# Partition Analysis [SFB F050-06]

### Project Lead

### Project Duration

01/03/2013 - 31/07/2022### Project URL

Go to Website## Partners

### The Austrian Science Fund (FWF)

## Publications

### 2023

[Schneider]

### 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]@

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}

}

**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

[Paule]

### MacMahon's Partition Analysis XIV: Partitions with n copies of n

#### G.E. Andrews, P. Paule

Technical report no. 22-14 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). October 2022. Research Article. Licensed under CC BY 4.0 International. [doi] [pdf]@

author = {G.E. Andrews and P. Paule},

title = {{MacMahon's Partition Analysis XIV: Partitions with n copies of n}},

language = {english},

abstract = {We apply the methods of partition analysis to partitions with~$n$ copies of~$n$. This allows us to obtain multivariable generatingfunctions related to classical Rogers-Ramanujan type identities. Also, partitionswith $n$ copies of $n$ are extended to partition diamonds yielding numerous new results including a natural connection to overpartitions and a variety of partitioncongruences.},

number = {22-14},

year = {2022},

month = {October},

keywords = {partitions, overpartitions, partitions with $n$ copies of $n$, partition analysis, $q$-series, modular forms and partition congruences, Radu's Ramanujan-Kolberg algorithm},

sponsor = {FWF},

length = {30},

type = {Research Article},

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**{RISC6626,author = {G.E. Andrews and P. Paule},

title = {{MacMahon's Partition Analysis XIV: Partitions with n copies of n}},

language = {english},

abstract = {We apply the methods of partition analysis to partitions with~$n$ copies of~$n$. This allows us to obtain multivariable generatingfunctions related to classical Rogers-Ramanujan type identities. Also, partitionswith $n$ copies of $n$ are extended to partition diamonds yielding numerous new results including a natural connection to overpartitions and a variety of partitioncongruences.},

number = {22-14},

year = {2022},

month = {October},

keywords = {partitions, overpartitions, partitions with $n$ copies of $n$, partition analysis, $q$-series, modular forms and partition congruences, Radu's Ramanujan-Kolberg algorithm},

sponsor = {FWF},

length = {30},

type = {Research Article},

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)}

}

### 2021

[Hemmecke]

### Construction of modular function bases for $Gamma_0(121)$ related to $p(11n+6)$

#### Ralf Hemmecke, Peter Paule, Silviu Radu

Integral Transforms and Special Functions 32(5-8), pp. 512-527. 2021. Taylor & Francis, 1065-2469. [doi]@

author = {Ralf Hemmecke and Peter Paule and Silviu Radu},

title = {{Construction of modular function bases for $Gamma_0(121)$ related to $p(11n+6)$}},

language = {english},

abstract = {Motivated by arithmetic properties of partitionnumbers $p(n)$, our goal is to find algorithmicallya Ramanujan type identity of the form$sum_{n=0}^{infty}p(11n+6)q^n=R$, where $R$ is apolynomial in products of the form$e_alpha:=prod_{n=1}^{infty}(1-q^{11^alpha n})$with $alpha=0,1,2$. To this end we multiply theleft side by an appropriate factor such the resultis a modular function for $Gamma_0(121)$ havingonly poles at infinity. It turns out thatpolynomials in the $e_alpha$ do not generate thefull space of such functions, so we were led tomodify our goal. More concretely, we give threedifferent ways to construct the space of modularfunctions for $Gamma_0(121)$ having only poles atinfinity. This in turn leads to three differentrepresentations of $R$ not solely in terms of the$e_alpha$ but, for example, by using as generatorsalso other functions like the modular invariant $j$.},

journal = {Integral Transforms and Special Functions},

volume = {32},

number = {5-8},

pages = {512--527},

publisher = {Taylor & Francis},

isbn_issn = {1065-2469},

year = {2021},

refereed = {yes},

keywords = {Ramanujan identities, bases for modular functions, integral bases},

sponsor = {FWF (SFB F50-06)},

length = {16},

url = {https://doi.org/10.1080/10652469.2020.1806261}

}

**article**{RISC6342,author = {Ralf Hemmecke and Peter Paule and Silviu Radu},

title = {{Construction of modular function bases for $Gamma_0(121)$ related to $p(11n+6)$}},

