# 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

### 2024

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

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

J. Symb. Comput. 120, pp. 1-50. 2024. ISSN: 0747-7171. arXiv:2011.08775 [cs.SC]. [doi]**article**{RISC6688,

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

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

language = {english},

journal = {J. Symb. Comput.},

volume = {120},

pages = {1--50},

isbn_issn = {ISSN: 0747-7171},

year = {2024},

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

refereed = {yes},

length = {50},

url = {https://doi.org/10.1016/j.jsc.2023.03.002}

}

### 2023

### Hypergeometric Structures in Feynman Integrals

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

Annals of Mathematics and Artificial Intelligence 91(5), pp. 591-649. 2023. ISSN 1573-7470. arXiv:2111.15501 [math-ph]. [doi]**article**{RISC6643,

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.},

journal = {Annals of Mathematics and Artificial Intelligence},

volume = { 91},

number = {5},

pages = {591--649},

isbn_issn = {ISSN 1573-7470},

year = {2023},

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

refereed = {yes},

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

length = {59},

url = {https://doi.org/10.1007/s10472-023-09831-8}

}

### Error bounds for the asymptotic expansion of the partition function

#### Koustav Banerjee, Peter Paule, Cristian-Silviu Radu, Carsten Schneider

Rocky Mt J Math to appear, pp. ?-?. 2023. ISSN: 357596. arXiv:2209.07887 [math.NT]. [doi]**article**{RISC6710,

author = {Koustav Banerjee and Peter Paule and Cristian-Silviu Radu and Carsten Schneider},

title = {{Error bounds for the asymptotic expansion of the partition function}},

language = {english},

abstract = {Asymptotic study on the partition function $p(n)$ began with the work of Hardy and Ramanujan. Later Rademacher obtained a convergent series for $p(n)$ and an error bound was given by Lehmer. Despite having this, a full asymptotic expansion for $p(n)$ with an explicit error bound is not known. Recently O'Sullivan studied the asymptotic expansion of $p^{k}(n)$-partitions into $k$th powers, initiated by Wright, and consequently obtained an asymptotic expansion for $p(n)$ along with a concise description of the coefficients involved in the expansion but without any estimation of the error term. Here we consider a detailed and comprehensive analysis on an estimation of the error term obtained by truncating the asymptotic expansion for $p(n)$ at any positive integer $n$. This gives rise to an infinite family of inequalities for $p(n)$ which finally answers to a question proposed by Chen. Our error term estimation predominantly relies on applications of algorithmic methods from symbolic summation. },

journal = {Rocky Mt J Math },

volume = {to appear},

pages = {?--?},

isbn_issn = {ISSN: 357596},

year = {2023},

note = {arXiv:2209.07887 [math.NT]},

refereed = {yes},

keywords = {partition function, asymptotic expansion, error bounds, symbolic summation},

length = {43},

url = {https://doi.org/10.35011/risc.22-13}

}

### 2022

### New 2– and 3–loop heavy flavor corrections to unpolarized and polarized deep-inelastic scattering

#### J. Ablinger, J. Blümlein, A. De Freitas, M. Saragnese, C. Schneider, K. Schönwald

SciPost Phys. Proc.(8), pp. 137.1-137.15. 2022. ISSN 2666-4003. DIS2021, arXiv:2107.09350 [hep-ph]. [doi]**article**{RISC6497,

author = {J. Ablinger and J. Blümlein and A. De Freitas and M. Saragnese and C. Schneider and K. Schönwald},

title = {{New 2– and 3–loop heavy flavor corrections to unpolarized and polarized deep-inelastic scattering}},

language = {english},

abstract = {A survey is given on the new 2-- and 3--loop results for the heavy flavor contributions to deep--inelastic scattering in the unpolarized and the polarized case. We also discuss related new mathematical aspectsapplied in these calculations.},

journal = {SciPost Phys. Proc.},

number = {8},

pages = {137.1--137.15},

isbn_issn = {ISSN 2666-4003},

year = {2022},

note = {DIS2021, arXiv:2107.09350 [hep-ph]},

refereed = {yes},

length = {15},

url = {https://www.doi.org/10.21468/SciPostPhysProc.8.137}

}

### The Two-Loop Massless Off-Shell QCD Operator Matrix Elements to Finite Terms

#### J. Blümlein, P. Marquard, C. Schneider, K. Schönwald

Nuclear Physics B 980, pp. 1-131. 2022. ISSN 0550-3213. arXiv:2202.03216 [hep-ph]. [doi]**article**{RISC6527,

