# Dr. Jakob Ablinger

## Ongoing Projects

## Software

### HarmonicSums

#### A Mathematica Package for dealing with Harmonic Sums, Generalized Harmonic Sums and Cyclotomic Sums and their related Integral Representations

The HarmonicSums package by Jakob Ablinger allows to deal with nested sums such as harmonic sums, S-sums, cyclotomic sums and cyclotmic S-sums as well as iterated integrals such as harmonic polylogarithms, multiple polylogarithms and cyclotomic polylogarithms in an algorithmic fashion. ...

### MultiIntegrate

#### The MultiIntegrate package allows to compute multi-dimensional integrals over hyperexponential integrands in terms of (generalized) harmonic sums.

The MultiIntegrate package allows to compute multi-dimensional integrals over hyperexponential integrands in terms of (generalized) harmonic sums. This package uses variations and extensions of the multivariate Alkmkvist-Zeilberger algorithm. Registration and Legal Notices The source code for this package is password ...

### 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

### Three loop heavy quark form factors and their asymptotic behavior

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

In: Proc.of 23rd DAE-BRNS High Energy Physics Symposium 2018, Behera, P.K., Bhatnagar, V., Shukla, P., Sinha, R. (ed.), Springer Proceedings in Physics 261, pp. 91-100. 2021. Springer, ISBN 978-981-33-4407-5. arXiv:1906.05829 [hep-ph]. [doi] [pdf]**inproceedings**{RISC6024,

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

title = {{Three loop heavy quark form factors and their asymptotic behavior}},

booktitle = {{Proc.of 23rd DAE-BRNS High Energy Physics Symposium 2018}},

language = {english},

series = {Springer Proceedings in Physics},

volume = {261},

pages = {91--100},

publisher = {Springer},

isbn_issn = {ISBN 978-981-33-4407-5},

year = {2021},

note = {arXiv:1906.05829 [hep-ph]},

editor = {Behera and P.K. and Bhatnagar and V. and Shukla and P. and Sinha and R.},

refereed = {yes},

length = {10},

url = {https://doi.org/10.1007/978-981-33-4408-2_14}

}

### Solving linear difference equations with coefficients in rings with idempotent representations

#### J. Ablinger, C. Schneider

In: Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation (Proc. ISSAC 21), Marc Mezzarobba (ed.), pp. 27-34. 2021. ISBN 978-1-4503-8382-0/21/06. arXiv:2102.03307 [cs.SC]. [doi] [pdf]**inproceedings**{RISC6302,

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

title = {{Solving linear difference equations with coefficients in rings with idempotent representations}},

booktitle = {{Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation (Proc. ISSAC 21)}},

language = {english},

pages = {27--34},

isbn_issn = {ISBN 978-1-4503-8382-0/21/06},

year = {2021},

note = {arXiv:2102.03307 [cs.SC]},

editor = {Marc Mezzarobba},

refereed = {yes},

length = {8},

url = {https://doi.org/10.1145/3452143.3465535}

}

### 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}

}

### 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}

}

### 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

Technical report no. 21-14 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). July 2021. Licensed under CC BY 4.0 International. [doi] [pdf]**techreport**{RISC6350,

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.},

number = {21-14},

year = {2021},

month = {July},

keywords = {deep inelastic scattering, 3-loop Feynman integrals, symbolic summation, large moment method, special functions},

length = {13},

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)}

}

### Extensions of the AZ-algorithm and the Package MultiIntegrate

#### J. Ablinger

In: Anti-Differentiation and the Calculation of Feynman Amplitudes, J. Blümlein, C. Schneider (ed.), Texts & Monographs in Symbolic Computation (A Series of the Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria) , pp. 35-61. 2021. Springer, ISBN 978-3-030-80218-9. arXiv:2101.11385 [cs.SC]. [doi]**incollection**{RISC6408,

author = {J. Ablinger},

title = {{Extensions of the AZ-algorithm and the Package MultiIntegrate}},

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

language = {english},

abstract = {We extend the (continuous) multivariate Almkvist-Zeilberger algorithm inorder to apply it for instance to special Feynman integrals emerging in renormalizable Quantum field Theories. We will consider multidimensional integrals overhyperexponential integrals and try to find closed form representations in terms ofnested sums and products or iterated integrals. In addition, if we fail to computea closed form solution in full generality, we may succeed in computing the firstcoeffcients of the Laurent series expansions of such integrals in terms of indefnitenested sums and products or iterated integrals. In this article we present the corresponding methods and algorithms. Our Mathematica package MultiIntegrate,can be considered as an enhanced implementation of the (continuous) multivariateAlmkvist Zeilberger algorithm to compute recurrences or differential equations forhyperexponential integrands and integrals. Together with the summation packageSigma and the package HarmonicSums our package provides methods to computeclosed form representations (or coeffcients of the Laurent series expansions) of multidimensional integrals over hyperexponential integrands in terms of nested sums oriterated integrals.},

series = {Texts & Monographs in Symbolic Computation (A Series of the Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria)},