language = {english},

abstract = {Motivated by arithmetic properties of partitionnumbers $p(n)$, our goal is to find algorithmicallya Ramanujan type identity of the form$sum_{n=0}^{infty}p(11n+6)q^n=R$, where $R$ is apolynomial in products of the form$e_alpha:=prod_{n=1}^{infty}(1-q^{11^alpha n})$with $alpha=0,1,2$. To this end we multiply theleft side by an appropriate factor such the resultis a modular function for $Gamma_0(121)$ havingonly poles at infinity. It turns out thatpolynomials in the $e_alpha$ do not generate thefull space of such functions, so we were led tomodify our goal. More concretely, we give threedifferent ways to construct the space of modularfunctions for $Gamma_0(121)$ having only poles atinfinity. This in turn leads to three differentrepresentations of $R$ not solely in terms of the$e_alpha$ but, for example, by using as generatorsalso other functions like the modular invariant $j$.},

journal = {Integral Transforms and Special Functions},

volume = {32},

number = {5-8},

pages = {512--527},

publisher = {Taylor & Francis},

isbn_issn = {1065-2469},

year = {2021},

refereed = {yes},

keywords = {Ramanujan identities, bases for modular functions, integral bases},

sponsor = {FWF (SFB F50-06)},

length = {16},

url = {https://doi.org/10.1080/10652469.2020.1806261}

}

[Paule]

### Contiguous Relations and Creative Telescoping

#### Peter Paule

In: Anti-Differentiation and the Calculation of Feynman Amplitudes, J. Bluemlein and C. Schneider (ed.), Texts and Monographs in Symbolic Computation , pp. 335-394. 2021. Springer, ISBN 978-3-030-80218-9. [doi] [pdf]@

author = {Peter Paule},

title = {{Contiguous Relations and Creative Telescoping}},

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

language = {english},

abstract = {This article presents an algorithmic theory of contiguous relations.Contiguous relations, first studied by Gauß, are a fundamental concept within the theory of hypergeometric series. In contrast to Takayama’s approach, which for elimination uses non-commutative Gröbner bases, our framework is based on parameterized telescoping and can be viewed as an extension of Zeilberger’s creative telescoping paradigm based on Gosper’s algorithm. The wide range of applications include elementary algorithmic explanations of the existence of classical formulas for non- terminating hypergeometric series such as Gauß, Pfaff-Saalschütz, or Dixon summation. The method can be used to derive new theorems, like a non-terminating extension of a classical recurrence established by Wilson between terminating 4F3-series. Moreover, our setting helps to explain the non-minimal order phenomenon of Zeilberger’s algorithm.},

series = {Texts and Monographs in Symbolic Computation},

pages = {335--394},

publisher = {Springer},

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

year = {2021},

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

refereed = {yes},

length = {61},

url = {https://doi.org/10.1007/978-3-030-80219-6_15 }

}

**incollection**{RISC6366,author = {Peter Paule},

title = {{Contiguous Relations and Creative Telescoping}},

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

language = {english},

abstract = {This article presents an algorithmic theory of contiguous relations.Contiguous relations, first studied by Gauß, are a fundamental concept within the theory of hypergeometric series. In contrast to Takayama’s approach, which for elimination uses non-commutative Gröbner bases, our framework is based on parameterized telescoping and can be viewed as an extension of Zeilberger’s creative telescoping paradigm based on Gosper’s algorithm. The wide range of applications include elementary algorithmic explanations of the existence of classical formulas for non- terminating hypergeometric series such as Gauß, Pfaff-Saalschütz, or Dixon summation. The method can be used to derive new theorems, like a non-terminating extension of a classical recurrence established by Wilson between terminating 4F3-series. Moreover, our setting helps to explain the non-minimal order phenomenon of Zeilberger’s algorithm.},

series = {Texts and Monographs in Symbolic Computation},

pages = {335--394},

publisher = {Springer},

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

year = {2021},

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

refereed = {yes},

length = {61},

url = {https://doi.org/10.1007/978-3-030-80219-6_15 }

}

[Paule]

