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

### Project Lead

### Project Duration

01/03/2013 - 31/08/2021### Project URL

Go to Website## Partners

### The Austrian Science Fund (FWF)

## Software

A Mathematica package based on Sigma that tries to evaluate automatically multi-sums to expressions in terms of indefinite nested sums defined over (q-)hypergeometric products. ...

### PLDESolver

#### The PLDESolver package is a Mathematica package to find solutions of parameterized linear difference equations in difference rings.

The PLDESolver package by Jakob Ablinger and Carsten Schneider is a Mathematica package that allows to compute solutions of non-degenerated linear difference operators in difference rings with zero-divisors by reducing it to finding solutions in difference rings that are integral ...

### 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 ...

## Publications

### 2021

### On Rational and Hypergeometric Solutions of Linear Ordinary Difference Equations in ΠΣ∗-field extensions

#### Sergei A. Abramov, Manuel Bronstein, Marko Petkovšek, Carsten Schneider

J. Symb. Comput. 107, pp. 23-66. 2021. ISSN 0747-7171. arXiv:2005.04944 [cs.SC]. [url]**article**{RISC6224,

author = {Sergei A. Abramov and Manuel Bronstein and Marko Petkovšek and Carsten Schneider},

title = {{On Rational and Hypergeometric Solutions of Linear Ordinary Difference Equations in ΠΣ∗-field extensions}},

language = {english},

journal = {J. Symb. Comput.},

volume = {107},

pages = {23--66},

isbn_issn = {ISSN 0747-7171},

year = {2021},

note = {arXiv:2005.04944 [cs.SC]},

refereed = {yes},

length = {44},

url = {https://doi.org/10.1016/j.jsc.2021.01.002}

}

### Term Algebras, Canonical Representations and Difference Ring Theory for Symbolic Summation

#### C. Schneider

In: Anti-Differentiation and the Calculation of Feynman Amplitudes, J. Blümlein and C. Schneider (ed.), Texts and Monographs in Symbolic Computuation to appear, pp. ?-?. 2021. Springer, arXiv:2102.01471 [cs.SC], RISC-Linz Report Series No. 21-03. [url]**incollection**{RISC6287,

author = {C. Schneider},

title = {{Term Algebras, Canonical Representations and Difference Ring Theory for Symbolic Summation}},

booktitle = {{Anti-Differentiation and the Calculation of Feynman Amplitudes}},

language = {english},

series = {Texts and Monographs in Symbolic Computuation},

volume = {to appear},

pages = {?--?},

publisher = {Springer},

isbn_issn = {?},

year = {2021},

note = {arXiv:2102.01471 [cs.SC], RISC-Linz Report Series No. 21-03},

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

refereed = {yes},

length = {55},

url = {https://arxiv.org/abs/2102.01471}

}

### Iterated integrals over letters induced by quadratic forms

#### J. Ablinger, J. Blümlein, C. Schneider

Physical Review D to appear, pp. -. 2021. ISSN 2470-0029. arXiv:2103.08330 [hep-th]. [url]**article**{RISC6294,

author = {J. Ablinger and J. Blümlein and C. Schneider},

title = {{Iterated integrals over letters induced by quadratic forms}},

language = {english},

journal = {Physical Review D },

volume = {to appear},

pages = {--},

isbn_issn = {ISSN 2470-0029},

year = {2021},

note = {arXiv:2103.08330 [hep-th]},

refereed = {yes},

length = {14},

url = {https://arxiv.org/abs/2103.08330}

}

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

#### J. Ablinger, A. Uncu

Journal of Symbolic Computation 107, pp. 145-166. 2021. ISSN 0747-7171. arXiv:1910.12410. [url]**article**{RISC6299,

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

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

language = {english},

journal = {Journal of Symbolic Computation},

volume = {107},

pages = {145--166},

isbn_issn = {ISSN 0747-7171},

year = {2021},

note = {arXiv:1910.12410},

refereed = {yes},

length = {22},

url = {https://doi.org/10.1016/j.jsc.2021.02.003}

}

### 2020

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

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

In: Proc. ACAT 2019, , J. Phys.: Conf. Ser. 1525012018, pp. 1-10. 2020. ISSN 1742-6596. arXiv:1905.03728 [hep-ph], doi:10.1088/1742-6596/1525/1/012018. [url]**inproceedings**{RISC4826,

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 = {{Proc. ACAT 2019}},

language = {english},

series = {J. Phys.: Conf. Ser.},

volume = {1525},

number = {012018},

pages = {1--10},

isbn_issn = {ISSN 1742-6596},

year = {2020},

note = {arXiv:1905.03728 [hep-ph], doi:10.1088/1742-6596/1525/1/012018},

editor = {?},

refereed = {yes},

length = {10},

url = {https://iopscience.iop.org/article/10.1088/1742-6596/1525/1/012018/}

}

### 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 , pp. 249-295. 2020. Springer, ISBN 978-3-030-44558-4. arXiv:1607.05314 [math.CO]. [url]**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},

pages = {249--295},

publisher = {Springer},

isbn_issn = {ISBN 978-3-030-44558-4},

year = {2020},

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

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

refereed = {yes},

length = {47},

url = {https://doi.org/10.1007/978-3-030-44559-1_14}

}

### Representation of hypergeometric products of higher nesting depths in difference rings

#### E.D. Ocansey, C. Schneider

Technical report no. 20-19 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2020. arXiv:2011.08775 [cs.SC]. [url] [pdf]**techreport**{RISC6221,

author = {E.D. Ocansey and C. Schneider},

title = {{Representation of hypergeometric products of higher nesting depths in difference rings}},

language = {english},

number = {20-19},

year = {2020},

note = {arXiv:2011.08775 [cs.SC]},

length = {48},

url = {https://arxiv.org/abs/2011.08775},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### 2019

