# Das Berechnen in der Jacobischen Varietät einer algebraischen Kurve über einen endlichen Körper

### Project Description

Projektnummer: M – 153

Forschungsprogramm: MEitner-Programm

### Project Lead

### Project Duration

01/01/1994 - 31/12/1995## Partners

### The Austrian Science Fund (FWF)

## Publications

### 2023

[Jimenez Pastor]

### An extension of holonomic sequences: $C^2$-finite sequences

#### A. Jimenez-Pastor, P. Nuspl, V. Pillwein

Journal of Symbolic Computation 116, pp. 400-424. 2023. ISSN: 0747-7171.@

author = {A. Jimenez-Pastor and P. Nuspl and V. Pillwein},

title = {{An extension of holonomic sequences: $C^2$-finite sequences}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {116},

pages = {400--424},

isbn_issn = {ISSN: 0747-7171},

year = {2023},

refereed = {yes},

length = {25}

}

**article**{RISC6636,author = {A. Jimenez-Pastor and P. Nuspl and V. Pillwein},

title = {{An extension of holonomic sequences: $C^2$-finite sequences}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {116},

pages = {400--424},

isbn_issn = {ISSN: 0747-7171},

year = {2023},

refereed = {yes},

length = {25}

}

[Mitteramskogler]

### General solutions of first-order algebraic ODEs in simple constant extensions

#### J. J. Mitteramskogler, F. Winkler

Journal of Systems Science and Complexity (JSSC), pp. 0-0. 2023. 1009-6124.@

author = {J. J. Mitteramskogler and F. Winkler},

title = {{General solutions of first-order algebraic ODEs in simple constant extensions}},

language = {english},

journal = {Journal of Systems Science and Complexity (JSSC)},

pages = {0--0},

isbn_issn = {1009-6124},

year = {2023},

refereed = {yes},

length = {0}

}

**article**{RISC6674,author = {J. J. Mitteramskogler and F. Winkler},

title = {{General solutions of first-order algebraic ODEs in simple constant extensions}},

language = {english},

journal = {Journal of Systems Science and Complexity (JSSC)},

pages = {0--0},

isbn_issn = {1009-6124},

year = {2023},

refereed = {yes},

length = {0}

}

[Schneider]

### Hypergeometric Structures in Feynman Integrals

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

Annals of Mathematics and Artificial Intelligence, Special issue on " Symbolic Computation in Software Science" to appear, pp. ?-?. 2023. ISSN 1573-7470. arXiv:2111.15501 [math-ph]. [doi]@

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, Special issue on " Symbolic Computation in Software Science"},

volume = {to appear},

pages = {?--?},

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 = {55},

url = {https://doi.org/10.48550/arXiv.2111.15501}

}

**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, Special issue on " Symbolic Computation in Software Science"},

volume = {to appear},

pages = {?--?},

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 = {55},

url = {https://doi.org/10.48550/arXiv.2111.15501}

}

[Smoot]

### A Congruence Family For 2-Elongated Plane Partitions: An Application of the Localization Method

#### N. Smoot

Journal of Number Theory 242, pp. 112-153. January 2023. ISSN 1096-1658. [doi]@

author = {N. Smoot},

title = {{A Congruence Family For 2-Elongated Plane Partitions: An Application of the Localization Method}},

language = {english},

abstract = {George Andrews and Peter Paule have recently conjectured an infinite family of congruences modulo powers of 3 for the 2-elongated plane partition function $d_2(n)$. This congruence family appears difficult to prove by classical methods. We prove a refined form of this conjecture by expressing the associated generating functions as elements of a ring of modular functions isomorphic to a localization of $mathbb{Z}[X]$.},

journal = {Journal of Number Theory},

volume = {242},

pages = {112--153},

isbn_issn = {ISSN 1096-1658},

year = {2023},

month = {January},

refereed = {yes},

length = {42},

url = {https://doi.org/10.1016/j.jnt.2022.07.014}

}

**article**{RISC6661,author = {N. Smoot},

title = {{A Congruence Family For 2-Elongated Plane Partitions: An Application of the Localization Method}},