### MacMahon's partition analysis XIII: Schmidt type partitions and modular forms

#### George E. Andrews and Peter Paule

Journal of Number Theory, pp. 95-119. 2021. Elsevier, ISSN 0022-314X. [doi] [pdf]@

author = {George E. Andrews and Peter Paule },

title = {{MacMahon's partition analysis XIII: Schmidt type partitions and modular forms}},

language = {english},

abstract = {In 1999, Frank Schmidt noted that the number of partitions of integers with distinct parts in which the first, third, fifth, etc., summands add to $n$ is equal to $p(n)$, the number of partitions of $n$. The object of this paper is to provide a context for this result which leads directly to many other theorems of this nature and which can be viewed as a continuation of our work on elongated partition diamonds. Again generating functions are infinite products built by the Dedekind eta function which, in turn, lead to interesting arithmetic theorems and conjectures for the related partition functions. },

journal = {Journal of Number Theory},

pages = {95--119},

publisher = {Elsevier},

isbn_issn = {ISSN 0022-314X},

year = {2021},

refereed = {yes},

length = {18},

url = {http://doi.org/10.1016/j.jnt.2021.09.008},

type = {open access}

}

**article**{RISC6373,author = {George E. Andrews and Peter Paule },

title = {{MacMahon's partition analysis XIII: Schmidt type partitions and modular forms}},

language = {english},

abstract = {In 1999, Frank Schmidt noted that the number of partitions of integers with distinct parts in which the first, third, fifth, etc., summands add to $n$ is equal to $p(n)$, the number of partitions of $n$. The object of this paper is to provide a context for this result which leads directly to many other theorems of this nature and which can be viewed as a continuation of our work on elongated partition diamonds. Again generating functions are infinite products built by the Dedekind eta function which, in turn, lead to interesting arithmetic theorems and conjectures for the related partition functions. },

journal = {Journal of Number Theory},

pages = {95--119},

publisher = {Elsevier},

isbn_issn = {ISSN 0022-314X},

year = {2021},

refereed = {yes},

length = {18},

url = {http://doi.org/10.1016/j.jnt.2021.09.008},

type = {open access}

}

[Radu]

### Infinite product formulae for generating functions for sequences of squares

#### Christian Krattenthaler, Mircea Merca, Cristian-Silviu Radu

In: Transcendence in Algebra, Combinatorics, Geometry and Number Theory. TRANS 2019, Bostan A., Raschel K. (ed.), Springer Proceedings in Mathematics & Statistics 373, pp. 193-236. 2021. Springer, Cham, 978-3-030-84303-8. [doi]@

author = {Christian Krattenthaler and Mircea Merca and Cristian-Silviu Radu},

title = {{Infinite product formulae for generating functions for sequences of squares}},

booktitle = {{Transcendence in Algebra, Combinatorics, Geometry and Number Theory. TRANS 2019}},

language = {english},

series = {Springer Proceedings in Mathematics & Statistics},

volume = {373},

pages = {193--236},

publisher = {Springer, Cham},

isbn_issn = {978-3-030-84303-8},

year = {2021},

editor = {Bostan A. and Raschel K.},

refereed = {yes},

length = {44},

url = {https://doi.org/10.1007/978-3-030-84304-5_8}

}

**incollection**{RISC6397,author = {Christian Krattenthaler and Mircea Merca and Cristian-Silviu Radu},

title = {{Infinite product formulae for generating functions for sequences of squares}},

booktitle = {{Transcendence in Algebra, Combinatorics, Geometry and Number Theory. TRANS 2019}},

language = {english},

series = {Springer Proceedings in Mathematics & Statistics},

volume = {373},

pages = {193--236},

publisher = {Springer, Cham},

isbn_issn = {978-3-030-84303-8},

year = {2021},

editor = {Bostan A. and Raschel K.},

refereed = {yes},

length = {44},

url = {https://doi.org/10.1007/978-3-030-84304-5_8}

}

### 2020

[Goswami]

### On sums of coefficients of polynomials related to the Borwein conjectures

#### Ankush Goswami, Venkata Raghu Tej Pantangi

Technical report no. 20-07 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). Ramanujan J. (to appear), May 2020. [pdf]@

author = {Ankush Goswami and Venkata Raghu Tej Pantangi},

title = {{On sums of coefficients of polynomials related to the Borwein conjectures}},

