pages = {--},

isbn_issn = {ISSN 1932-2240},

year = {2018},

note = {Extended abstract of the poster presentation at 43st International Symposium on Symbolic and Algebraic Computation (ISSAC'18)},

refereed = {yes},

length = {0}

}

**article**{RISC5736,author = {J. Middeke and C.~Schneider},

title = {{Towards a Direct Method for Finding Hypergeometric Solutions of Linear First Order Recurrence Systems}},

language = {english},

journal = {To appear in ACM Communications in Computer Algebra},

pages = {--},

isbn_issn = {ISSN 1932-2240},

year = {2018},

note = {Extended abstract of the poster presentation at 43st International Symposium on Symbolic and Algebraic Computation (ISSAC'18)},

refereed = {yes},

length = {0}

}

### 2017

[Ablinger]

### Discovering and Proving Infinite Binomial Sums Identities

#### J. Ablinger

Experimental Mathematics 26(1), pp. 62-71. 2017. ISSN 1058-6458. 10.1080/10586458.2015.1116028. [url]@

author = {J. Ablinger},

title = {{Discovering and Proving Infinite Binomial Sums Identities}},

language = {english},

journal = {Experimental Mathematics},

volume = {26},

number = {1},

pages = {62--71},

isbn_issn = {ISSN 1058-6458},

year = {2017},

note = {10.1080/10586458.2015.1116028},

refereed = {yes},

length = {10},

url = {http://arxiv.org/abs/1507.01703}

}

**article**{RISC5159,author = {J. Ablinger},

title = {{Discovering and Proving Infinite Binomial Sums Identities}},

language = {english},

journal = {Experimental Mathematics},

volume = {26},

number = {1},

pages = {62--71},

isbn_issn = {ISSN 1058-6458},

year = {2017},

note = {10.1080/10586458.2015.1116028},

refereed = {yes},

length = {10},

url = {http://arxiv.org/abs/1507.01703}

}

[Ablinger]

### Computing the Inverse Mellin Transform of Holonomic Sequences using Kovacic's Algorithm

#### J. Ablinger

In: Proc. of the 13th International Symposium on Radiative Corrections (Applications of Quantum Field Theory to Phenomenology), A. Hoang and C. Schneider (ed.)PoS (RADCOR2017) 069, pp. 1-8. 2017. ISSN 1824-8039. arXiv:1801.01039 [cs.SC]. [url]@

author = {J. Ablinger},

title = {{Computing the Inverse Mellin Transform of Holonomic Sequences using Kovacic's Algorithm}},

booktitle = {{Proc. of the 13th International Symposium on Radiative Corrections (Applications of Quantum Field Theory to Phenomenology)}},

language = {english},

volume = {PoS (RADCOR2017) 069},

pages = {1--8},

isbn_issn = {ISSN 1824-8039},

year = {2017},

note = {arXiv:1801.01039 [cs.SC]},

editor = {A. Hoang and C. Schneider},

refereed = {no},

length = {8},

url = {https://pos.sissa.it/290/001/pdf}

}

**inproceedings**{RISC5527,author = {J. Ablinger},

title = {{Computing the Inverse Mellin Transform of Holonomic Sequences using Kovacic's Algorithm}},

booktitle = {{Proc. of the 13th International Symposium on Radiative Corrections (Applications of Quantum Field Theory to Phenomenology)}},

language = {english},

volume = {PoS (RADCOR2017) 069},

pages = {1--8},

isbn_issn = {ISSN 1824-8039},

year = {2017},

note = {arXiv:1801.01039 [cs.SC]},

editor = {A. Hoang and C. Schneider},

refereed = {no},

length = {8},

url = {https://pos.sissa.it/290/001/pdf}

}

[Hasselhuhn]