language = {english},

abstract = {George Andrews and Peter Paule have recently conjectured an infinite family of congruences modulo powers of 3 for the 2-elongated plane partition function $d_2(n)$. This congruence family appears difficult to prove by classical methods. We prove a refined form of this conjecture by expressing the associated generating functions as elements of a ring of modular functions isomorphic to a localization of $mathbb{Z}[X]$.},

journal = {Journal of Number Theory},

volume = {242},

pages = {112--153},

isbn_issn = {ISSN 1096-1658},

year = {2023},

month = {January},

refereed = {yes},

length = {42},

url = {https://doi.org/10.1016/j.jnt.2022.07.014}

}

### 2022

[Banerjee]

### Hook Type enumeration and parity of parts in partitions

#### K. Banerjee, M. G. Dastidar

Research Institute for Symbolic Computation, JKU, Linz. Technical report no. RISC6596, 2022. [pdf]@

author = {K. Banerjee and M. G. Dastidar},

title = {{Hook Type enumeration and parity of parts in partitions}},

language = {english},

abstract = {This paper is devoted to study an association between hook type enumeration and counting integer partitions subject to parity of its parts. We shall primarily focus on a result of Andrews in two possible direction. First, we confirm a conjecture of Rubey and secondly, we extend the theorem of Andrews in a more general set up. },

number = {RISC6596},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {8}

}

**techreport**{RISC6596,author = {K. Banerjee and M. G. Dastidar},

title = {{Hook Type enumeration and parity of parts in partitions}},

language = {english},

abstract = {This paper is devoted to study an association between hook type enumeration and counting integer partitions subject to parity of its parts. We shall primarily focus on a result of Andrews in two possible direction. First, we confirm a conjecture of Rubey and secondly, we extend the theorem of Andrews in a more general set up. },

number = {RISC6596},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {8}

}

[Banerjee]

### Hook type tableaux and partition identities

#### K. Banerjee, M. G. Dastidar

Research Institute for Symbolic Computation, JKU, Linz. Technical report no. RISC6597, 2022. [pdf]@

author = {K. Banerjee and M. G. Dastidar},

title = {{Hook type tableaux and partition identities}},

language = {english},

abstract = {In this paper we exhibit the box-stacking principle (BSP) in conjunction with Young diagrams to prove generalizations of Stanley's and Elder's theorems without even the use of partition statistics in general. We primarily focus on to study Stanley's theorem in color partition context.},

number = {RISC6597},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {13}

}

**techreport**{RISC6597,author = {K. Banerjee and M. G. Dastidar},

title = {{Hook type tableaux and partition identities}},

language = {english},

abstract = {In this paper we exhibit the box-stacking principle (BSP) in conjunction with Young diagrams to prove generalizations of Stanley's and Elder's theorems without even the use of partition statistics in general. We primarily focus on to study Stanley's theorem in color partition context.},

number = {RISC6597},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {13}

}

[Banerjee]

### Positivity of the second shifted difference of partitions and overpartitions: a combinatorial approach

#### Koustav Banerjee

Research Institute for Symbolic Computation, JKU, Linz. Technical report no. RISC6592, 2022. [pdf]@

author = {Koustav Banerjee},

title = {{Positivity of the second shifted difference of partitions and overpartitions: a combinatorial approach}},

language = {english},

number = {RISC6592},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {5}

}

**techreport**{RISC6592,author = {Koustav Banerjee},

title = {{Positivity of the second shifted difference of partitions and overpartitions: a combinatorial approach}},

language = {english},

number = {RISC6592},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {5}

}

[Banerjee]