language = {english},

abstract = {Recently, Li (Int. J. Number Theory 2020) obtained an asymptotic formula for a certain partial sum involving coefficients for the polynomial in the First Borwein conjecture. As a consequence, he showed the positivity of this sum. His result was based on a sieving principle discovered by himself and Wan (Sci. China. Math. 2010). In fact, Li points out in his paper that his method can be generalized to prove an asymptotic formula for a general partial sum involving coefficients for any prime $p>3$. In this work, we extend Li's method to obtain asymptotic formula for several partial sums of coefficients of a very general polynomial. We find that in the special cases $p=3, 5$, the signs of these sums are consistent with the three famous Borwein conjectures. Similar sums have been studied earlier by Zaharescu (Ramanujan J. 2006) using a completely different method. We also improve on the error terms in the asymptotic formula for Li and Zaharescu. Using a recent result of Borwein (JNT 1993), we also obtain an asymptotic estimate for the maximum of the absolute value of these coefficients for primes $p=2, 3, 5, 7, 11, 13$ and for $p>15$, we obtain a lower bound on the maximum absolute value of these coefficients for sufficiently large $n$.},

number = {20-07},

year = {2020},

month = {May},

howpublished = {Ramanujan J. (to appear)},

length = {13},

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**{RISC6113,author = {Ankush Goswami and Venkata Raghu Tej Pantangi},

title = {{On sums of coefficients of polynomials related to the Borwein conjectures}},

language = {english},

abstract = {Recently, Li (Int. J. Number Theory 2020) obtained an asymptotic formula for a certain partial sum involving coefficients for the polynomial in the First Borwein conjecture. As a consequence, he showed the positivity of this sum. His result was based on a sieving principle discovered by himself and Wan (Sci. China. Math. 2010). In fact, Li points out in his paper that his method can be generalized to prove an asymptotic formula for a general partial sum involving coefficients for any prime $p>3$. In this work, we extend Li's method to obtain asymptotic formula for several partial sums of coefficients of a very general polynomial. We find that in the special cases $p=3, 5$, the signs of these sums are consistent with the three famous Borwein conjectures. Similar sums have been studied earlier by Zaharescu (Ramanujan J. 2006) using a completely different method. We also improve on the error terms in the asymptotic formula for Li and Zaharescu. Using a recent result of Borwein (JNT 1993), we also obtain an asymptotic estimate for the maximum of the absolute value of these coefficients for primes $p=2, 3, 5, 7, 11, 13$ and for $p>15$, we obtain a lower bound on the maximum absolute value of these coefficients for sufficiently large $n$.},

number = {20-07},

year = {2020},

month = {May},

howpublished = {Ramanujan J. (to appear)},

length = {13},

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)}

}

[Goswami]

### Some formulae for coefficients in restricted $q$-products

#### Ankush Goswami, Venkata Raghu Tej Pantangi

Technical report no. 20-08 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). May 2020. [pdf]@

author = {Ankush Goswami and Venkata Raghu Tej Pantangi},

title = {{Some formulae for coefficients in restricted $q$-products}},

language = {english},

abstract = {In this paper, we derive some formulae involving coefficients of polynomials which occur quite naturally in the study of restricted partitions. Our method involves a recently discovered sieve technique by Li and Wan (Sci. China. Math. 2010). Based on this method, by considering cyclic groups of different orders we obtain some new results for these coefficients. The general result (see Theorem \ref{main00}) holds for any group of the form $\mathbb{Z}_{N}$ where $N\in\mathbb{N}$ and expresses certain partial sums of coefficients in terms of expressions involving roots of unity. By specializing $N$ to different values, we see that these expressions simplify in some cases and we obtain several nice identities involving these coefficients. We also use a result of Sudler (Quarterly J. Math. 1964) to obtain an asymptotic formula for the maximum absolute value of these coefficients.},

number = {20-08},

year = {2020},

month = {May},

length = {11},

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**{RISC6114,author = {Ankush Goswami and Venkata Raghu Tej Pantangi},

title = {{Some formulae for coefficients in restricted $q$-products}},

language = {english},

abstract = {In this paper, we derive some formulae involving coefficients of polynomials which occur quite naturally in the study of restricted partitions. Our method involves a recently discovered sieve technique by Li and Wan (Sci. China. Math. 2010). Based on this method, by considering cyclic groups of different orders we obtain some new results for these coefficients. The general result (see Theorem \ref{main00}) holds for any group of the form $\mathbb{Z}_{N}$ where $N\in\mathbb{N}$ and expresses certain partial sums of coefficients in terms of expressions involving roots of unity. By specializing $N$ to different values, we see that these expressions simplify in some cases and we obtain several nice identities involving these coefficients. We also use a result of Sudler (Quarterly J. Math. 1964) to obtain an asymptotic formula for the maximum absolute value of these coefficients.},

number = {20-08},

year = {2020},

month = {May},

length = {11},

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)}

}

[Goswami]