pages = {35--61},

publisher = {Springer},

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

year = {2021},

note = {arXiv:2101.11385 [cs.SC]},

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

refereed = {yes},

keywords = {multivariate Almkvist-Zeilberger algorithm, hyperexponential integrals, iterated integrals, nested sums},

length = {27},

url = {https://doi.org/10.1007/978-3-030-80219-6_2}

}

### Extensions of the AZ-algorithm and the Package MultiIntegrate

#### J. Ablinger

In: Anti-Differentiation and the Calculation of Feynman Amplitudes, J. Blümlein, C. Schneider (ed.), Texts & Monographs in Symbolic Computation (A Series of the Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria) , pp. 35-61. 2021. Springer, ISBN 978-3-030-80218-9. arXiv:2101.11385 [cs.SC]. [doi]**incollection**{RISC6409,

author = {J. Ablinger},

title = {{Extensions of the AZ-algorithm and the Package MultiIntegrate}},

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

language = {english},

abstract = {We extend the (continuous) multivariate Almkvist-Zeilberger algorithm inorder to apply it for instance to special Feynman integrals emerging in renormalizable Quantum field Theories. We will consider multidimensional integrals overhyperexponential integrals and try to find closed form representations in terms ofnested sums and products or iterated integrals. In addition, if we fail to computea closed form solution in full generality, we may succeed in computing the firstcoeffcients of the Laurent series expansions of such integrals in terms of indefnitenested sums and products or iterated integrals. In this article we present the corresponding methods and algorithms. Our Mathematica package MultiIntegrate,can be considered as an enhanced implementation of the (continuous) multivariateAlmkvist Zeilberger algorithm to compute recurrences or differential equations forhyperexponential integrands and integrals. Together with the summation packageSigma and the package HarmonicSums our package provides methods to computeclosed form representations (or coeffcients of the Laurent series expansions) of multidimensional integrals over hyperexponential integrands in terms of nested sums oriterated integrals.},

series = {Texts & Monographs in Symbolic Computation (A Series of the Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria)},

pages = {35--61},

publisher = {Springer},

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

year = {2021},

note = {arXiv:2101.11385 [cs.SC]},

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

refereed = {yes},

keywords = {multivariate Almkvist-Zeilberger algorithm, hyperexponential integrals, iterated integrals, nested sums},

length = {27},

url = {https://doi.org/10.1007/978-3-030-80219-6_2}

}

### 2020

### The three-loop polarized pure singlet operator matrix element with two different masses

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

Nuclear Physics B 952(114916), pp. 1-18. 2020. ISSN 0550-3213. arXiv:1911.11630 [hep-ph]. [doi]**article**{RISC6017,

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

title = {{The three-loop polarized pure singlet operator matrix element with two different masses}},

language = {english},

journal = {Nuclear Physics B },

volume = {952},

number = {114916},

pages = {1--18},

isbn_issn = {ISSN 0550-3213},

year = {2020},

note = {arXiv:1911.11630 [hep-ph]},

refereed = {yes},

length = {18},

url = {https://doi.org/10.1016/j.nuclphysb.2020.114916}

}

### The three-loop single mass polarized pure singlet operator matrix element

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

Nuclear Physics B 953(114945), pp. 1-25. 2020. ISSN 0550-3213. arXiv:1912.02536 [hep-ph]. [doi]**article**{RISC6075,

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

title = {{The three-loop single mass polarized pure singlet operator matrix element}},

language = {english},

journal = {Nuclear Physics B},

volume = {953},

number = {114945},

pages = {1--25},

isbn_issn = {ISSN 0550-3213},

year = {2020},

note = {arXiv:1912.02536 [hep-ph]},

refereed = {yes},

length = {25},

url = {https://doi.org/10.1016/j.nuclphysb.2020.114945}

}

### Proving Two Conjectural Series for $\zeta(7)$ and Discovering More Series for $\zeta(7)$

#### J. Ablinger

In: Mathematical Aspects of Computer and Information Science, D. Slamanig, E. Tsigaridas, Z. Zafeirakopoulos (ed.), pp. 42-47. 2020. Springer International Publishing, 978-3-030-43120-4. [url]**inproceedings**{RISC6102,

author = {J. Ablinger},

title = {{Proving Two Conjectural Series for $\zeta(7)$ and Discovering More Series for $\zeta(7)$}},

booktitle = {{Mathematical Aspects of Computer and Information Science}},

language = {english},

pages = {42--47},

publisher = {Springer International Publishing},

isbn_issn = {978-3-030-43120-4},

year = {2020},

editor = {D. Slamanig and E. Tsigaridas and Z. Zafeirakopoulos},

refereed = {yes},

length = {6},

url = {https://arxiv.org/abs/1908.06631v1}

}

### Subleading logarithmic QED initial state corrections to $e^+e^−\to γ^⁎/Z^{0⁎}$ to $O(\alpha^6L^5)$

#### J. Ablinger, J. Blümlein, A. De Freitas, K. Schönwald

Nuclear Physics B 955, pp. 115045-115045. 2020. ISSN 0550-3213. [url]**article**{RISC6111,