### Ramanujan's theta functions and parity of parts and cranks of partitions

#### K. Banerjee, M. G. Dastidar

Research Institute for Symbolic Computation, JKU, Linz. Technical report no. RISC6595, 2022. [pdf]@

author = {K. Banerjee and M. G. Dastidar},

title = {{Ramanujan's theta functions and parity of parts and cranks of partitions}},

language = {english},

abstract = {In this paper we explore intricate connections between Ramanujan's theta functions and a class of partition functions defined by the nature of the parity of their parts. This consequently leads us to the parity analysis of the crank of a partition and its correlation to the number of partitions with odd number of parts, self-conjugate partitions, and also with Durfee squares and Frobenius symbols.},

number = {RISC6595},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {13}

}

**techreport**{RISC6595,author = {K. Banerjee and M. G. Dastidar},

title = {{Ramanujan's theta functions and parity of parts and cranks of partitions}},

language = {english},

abstract = {In this paper we explore intricate connections between Ramanujan's theta functions and a class of partition functions defined by the nature of the parity of their parts. This consequently leads us to the parity analysis of the crank of a partition and its correlation to the number of partitions with odd number of parts, self-conjugate partitions, and also with Durfee squares and Frobenius symbols.},

number = {RISC6595},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {13}

}

[Banerjee]

### Inequalities for the partition function arising from truncated theta series

#### K. Banerjee, M. G. Dastidar

Research Institute for Symbolic Computation, JKU, Linz. Technical report no. RISC6622, 2022. [pdf]@

author = {K. Banerjee and M. G. Dastidar},

title = {{Inequalities for the partition function arising from truncated theta series}},

language = {english},

number = {RISC6622},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {12}

}

**techreport**{RISC6622,author = {K. Banerjee and M. G. Dastidar},

title = {{Inequalities for the partition function arising from truncated theta series}},

language = {english},

number = {RISC6622},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {12}

}

[Banerjee]

### Parity biases in partitions and restricted partitions

#### Banerjee Koustav, Bhattacharjee Sreerupa, Dastidar Manosij Ghosh, Mahanta Pankaj Jyoti, Saikia Manjil P

European Journal of Combinatorics 103, pp. 103522-103522. 2022. Elsevier, ISSN 0195-6698. [pdf]@

author = {Banerjee Koustav and Bhattacharjee Sreerupa and Dastidar Manosij Ghosh and Mahanta Pankaj Jyoti and Saikia Manjil P},

title = {{Parity biases in partitions and restricted partitions}},

language = {english},

journal = {European Journal of Combinatorics},

volume = {103},

pages = {103522--103522},

publisher = {Elsevier},

isbn_issn = {ISSN 0195-6698},

year = {2022},

refereed = {yes},

length = {19}

}

**article**{RISC6606,author = {Banerjee Koustav and Bhattacharjee Sreerupa and Dastidar Manosij Ghosh and Mahanta Pankaj Jyoti and Saikia Manjil P},

title = {{Parity biases in partitions and restricted partitions}},

language = {english},

journal = {European Journal of Combinatorics},

volume = {103},

pages = {103522--103522},

publisher = {Elsevier},

isbn_issn = {ISSN 0195-6698},

year = {2022},

refereed = {yes},

length = {19}

}

[Banerjee]

### New inequalities for p(n) and log p(n)

#### K. Banerjee, P. Paule, C. S. Radu, W. H. Zeng

Research Institute for Symbolic Computation, JKU, Linz. Technical report no. RISC6607, 2022. [pdf]@

author = {K. Banerjee and P. Paule and C. S. Radu and W. H. Zeng},

title = {{New inequalities for p(n) and log p(n)}},

language = {english},

number = {RISC6607},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {37}

}

**techreport**{RISC6607,author = {K. Banerjee and P. Paule and C. S. Radu and W. H. Zeng},

title = {{New inequalities for p(n) and log p(n)}},

language = {english},

number = {RISC6607},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {37}

}

[Banerjee]