### Congruences for generalized Fishburn numbers at roots of unity

#### Ankush Goswami

Technical report no. 20-09 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). 2020. [pdf]@

author = {Ankush Goswami},

title = {{Congruences for generalized Fishburn numbers at roots of unity}},

language = {english},

abstract = {There has been significant recent interest in the arithmeticproperties of the coefficients of $F(1-q)$ and $\mathcal{F}_t(1-q)$where $F(q)$ is the Kontsevich-Zagier strange series and $\mathcal{F}_t(q)$ is the strange series associated to a family of torus knots as studied by Bijaoui, Boden, Myers, Osburn, Rushworth, Tronsgardand Zhou. In this paper, we prove prime power congruences for two families of generalized Fishburn numbers, namely, for the coefficients of $(\zeta_N - q)^s F((\zeta_N - q)^r)$ and $(\zeta_N - q)^s \mathcal{F}_t((\zeta_N -q)^r)$, where $\zeta_N$ is an $N$th root of unity and $r$, $s$ are certain integers.},

number = {20-09},

year = {2020},

length = {17},

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**{RISC6119,author = {Ankush Goswami},

title = {{Congruences for generalized Fishburn numbers at roots of unity}},

language = {english},

abstract = {There has been significant recent interest in the arithmeticproperties of the coefficients of $F(1-q)$ and $\mathcal{F}_t(1-q)$where $F(q)$ is the Kontsevich-Zagier strange series and $\mathcal{F}_t(q)$ is the strange series associated to a family of torus knots as studied by Bijaoui, Boden, Myers, Osburn, Rushworth, Tronsgardand Zhou. In this paper, we prove prime power congruences for two families of generalized Fishburn numbers, namely, for the coefficients of $(\zeta_N - q)^s F((\zeta_N - q)^r)$ and $(\zeta_N - q)^s \mathcal{F}_t((\zeta_N -q)^r)$, where $\zeta_N$ is an $N$th root of unity and $r$, $s$ are certain integers.},

number = {20-09},

year = {2020},

length = {17},

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)}

}

[Radu]

### A reduction theorem of certain relations modulo p involving modular forms

#### Radu, Cristian-Silviu

In: Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra, in Honour of Peter Paule on his 60th Birthday, V. Pillwein, C. Schneider (ed.), pp. 317-337. 2020. Springer, 978-3-030-44558-4. [doi] [pdf]@

author = {Radu and Cristian-Silviu},

title = {{A reduction theorem of certain relations modulo p involving modular forms}},

booktitle = {{Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra, in Honour of Peter Paule on his 60th Birthday}},

language = {english},

pages = {317--337},

publisher = {Springer},

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

year = {2020},

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

refereed = {yes},

length = {21},

url = {https://doi.org/10.1007/978-3-030-44559-1_16}

}

**incollection**{RISC6110,author = {Radu and Cristian-Silviu},

title = {{A reduction theorem of certain relations modulo p involving modular forms}},

booktitle = {{Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra, in Honour of Peter Paule on his 60th Birthday}},

language = {english},

pages = {317--337},

publisher = {Springer},

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

year = {2020},

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

refereed = {yes},

length = {21},

url = {https://doi.org/10.1007/978-3-030-44559-1_16}

}

### 2019

[Goswami]

### A q-analogue for Euler’s evaluations of the Riemann zeta function

#### Ankush Goswami

Research in Number Theory 5:3, pp. 1-11. 2019. Springer, 10.1007.@

author = {Ankush Goswami},

title = {{A q-analogue for Euler’s evaluations of the Riemann zeta function}},

language = {english},

journal = {Research in Number Theory},

volume = {5:3},

pages = {1--11},

publisher = {Springer},

isbn_issn = {10.1007},

year = {2019},

refereed = {yes},

length = {11}

}

**article**{RISC5959,author = {Ankush Goswami},

title = {{A q-analogue for Euler’s evaluations of the Riemann zeta function}},

language = {english},

journal = {Research in Number Theory},

volume = {5:3},

pages = {1--11},

publisher = {Springer},

isbn_issn = {10.1007},

year = {2019},

refereed = {yes},

length = {11}

}

[Goswami]