### Three Loop Massive Operator Matrix Elements and Asymptotic Wilson Coefficients with Two Different Masses

#### J. Ablinger, J. Blümlein, A. De Freitas, A. Hasselhuhn, C. Schneider, F. Wißbrock

Nucl. Phys. B. 921, pp. 585-688. 2017. ISSN 0550-3213. arXiv:1705.07030 [hep-ph]. [url]@

author = {J. Ablinger and J. Blümlein and A. De Freitas and A. Hasselhuhn and C. Schneider and F. Wißbrock},

title = {{Three Loop Massive Operator Matrix Elements and Asymptotic Wilson Coefficients with Two Different Masses}},

language = {english},

journal = {Nucl. Phys. B.},

volume = {921},

pages = {585--688},

isbn_issn = {ISSN 0550-3213},

year = {2017},

note = {arXiv:1705.07030 [hep-ph]},

refereed = {yes},

length = {99},

url = {https://doi.org/10.1016/j.nuclphysb.2017.12.018}

}

**article**{RISC5451,author = {J. Ablinger and J. Blümlein and A. De Freitas and A. Hasselhuhn and C. Schneider and F. Wißbrock},

title = {{Three Loop Massive Operator Matrix Elements and Asymptotic Wilson Coefficients with Two Different Masses}},

language = {english},

journal = {Nucl. Phys. B.},

volume = {921},

pages = {585--688},

isbn_issn = {ISSN 0550-3213},

year = {2017},

note = {arXiv:1705.07030 [hep-ph]},

refereed = {yes},

length = {99},

url = {https://doi.org/10.1016/j.nuclphysb.2017.12.018}

}

[Jiu]

### Integral representations of equally positive integer-indexed harmonic sums at infinity

#### L. Jiu

Research in Number Theory 3(10), pp. 1-4. 2017. 2363-9555. [url]@

author = {L. Jiu},

title = {{Integral representations of equally positive integer-indexed harmonic sums at infinity}},

language = {English},

abstract = {We identify a partition-theoretic generalization of Riemann zeta function and the equally positive integer-indexed harmonic sums at infinity, to obtain the generating function and the integral representations of the latter. The special cases coincide with zeta values at positive integer arguments.},

journal = {Research in Number Theory},

volume = {3},

number = {10},

pages = {1--4},

isbn_issn = {2363-9555},

year = {2017},

refereed = {no},

length = {4},

url = {https://resnumtheor.springeropen.com/articles/10.1007/s40993-017-0074-x}

}

**article**{RISC5385,author = {L. Jiu},

title = {{Integral representations of equally positive integer-indexed harmonic sums at infinity}},

language = {English},

abstract = {We identify a partition-theoretic generalization of Riemann zeta function and the equally positive integer-indexed harmonic sums at infinity, to obtain the generating function and the integral representations of the latter. The special cases coincide with zeta values at positive integer arguments.},

journal = {Research in Number Theory},

volume = {3},

number = {10},

pages = {1--4},

isbn_issn = {2363-9555},

year = {2017},

refereed = {no},

length = {4},

url = {https://resnumtheor.springeropen.com/articles/10.1007/s40993-017-0074-x}

}

[Jiu]

### An extension of the method of brackets. Part 1

#### Ivan Gonzalez, Karen Kohl, Lin Jiu, and Victor H. Moll

Open Mathematics (formerly Central European Journal of Mathematics) 15, pp. 1181-1211. 2017. 2391-5455. [url]@

author = {Ivan Gonzalez and Karen Kohl and Lin Jiu and and Victor H. Moll},

title = {{An extension of the method of brackets. Part 1}},

language = {english},

abstract = {The method of brackets is an efficient method for the evaluation of a large class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the concepts of null and divergent series. These are formal representations of functions, whose coefficients $a_n$ have meromorphic representations for $n\in\mathbb{C}$, but might vanish or blow up when $n\in\mathbb{N}$. These ideas are illustrated with the evaluation of a variety of entries from the classical table of integrals by Gradshteyn and Ryzhik.},

journal = {Open Mathematics (formerly Central European Journal of Mathematics)},

volume = {15},

pages = {1181--1211},

isbn_issn = {2391-5455},

year = {2017},

refereed = {no},

length = {31},

url = {https://www.degruyter.com/view/j/math.2017.15.issue-1/math-2017-0100/math-2017-0100.xml?format=INT}