### Inequalities for the modified Bessel function of first kind of non-negative order

#### K. Banerjee

Research Institute for Symbolic Computation, JKU, Linz. Technical report no. RISC6608, 2022. [pdf]@

author = {K. Banerjee},

title = {{Inequalities for the modified Bessel function of first kind of non-negative order}},

language = {english},

number = {RISC6608},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {28}

}

**techreport**{RISC6608,author = {K. Banerjee},

title = {{Inequalities for the modified Bessel function of first kind of non-negative order}},

language = {english},

number = {RISC6608},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {28}

}

[Banerjee]

### An unified framework to prove multiplicative inequalities for the partition function

#### K. Banerjee

Research Institute for Symbolic Computation, JKU, Linz. Technical report no. RISC6614, 2022. [pdf]@

author = {K. Banerjee},

title = {{An unified framework to prove multiplicative inequalities for the partition function}},

language = {english},

number = {RISC6614},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {24}

}

**techreport**{RISC6614,author = {K. Banerjee},

title = {{An unified framework to prove multiplicative inequalities for the partition function}},

language = {english},

number = {RISC6614},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {24}

}

[Banerjee]

### The localization method applied to k-elognated plane partitions and divisibily by 5

#### K. Banerjee, N. A. Smoot

Research Institute for Symbolic Computation, JKU, Linz. Technical report no. RISC6611, 2022. [pdf]@

author = {K. Banerjee and N. A. Smoot},

title = {{The localization method applied to k-elognated plane partitions and divisibily by 5}},

language = {english},

number = {RISC6611},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {40}

}

**techreport**{RISC6611,author = {K. Banerjee and N. A. Smoot},

title = {{The localization method applied to k-elognated plane partitions and divisibily by 5}},

language = {english},

number = {RISC6611},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {40}

}

[Banerjee]

### Invariants of the quartic binary form and proofs of Chen's conjectures on partition function inequalities

#### K. Banerjee

Research Institute for Symbolic Computation, JKU, Linz. Technical report no. RISC6615, 2022. [pdf]@

author = {K. Banerjee},

title = {{Invariants of the quartic binary form and proofs of Chen's conjectures on partition function inequalities}},

language = {english},

number = {RISC6615},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {31}

}

**techreport**{RISC6615,author = {K. Banerjee},

title = {{Invariants of the quartic binary form and proofs of Chen's conjectures on partition function inequalities}},

language = {english},

number = {RISC6615},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {31}

}

[Cerna]

### Learning Higher-Order Programs without Meta-Interpretive Learning

#### Stanislav Purgal and David Cerna and Cezary Kalisyk

In: Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence, {IJCAI-22}, Lud De Raedt (ed.), Proceedings of International Joint Conference on Artificial Intelligence, Main Track , pp. 2726-2733. july 2022. International Joint Conferences on Artificial Intelligence Organization, 10.24963/ijcai.2022/378. [doi]@

author = {Stanislav Purgal and David Cerna and Cezary Kalisyk},

title = {{Learning Higher-Order Programs without Meta-Interpretive Learning}},

booktitle = {{Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence, {IJCAI-22}}},

language = {english},

abstract = {Learning complex programs through textit{inductive logic programming} (ILP) remains a formidable challenge. Existing higher-order enabled ILP systems show improved accuracy and learning performance, though remain hampered by the limitations of the underlying learning mechanism. Experimental results show that our extension of the versatile textit{Learning From Failures} paradigm by higher-order definitions significantly improves learning performance without the burdensome human guidance required by existing systems. Furthermore, we provide a theoretical framework capturing the class of higher-order definitions handled by our extension.},

series = {Main Track},

pages = {2726--2733},

publisher = {International Joint Conferences on Artificial Intelligence Organization},