author = {J. Blümlein and P. Marquard and C. Schneider and K. Schönwald},

title = {{The Two-Loop Massless Off-Shell QCD Operator Matrix Elements to Finite Terms}},

language = {english},

abstract = {We calculate the unpolarized and polarized two--loop massless off--shell operator matrix elements in QCD to $O(ep)$ in the dimensional parameter in an automated way. Here we use the method of arbitrary high Mellin moments and difference ring theory, based on integration-by-parts relations. This method also constitutes one way to compute the QCD anomalous dimensions. The presented higher order contributions to these operator matrix elements occur as building blocks in the corresponding higher order calculations upto four--loop order. All contributing quantities can be expressed in terms of harmonic sums in Mellin--$N$ space or by harmonic polylogarithms in $z$--space. We also perform comparisons to the literature. },

journal = {Nuclear Physics B},

volume = {980},

pages = {1--131},

isbn_issn = {ISSN 0550-3213},

year = {2022},

note = {arXiv:2202.03216 [hep-ph]},

refereed = {yes},

keywords = {QCD, Operator Matrix Element, 2-loop Feynman diagrams, computer algebra, large moment method},

length = {131},

url = {https://www.doi.org/10.1016/j.nuclphysb.2022.115794}

}

### The 3-loop anomalous dimensions from off-shell operator matrix elements

#### J. Blümlein, P. Marquard, C. Schneider, K. Schönwald

In: Proceedings of Loops and Legs in Quantum Field Theory, PoS(LL2022) 48, P. Marquard, M. Steinhauser (ed.)416, pp. 1-12. July 2022. ISSN 1824-8039. arXiv:2207.07943 [hep-ph]. [doi]**inproceedings**{RISC6528,

author = {J. Blümlein and P. Marquard and C. Schneider and K. Schönwald},

title = {{The 3-loop anomalous dimensions from off-shell operator matrix elements}},

booktitle = {{Proceedings of Loops and Legs in Quantum Field Theory, PoS(LL2022) 48}},

language = {english},

abstract = {We report on the calculation of the three--loop polarized and unpolarized flavor non--singlet and the polarized singlet anomalous dimensions using massless off--shell operator matrix elements in a gauge--variant framework. We also reconsider the unpolarized two--loop singlet anomalous dimensions and correct errors in the foregoing literature.},

volume = {416},

pages = {1--12},

isbn_issn = {ISSN 1824-8039},

year = {2022},

month = {July},

note = { arXiv:2207.07943 [hep-ph]},

editor = {P. Marquard and M. Steinhauser},

refereed = {no},

keywords = {hypergeometric series, Appell-series, symbolic summation, coupled systems, partial linear difference equations},

length = {12},

url = {https://doi.org/10.22323/1.416.0048 }

}

### Computer Algebra and Hypergeometric Structures for Feynman Integrals

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

In: Proceedings of Loops and Legs in Quantum Field Theory, PoS(LL2022) 041, P. Marquard, M. Steinhauser (ed.)416, pp. 1-11. 2022. ISSN 1824-8039. arXiv:2207.08524 [math-ph]. [doi]**inproceedings**{RISC6619,

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

title = {{Computer Algebra and Hypergeometric Structures for Feynman Integrals}},

booktitle = {{Proceedings of Loops and Legs in Quantum Field Theory, PoS(LL2022) 041}},

language = {english},

abstract = {We present recent computer algebra methods that support the calculations of (multivariate) series solutions for (certain coupled systems of partial) linear differential equations. The summand of the series solutions may be built by hypergeometric products and more generally by indefinite nested sums defined over such products. Special cases are hypergeometric structures such as Appell-functions or generalizations of them that arise frequently when dealing with parameter Feynman integrals.},

volume = {416},

pages = {1--11},

isbn_issn = {ISSN 1824-8039},

year = {2022},

note = { arXiv:2207.08524 [math-ph]},

editor = {P. Marquard and M. Steinhauser},

refereed = {no},

keywords = {hypergeometric series, Appell-series, symbolic summation, coupled systems, partial linear difference equations},

length = {11},

url = {https://doi.org/10.22323/1.416.0041 }

}

### 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}

}

### 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}

}

### 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}

}