}

**article**{RISC5483,author = {Ivan Gonzalez and Karen Kohl and Lin Jiu and and Victor H. Moll},

title = {{An extension of the method of brackets. Part 1}},

language = {english},

abstract = {The method of brackets is an efficient method for the evaluation of a large class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the concepts of null and divergent series. These are formal representations of functions, whose coefficients $a_n$ have meromorphic representations for $n\in\mathbb{C}$, but might vanish or blow up when $n\in\mathbb{N}$. These ideas are illustrated with the evaluation of a variety of entries from the classical table of integrals by Gradshteyn and Ryzhik.},

journal = {Open Mathematics (formerly Central European Journal of Mathematics)},

volume = {15},

pages = {1181--1211},

isbn_issn = {2391-5455},

year = {2017},

refereed = {no},

length = {31},

url = {https://www.degruyter.com/view/j/math.2017.15.issue-1/math-2017-0100/math-2017-0100.xml?format=INT}

}

[Middeke]

### Denominator Bounds and Polynomial Solutions for Systems of q-Recurrences over K(t) for Constant K

#### Johannes Middeke

In: Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation, Michael Burr (ed.), Proceedings of International Symposium on Symbolic and Algebraic Computation, pp. 325-332. 2017. 978-1-4503-5064-8.@

author = {Johannes Middeke},

title = {{Denominator Bounds and Polynomial Solutions for Systems of q-Recurrences over K(t) for Constant K}},

booktitle = {{Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation}},

language = {english},

abstract = {We consider systems A_ell(t )y(q^ell t ) + . . . + A 0 (t )y(t ) = b (t ) of higher order q-recurrence equations with rational coefficients. We extend a method for finding a bound on the maximal power of t in the denominator of arbitrary rational solutions y(t ) as well as a method for bounding the degree of polynomial solutions from the scalar case to the systems case. The approach is direct and does not rely on uncoupling or reduction to a first order system. Unlike in the scalar case this usually requires an initial transformation of the system.},

pages = {325--332},

isbn_issn = {978-1-4503-5064-8},

year = {2017},

editor = {Michael Burr},

refereed = {yes},

length = {7},

conferencename = {International Symposium on Symbolic and Algebraic Computation}

}

**inproceedings**{RISC5489,author = {Johannes Middeke},

title = {{Denominator Bounds and Polynomial Solutions for Systems of q-Recurrences over K(t) for Constant K}},

booktitle = {{Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation}},

language = {english},

abstract = {We consider systems A_ell(t )y(q^ell t ) + . . . + A 0 (t )y(t ) = b (t ) of higher order q-recurrence equations with rational coefficients. We extend a method for finding a bound on the maximal power of t in the denominator of arbitrary rational solutions y(t ) as well as a method for bounding the degree of polynomial solutions from the scalar case to the systems case. The approach is direct and does not rely on uncoupling or reduction to a first order system. Unlike in the scalar case this usually requires an initial transformation of the system.},

pages = {325--332},

isbn_issn = {978-1-4503-5064-8},

year = {2017},

editor = {Michael Burr},

refereed = {yes},

length = {7},

conferencename = {International Symposium on Symbolic and Algebraic Computation}

}

[Raab]

### Iterative and Iterative-Noniterative Integral Solutions in 3-Loop Massive QCD Calculations

#### J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, E. Imamoglu, M. van Hoeij A. von Manteuffel, C.G. Raab, C.-S. Radu, C. Schneider