isbn_issn = {10.24963/ijcai.2022/378},

year = {2022},

month = {july},

editor = {Lud De Raedt},

refereed = {yes},

keywords = {Inductive Logic Programming, Higher order definitions},

length = {8},

conferencename = {International Joint Conference on Artificial Intelligence},

url = {https://doi.org/10.35011/risc.21-22}

}

**inproceedings**{RISC6646,author = {Stanislav Purgal and David Cerna and Cezary Kalisyk},

title = {{Learning Higher-Order Programs without Meta-Interpretive Learning}},

booktitle = {{Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence, {IJCAI-22}}},

language = {english},

abstract = {Learning complex programs through textit{inductive logic programming} (ILP) remains a formidable challenge. Existing higher-order enabled ILP systems show improved accuracy and learning performance, though remain hampered by the limitations of the underlying learning mechanism. Experimental results show that our extension of the versatile textit{Learning From Failures} paradigm by higher-order definitions significantly improves learning performance without the burdensome human guidance required by existing systems. Furthermore, we provide a theoretical framework capturing the class of higher-order definitions handled by our extension.},

series = {Main Track},

pages = {2726--2733},

publisher = {International Joint Conferences on Artificial Intelligence Organization},

isbn_issn = {10.24963/ijcai.2022/378},

year = {2022},

month = {july},

editor = {Lud De Raedt},

refereed = {yes},

keywords = {Inductive Logic Programming, Higher order definitions},

length = {8},

conferencename = {International Joint Conference on Artificial Intelligence},

url = {https://doi.org/10.35011/risc.21-22}

}

[de Freitas]

### The Unpolarized and Polarized Single-Mass Three-Loop Heavy Flavor Operator Matrix Elements $A_{gg, Q}$ and $Delta A_{gg, Q}$

#### J. Ablinger, A. Behring, J. Bluemlein, A. De Freitas, A. Goedicke, A. von Manteuffel, C. Schneider, K. Schoenwald

Journal of High Energy Physics 2022(12, Article 134), pp. 1-55. 2022. ISSN 1029-8479. arXiv:2211.05462 [hep-ph]. [doi]@

author = {J. Ablinger and A. Behring and J. Bluemlein and A. De Freitas and A. Goedicke and A. von Manteuffel and C. Schneider and K. Schoenwald},

title = {{The Unpolarized and Polarized Single-Mass Three-Loop Heavy Flavor Operator Matrix Elements $A_{gg, Q}$ and $Delta A_{gg, Q}$}},

language = {english},

abstract = {We calculate the gluonic massive operator matrix elements in the unpolarized and polarized cases, $A_{gg,Q}(x,mu^2)$ and $Delta A_{gg,Q}(x,mu^2)$, at three--loop order for a single mass. These quantities contribute to the matching of the gluon distribution in the variable flavor number scheme. The polarized operator matrix element is calculated in the Larin scheme. These operator matrix elements contain finite binomial and inverse binomial sums in Mellin $N$--space and iterated integrals over square root--valued alphabets in momentum fraction $x$--space. We derive the necessary analytic relations for the analytic continuation of these quantities from the even or odd Mellin moments into the complex plane, present analytic expressions in momentum fraction $x$--space and derive numerical results. The present results complete the gluon transition matrix elements both of the single-- and double--mass variable flavor number scheme to three--loop order.},

journal = {Journal of High Energy Physics},

volume = {2022},

number = {12, Article 134},

pages = {1--55},

isbn_issn = { ISSN 1029-8479},

year = {2022},

note = {arXiv:2211.05462 [hep-ph]},

refereed = {yes},

keywords = {Feynman integrals, linear difference equations, linear differential equations, binomial sums, harmonic sums, iterative integrals, computer algebra},

length = {48},

url = {https://doi.org/10.1007/JHEP12(2022)134}

}

**article**{RISC6632,author = {J. Ablinger and A. Behring and J. Bluemlein and A. De Freitas and A. Goedicke and A. von Manteuffel and C. Schneider and K. Schoenwald},

title = {{The Unpolarized and Polarized Single-Mass Three-Loop Heavy Flavor Operator Matrix Elements $A_{gg, Q}$ and $Delta A_{gg, Q}$}},