### Discovering and Proving Infinite Pochhammer Sum Identities

#### J. Ablinger

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

}

### 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]**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}

}

### Difference Ring Algorithms for Nested Products

#### Evans Doe Ocansey

RISC, Johannes Kepler University. PhD Thesis. November 2019. [pdf]**phdthesis**{RISC6199,

author = {Evans Doe Ocansey},

title = {{Difference Ring Algorithms for Nested Products}},

language = {english},

year = {2019},

month = {November},

translation = {0},

school = {RISC, Johannes Kepler University},

length = {178}

}

### Difference Ring Algorithms for Nested Products

#### Evans Doe Ocansey

Technical report no. 20-08 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. November 2019.**techreport**{RISC6200,

author = {Evans Doe Ocansey},

title = {{Difference Ring Algorithms for Nested Products}},

language = {english},

number = {20-08},

year = {2019},

month = {November},

length = {178},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

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

#### Ali Kemal Uncu

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

}

### 2018

### 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]**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}

}

### Elementary Polynomial Identities Involving q-Trinomial Coefficients

#### Ali Kemal Uncu, Alexander Berkovich

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

}

### 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]**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}

}

### 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]**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}

}

### 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. 2018. ISSN 1824-8039. arXiv:1711.09742 [hep-ph]. [url]**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 = {2018},

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

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

refereed = {no},

length = {13},

url = {https://doi.org/10.22323/1.290.0069 }

}

### Heavy Flavor Wilson Coefficients in Deep-Inelastic Scattering: Recent Results

#### J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, C. Schneider

In: Proceedings of QCDEV2017, L. Gamberg, A. Prokudin, J.W. Qiu and A. Radyushkin (ed.)PoS(QCDEV2017)031 , pp. 1-10. 2018. ISSN 1824-8039. arXiv:1711.07957 [hep-ph]. [url]**inproceedings**{RISC5506,

author = {J. Ablinger and A. Behring and J. Blümlein and A. De Freitas and A. von Manteuffel and C. Schneider},

title = {{Heavy Flavor Wilson Coefficients in Deep-Inelastic Scattering: Recent Results}},

booktitle = {{Proceedings of QCDEV2017}},

language = {english},

volume = {PoS(QCDEV2017)031 },

pages = {1--10},

isbn_issn = {ISSN 1824-8039},

year = {2018},

note = {arXiv:1711.07957 [hep-ph]},

editor = {L. Gamberg and A. Prokudin and J.W. Qiu and A. Radyushkin},

refereed = {no},

length = {10},

url = {https://doi.org/10.22323/1.308.0031 }

}

### The massive 3-loop operator matrix elements with two masses and the generalized variable flavor number scheme

#### J. Ablinger, J. Blümlein, A. De Freitas, A. Goedicke, C. Schneider, K. Schönwald, F. Wißbrock

In: Proc. of the 13th International Symposium on Radiative Corrections (Applications of Quantum Field Theory to Phenomenology), A. Hoang, C. Schneider (ed.)PoS (RADCOR2017) 071, pp. 1-20. 2018. ISSN 1824-8039. arXiv:1712.00745 [hep-ph]. [url]**inproceedings**{RISC5513,

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

title = {{The massive 3-loop operator matrix elements with two masses and the generalized variable flavor number scheme}},

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

language = {english},

volume = {PoS (RADCOR2017) 071},

pages = {1--20},

isbn_issn = {ISSN 1824-8039},

year = {2018},

note = { arXiv:1712.00745 [hep-ph]},

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

refereed = {no},

length = {20},

url = {https://doi.org/10.22323/1.290.0071 }

}

### Special functions, transcendentals and their numerics

#### Jakob Ablinger, Johannes Blümlein, Mark Round, Carsten 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) 010, pp. 1-7. 2018. ISSN 1824-8039. arXiv:1712.08541 [hep-th]. [url]**inproceedings**{RISC5521,

author = {Jakob Ablinger and Johannes Blümlein and Mark Round and Carsten Schneider},

title = {{Special functions, transcendentals and their numerics}},

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

language = {english},

volume = {PoS (RADCOR2017) 010},

pages = {1--7},

isbn_issn = {ISSN 1824-8039},

year = {2018},

note = { arXiv:1712.08541 [hep-th]},

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

refereed = {no},

length = {7},

url = {https://doi.org/10.22323/1.290.0010 }

}