author = {J. Ablinger and J. Blümlein and A. De Freitas and K. Schönwald},

title = {{Subleading logarithmic QED initial state corrections to $e^+e^−\to γ^⁎/Z^{0⁎}$ to $O(\alpha^6L^5)$}},

language = {english},

journal = {Nuclear Physics B},

volume = {955},

pages = {115045--115045},

isbn_issn = { ISSN 0550-3213},

year = {2020},

refereed = {yes},

length = {0},

url = {http://www.sciencedirect.com/science/article/pii/S0550321320301310}

}

### The Two-mass Contribution to the Three-Loop Polarized Operator Matrix Element $A_{gg, Q}^{(3)}$

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

Nuclear Physics B 955, pp. 1-70. 2020. ISSN 0550-3213. arXiv:2004.08916 [hep-ph]. [doi]**article**{RISC6115,

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

title = {{The Two-mass Contribution to the Three-Loop Polarized Operator Matrix Element $A_{gg,Q}^{(3)}$ }},

language = {english},

journal = {Nuclear Physics B},

volume = {955},

pages = {1--70},

isbn_issn = {ISSN 0550-3213},

year = {2020},

note = {arXiv:2004.08916 [hep-ph]},

refereed = {no},

length = {59},

url = {https://doi.org/10.1016/j.nuclphysb.2020.115059}

}

### 2019

### Automated Solution of First Order Factorizable Systems of Differential Equations in One Variable

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

Nucl. Phys. B 939, pp. 253-291. 2019. ISSN 0550-3213. arXiv:1810.12261 [hep-ph]. [doi]**article**{RISC5795,

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

title = {{Automated Solution of First Order Factorizable Systems of Differential Equations in One Variable}},

language = {english},

journal = {Nucl. Phys. B},

volume = {939},

pages = {253--291},

isbn_issn = {ISSN 0550-3213},

year = {2019},

note = {arXiv:1810.12261 [hep-ph]},

refereed = {yes},

length = {39},

url = {https://doi.org/10.1016/j.nuclphysb.2018.12.010}

}

### 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}

}

### Numerical Implementation of Harmonic Polylogarithms to Weight w = 8

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

Comput. Phys. Comm. 240, pp. 189-201. 2019. ISSN 0010-4655. arXiv:1809.07084 [hep-ph]. [doi] [pdf]**article**{RISC5938,

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

title = {{Numerical Implementation of Harmonic Polylogarithms to Weight w = 8}},

language = {english},

journal = {Comput. Phys. Comm.},

volume = {240},

pages = {189--201},

isbn_issn = {ISSN 0010-4655},

year = {2019},

note = {arXiv:1809.07084 [hep-ph]},

refereed = {yes},

length = {12},

url = {https://www.doi.org/10.1016/j.cpc.2019.02.005}

}

### 2018

### The Two-mass Contribution to the Three-Loop Gluonic Operator Matrix Element $A_{gg, Q}^{(3)}$

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

Nucl. Phys. B 932, pp. 129-240. 2018. ISSN 0550-3213. arXiv:1804.02226 [hep-ph]. [doi]**article**{RISC5618,

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

title = {{The Two-mass Contribution to the Three-Loop Gluonic Operator Matrix Element $A_{gg,Q}^{(3)}$}},

language = {english},

journal = {Nucl. Phys. B},

volume = {932},

pages = {129--240},

isbn_issn = {ISSN 0550-3213},

year = {2018},

note = {arXiv:1804.02226 [hep-ph]},

refereed = {yes},

length = {112},

url = {https://doi.org/10.1016/j.nuclphysb.2018.04.023}

}

### 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 }

}

### Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams

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

J. Math. Phys. 59(062305), pp. 1-55. 2018. ISSN 0022-2488. arXiv:1706.01299 [hep-th]. [doi] [pdf]**article**{RISC5456,

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

title = {{Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams}},

language = {english},

journal = {J. Math. Phys.},

volume = {59},

number = {062305},

pages = {1--55},

isbn_issn = {ISSN 0022-2488},

year = {2018},

note = {arXiv:1706.01299 [hep-th]},

refereed = {yes},

length = {55},

url = {https://www.doi.org/10.1063/1.4986417}

}

### The two-mass contribution to the three-loop pure singlet operator matrix element

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

Nucl. Phys. B(927), pp. 339-367. 2018. ISSN 0550-3213. arXiv:1711.06717 [hep-ph]. [doi]**article**{RISC5504,

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

title = {{The two-mass contribution to the three-loop pure singlet operator matrix element}},

language = {english},

journal = {Nucl. Phys. B},

number = {927},

pages = {339--367},

isbn_issn = {ISSN 0550-3213},

year = {2018},

note = {arXiv:1711.06717 [hep-ph]},

refereed = {yes},

length = {29},

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

}

### 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 }

}