language = {english},

abstract = {We calculate the gluonic massive operator matrix elements in the unpolarized and polarized cases, $A_{gg,Q}(x,mu^2)$ and $Delta A_{gg,Q}(x,mu^2)$, at three--loop order for a single mass. These quantities contribute to the matching of the gluon distribution in the variable flavor number scheme. The polarized operator matrix element is calculated in the Larin scheme. These operator matrix elements contain finite binomial and inverse binomial sums in Mellin $N$--space and iterated integrals over square root--valued alphabets in momentum fraction $x$--space. We derive the necessary analytic relations for the analytic continuation of these quantities from the even or odd Mellin moments into the complex plane, present analytic expressions in momentum fraction $x$--space and derive numerical results. The present results complete the gluon transition matrix elements both of the single-- and double--mass variable flavor number scheme to three--loop order.},

journal = {Journal of High Energy Physics},

volume = {2022},

number = {12, Article 134},

pages = {1--55},

isbn_issn = { ISSN 1029-8479},

year = {2022},

note = {arXiv:2211.05462 [hep-ph]},

refereed = {yes},

keywords = {Feynman integrals, linear difference equations, linear differential equations, binomial sums, harmonic sums, iterative integrals, computer algebra},

length = {48},

url = {https://doi.org/10.1007/JHEP12(2022)134}

}

[Dominici]

### Truncated Hermite polynomials

#### Diego Dominici and Francisco Marcell{\'a}n

Technical report no. 22-10 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). August 2022. Licensed under CC BY 4.0 International. [doi] [pdf]@

author = {Diego Dominici and Francisco Marcell{\'a}n},

title = {{Truncated Hermite polynomials}},

language = {english},

abstract = {We define the family of truncated Hermite polynomials $P_{n}left(x;zright) $, orthogonal with respect to the linear functional[Lleft[ pright] = int_{-z}^{z} pleft( xright) e^{-x^{2}} ,dx. ]The connection of $P_{n}left( x;zright) $ with the Hermite and Rys polynomialsis stated. The semiclassical character of $P_{n}left( x;zright) $ aspolynomials of class $2$ is emphasized.As a consequence, several properties of $P_{n}left( x;zright) $ concerningthe coefficients $gamma_{n}left( zright) $ in the three-term recurrencerelation they satisfy as well as the moments and the Stieltjes function of $L$are given. Ladder operators associated with the linear functional $L$, aholonomic differential equation (in $x)$ for the polynomials $P_{n}left(x;zright) $, and a nonlinear ODE for the functions $gamma_{n}left(zright) $ are deduced. },

number = {22-10},

year = {2022},

month = {August},

keywords = {Orthogonal polynomials, Gaussian distribution},

length = {37},

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)}

}

**techreport**{RISC6531,author = {Diego Dominici and Francisco Marcell{\'a}n},

title = {{Truncated Hermite polynomials}},

language = {english},

abstract = {We define the family of truncated Hermite polynomials $P_{n}left(x;zright) $, orthogonal with respect to the linear functional[Lleft[ pright] = int_{-z}^{z} pleft( xright) e^{-x^{2}} ,dx. ]The connection of $P_{n}left( x;zright) $ with the Hermite and Rys polynomialsis stated. The semiclassical character of $P_{n}left( x;zright) $ aspolynomials of class $2$ is emphasized.As a consequence, several properties of $P_{n}left( x;zright) $ concerningthe coefficients $gamma_{n}left( zright) $ in the three-term recurrencerelation they satisfy as well as the moments and the Stieltjes function of $L$are given. Ladder operators associated with the linear functional $L$, aholonomic differential equation (in $x)$ for the polynomials $P_{n}left(x;zright) $, and a nonlinear ODE for the functions $gamma_{n}left(zright) $ are deduced. },

number = {22-10},

year = {2022},

month = {August},

keywords = {Orthogonal polynomials, Gaussian distribution},

length = {37},

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)}

}

[Dominici]