In: Proc. of the 13th International Symposium on Radiative Corrections (Applications of Quantum Field Theory to Phenomenology), A. Hoang, C. Schneider (ed.)PoS (RADCOR2017) 069, pp. 1-13. 2017. ISSN 1824-8039. arXiv:1711.09742 [hep-ph]. [url]@

author = {J. Ablinger and A. Behring and J. Blümlein and A. De Freitas and E. Imamoglu and M. van Hoeij A. von Manteuffel and C.G. Raab and C.-S. Radu and C. Schneider},

title = {{Iterative and Iterative-Noniterative Integral Solutions in 3-Loop Massive QCD Calculations}},

booktitle = {{Proc. of the 13th International Symposium on Radiative Corrections (Applications of Quantum Field Theory to Phenomenology)}},

language = {english},

volume = {PoS (RADCOR2017) 069},

pages = {1--13},

isbn_issn = {ISSN 1824-8039},

year = {2017},

note = {arXiv:1711.09742 [hep-ph]},

editor = {A. Hoang and C. Schneider},

refereed = {no},

length = {13},

url = {https://pos.sissa.it/290/069/pdf}

}

**inproceedings**{RISC5350,author = {J. Ablinger and A. Behring and J. Blümlein and A. De Freitas and E. Imamoglu and M. van Hoeij A. von Manteuffel and C.G. Raab and C.-S. Radu and C. Schneider},

title = {{Iterative and Iterative-Noniterative Integral Solutions in 3-Loop Massive QCD Calculations}},

booktitle = {{Proc. of the 13th International Symposium on Radiative Corrections (Applications of Quantum Field Theory to Phenomenology)}},

language = {english},

volume = {PoS (RADCOR2017) 069},

pages = {1--13},

isbn_issn = {ISSN 1824-8039},

year = {2017},

note = {arXiv:1711.09742 [hep-ph]},

editor = {A. Hoang and C. Schneider},

refereed = {no},

length = {13},

url = {https://pos.sissa.it/290/069/pdf}

}

[Schneider]

### Summation Theory II: Characterizations of $R\Pi\Sigma$-extensions and algorithmic aspects

#### C. Schneider

J. Symb. Comput. 80(3), pp. 616-664. 2017. ISSN 0747-7171. arXiv:1603.04285 [cs.SC]. [url]@

author = {C. Schneider},

title = {{Summation Theory II: Characterizations of $R\Pi\Sigma$-extensions and algorithmic aspects}},

language = {english},

journal = {J. Symb. Comput.},

volume = {80},

number = {3},

pages = {616--664},

isbn_issn = {ISSN 0747-7171},

year = {2017},

note = {arXiv:1603.04285 [cs.SC]},

refereed = {yes},

length = {54},

url = {http://arxiv.org/abs/1603.04285}

}

**article**{RISC5270,author = {C. Schneider},

title = {{Summation Theory II: Characterizations of $R\Pi\Sigma$-extensions and algorithmic aspects}},

language = {english},

journal = {J. Symb. Comput.},

volume = {80},

number = {3},

pages = {616--664},

isbn_issn = {ISSN 0747-7171},

year = {2017},

note = {arXiv:1603.04285 [cs.SC]},

refereed = {yes},

length = {54},

url = {http://arxiv.org/abs/1603.04285}

}

[Schneider]

### Asymptotic and exact results on the complexity of the Novelli-Pak-Stoyanovskii algorithm

#### C. Schneider, R. Sulzgruber

Electron. J. Combin. 24(2), pp. 1-33. 2017. ISSN: 1077-8926. #P2.28, arXiv:1606.07597. [url]@

author = {C. Schneider and R. Sulzgruber},

title = {{Asymptotic and exact results on the complexity of the Novelli--Pak--Stoyanovskii algorithm}},

language = {english},

journal = {Electron. J. Combin. },

volume = {24},

number = {2},

pages = {1--33},

isbn_issn = {ISSN: 1077-8926},

year = {2017},

note = {#P2.28, arXiv:1606.07597},

refereed = {yes},

length = {25},

url = {http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i2p28}

}

**article**{RISC5318,author = {C. Schneider and R. Sulzgruber},

title = {{Asymptotic and exact results on the complexity of the Novelli--Pak--Stoyanovskii algorithm}},

language = {english},

journal = {Electron. J. Combin. },

volume = {24},

number = {2},

pages = {1--33},

isbn_issn = {ISSN: 1077-8926},

year = {2017},

note = {#P2.28, arXiv:1606.07597},

refereed = {yes},

length = {25},

url = {http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i2p28}

}