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

### Project Lead

### Project Duration

01/03/2013 - 31/07/2022### 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]. [doi] [pdf]**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 , pp. 423-485. 2021. Springer, ISBN 978-3-030-80218-9. arXiv:2102.01471 [cs.SC], RISC-Linz Report Series No. 21-03. [doi] [pdf]**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},

pages = {423--485},

publisher = {Springer},

isbn_issn = {ISBN 978-3-030-80218-9},

year = {2021},

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

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

refereed = {yes},

length = {63},

url = {https://doi.org/10.1007/978-3-030-80219-6_17}

}

### Iterated integrals over letters induced by quadratic forms

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

Physical Review D 103(9), pp. 096025-096035. 2021. ISSN 2470-0029. arXiv:2103.08330 [hep-th]. [doi]**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 = {103},

number = {9},

pages = {096025--096035},

isbn_issn = {ISSN 2470-0029},

year = {2021},

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

refereed = {yes},

length = {11},

url = {https://www.doi.org/10.1103/PhysRevD.103.096025}

}

### Hypergeometric Structures in Feynman Integrals

#### J. Blümlein, C. Schneider, M. Saragnese

Technical report no. 21-17 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). December 2021. arXiv:2111.15501 [math-ph]. Licensed under CC BY 4.0 International. [doi] [pdf]**techreport**{RISC6380,

author = {J. Blümlein and C. Schneider and M. Saragnese},

title = {{Hypergeometric Structures in Feynman Integrals}},

language = {english},

abstract = {Hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package {tt Sigma} in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code {tt HypSeries} transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code {tt solvePartialLDE} is designed. Generalized hypergeometric functions, Appell-,~Kamp'e de F'eriet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton--type functions are considered. We illustrate the algorithms by examples.},

number = {21-17},

year = {2021},

month = {December},

note = {arXiv:2111.15501 [math-ph]},

keywords = {hypergeometric functions, symbolic summation, expansion, partial linear difference equations, partial linear differential equations},

length = {55},

license = {CC BY 4.0 International},

type = {RISC Report Series},

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

address = {Altenberger Straße 69, 4040 Linz, Austria},

issn = {2791-4267 (online)}

}

### 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. [doi]**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]**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]},

editor = {?},

refereed = {yes},

length = {10},

url = {https://www.doi.org/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]. [doi] [pdf]**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, Austria. ISSN 2791-4267 (online). 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},

type = {RISC Report Series},

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

address = {Altenberger Straße 69, 4040 Linz, Austria},

issn = {2791-4267 (online)}

}

### 2019

### Discovering and Proving Infinite Pochhammer Sum Identities

#### J. Ablinger

Experimental Mathematics, pp. 1-15. 2019. Taylor & Francis, 10.1080/10586458.2019.1627254. [doi]**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, Austria. ISSN 2791-4267 (online). 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 = {Altenberger Straße 69, 4040 Linz, Austria},

issn = {2791-4267 (online)}

}

### 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}

}

### 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]. [doi]**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]. [doi]**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]. [doi]**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]. [doi]**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 }

}