### Comparative asymptotics for discrete semiclassical orthogonal polynomials

#### Diego Dominici

Technical report no. 22-11 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). August 2022. Licensed under CC BY 4.0 International. [doi] [pdf]@

author = {Diego Dominici},

title = {{Comparative asymptotics for discrete semiclassical orthogonal polynomials}},

language = {english},

abstract = {We study the ratio $frac{P_{n}left( x;zright) }{phi_{n}left( xright)}$ asymptotically as $nrightarrowinfty,$ where the polynomials $P_{n}left(x;zright) $ are orthogonal with respect to a discrete linear functional and$phi_{n}left( xright) $ denote the falling factorial polynomials.We give recurrences that allow the computation of high order asymptoticexpansions of $P_{n}left( x;zright) $ and give examples for most discretesemiclassical polynomials of class $sleq2.$We show several plots illustrating the accuracy of our results.},

number = {22-11},

year = {2022},

month = {August},

keywords = {Orthogonal polynomials, asymptotic analysis },

length = {53},

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)}

}

**techreport**{RISC6579,author = {Diego Dominici},

title = {{Comparative asymptotics for discrete semiclassical orthogonal polynomials}},

language = {english},

abstract = {We study the ratio $frac{P_{n}left( x;zright) }{phi_{n}left( xright)}$ asymptotically as $nrightarrowinfty,$ where the polynomials $P_{n}left(x;zright) $ are orthogonal with respect to a discrete linear functional and$phi_{n}left( xright) $ denote the falling factorial polynomials.We give recurrences that allow the computation of high order asymptoticexpansions of $P_{n}left( x;zright) $ and give examples for most discretesemiclassical polynomials of class $sleq2.$We show several plots illustrating the accuracy of our results.},

number = {22-11},

year = {2022},

month = {August},

keywords = {Orthogonal polynomials, asymptotic analysis },

length = {53},

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)}

}

[Dominici]

### Asymptotic analysis of a family of Sobolev orthogonal polynomials related to the generalized Charlier polynomials

#### Diego Dominici and Juan José Moreno Balcázar

Technical report no. 22-16 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). November 2022. Licensed under CC BY 4.0 International. [doi] [pdf]@

author = {Diego Dominici and Juan José Moreno Balcázar},

title = {{Asymptotic analysis of a family of Sobolev orthogonal polynomials related to the generalized Charlier polynomials}},

language = {english},

abstract = {In this paper we tackle the asymptotic behaviour of a family of orthogonalpolynomials with respect to a nonstandard inner product involving the forwardoperator $Delta$. Concretely, we treat the generalized Charlier weights inthe framework of $Delta$--Sobolev orthogonality. We obtain an asymptoticexpansion for this orthogonal polynomials where the falling factorialpolynomials play an important role.},

number = {22-16},

year = {2022},

month = {November},

keywords = {Sobolev orthogonal polynomials, asymptotic analysis, discrete semiclassical orthogonal polynomials},

length = {18},

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)}

}

**techreport**{RISC6625,author = {Diego Dominici and Juan José Moreno Balcázar},

title = {{Asymptotic analysis of a family of Sobolev orthogonal polynomials related to the generalized Charlier polynomials}},

language = {english},

abstract = {In this paper we tackle the asymptotic behaviour of a family of orthogonalpolynomials with respect to a nonstandard inner product involving the forwardoperator $Delta$. Concretely, we treat the generalized Charlier weights inthe framework of $Delta$--Sobolev orthogonality. We obtain an asymptoticexpansion for this orthogonal polynomials where the falling factorialpolynomials play an important role.},

number = {22-16},

year = {2022},

month = {November},

keywords = {Sobolev orthogonal polynomials, asymptotic analysis, discrete semiclassical orthogonal polynomials},

length = {18},

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)}

}