### A q-analogue for Euler’s $\zeta(6)=\pi^6/6$

#### Ankush Goswami

In: Combinatory Analysis 2018: A Conference in Honor of George Andrews' 80th Birthday, Springer-Birkhauser (ed.), Proceedings of Combinatory Analysis 2018: A Conference in Honor of George Andrews' 80th Birthday, pp. 1-5. 2019. none.@

author = {Ankush Goswami},

title = {{A q-analogue for Euler’s $\zeta(6)=\pi^6/6$}},

booktitle = {{Combinatory Analysis 2018: A Conference in Honor of George Andrews' 80th Birthday}},

language = {english},

pages = {1--5},

isbn_issn = {none},

year = {2019},

editor = {Springer-Birkhauser},

refereed = {yes},

length = {5},

conferencename = {Combinatory Analysis 2018: A Conference in Honor of George Andrews' 80th Birthday}

}

**inproceedings**{RISC5960,author = {Ankush Goswami},

title = {{A q-analogue for Euler’s $\zeta(6)=\pi^6/6$}},

booktitle = {{Combinatory Analysis 2018: A Conference in Honor of George Andrews' 80th Birthday}},

language = {english},

pages = {1--5},

isbn_issn = {none},

year = {2019},

editor = {Springer-Birkhauser},

refereed = {yes},

length = {5},

conferencename = {Combinatory Analysis 2018: A Conference in Honor of George Andrews' 80th Birthday}

}

[Goswami]

### Some Problems in Analytic Number Theory

#### Ankush Goswami

University of Florida. PhD Thesis. 2019. First part of this thesis is to appear in Proceedings of Analytic and Combinatorial Number Theory: The Legacy of Ramanujan - A CONFERENCE IN HONOR OF BRUCE C. BERNDT'S 80TH BIRTHDAY.@

author = {Ankush Goswami},

title = {{Some Problems in Analytic Number Theory}},

language = {english},

year = {2019},

note = {First part of this thesis is to appear in Proceedings of Analytic and Combinatorial Number Theory: The Legacy of Ramanujan -- A CONFERENCE IN HONOR OF BRUCE C. BERNDT'S 80TH BIRTHDAY},

translation = {0},

school = {University of Florida},

length = {90}

}

**phdthesis**{RISC5961,author = {Ankush Goswami},

title = {{Some Problems in Analytic Number Theory}},

language = {english},

year = {2019},

note = {First part of this thesis is to appear in Proceedings of Analytic and Combinatorial Number Theory: The Legacy of Ramanujan -- A CONFERENCE IN HONOR OF BRUCE C. BERNDT'S 80TH BIRTHDAY},

translation = {0},

school = {University of Florida},

length = {90}

}

[Goswami]

### On the parity of some partition functions

#### Ankush Goswami, Abhash Kumar Jha

Technical report no. 20-06 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). 2019. [pdf]@

author = {Ankush Goswami and Abhash Kumar Jha},

title = {{On the parity of some partition functions}},

language = {english},

abstract = {Recently, Andrews carried out a thorough investigation of integer partitions in which all parts of a given parity are smaller than those of the opposite parity. Further, considering a subset of this set of partitions, he obtains several interesting arithmetic and combinatorial properties and its connections to the third order mock theta function $\nu(q)$. In fact, he shows the existence of a Dyson-type crank that explains a mod $5$ congruence in this subset. At the end of his paper, one of the problems he poses is to undertake a more extensive investigation on the properties of the subset of partitions. Since then there have been several investigations in various ways, including works of Jennings-Shaffer and Bringmann (Ann. Comb. 2019), Barman and Ray (2019), and Uncu (2019). In this paper, we study certain congruences satisfied by the above set of partitions (and the subset above) along with a certain subset of partitions (of Andrews' partitions above) studied by Uncu and also establish a connection between one of Andrews' partition function above with $p(n)$, the number of unrestricted partitions of $n$. Besides, we provide a combinatorial description of Uncu's partition function. },

number = {20-06},

year = {2019},

sponsor = {First author: SFB F50-06 of the Austrian Science Fund (FWF)},

length = {15},

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**{RISC6112,author = {Ankush Goswami and Abhash Kumar Jha},

title = {{On the parity of some partition functions}},

language = {english},

abstract = {Recently, Andrews carried out a thorough investigation of integer partitions in which all parts of a given parity are smaller than those of the opposite parity. Further, considering a subset of this set of partitions, he obtains several interesting arithmetic and combinatorial properties and its connections to the third order mock theta function $\nu(q)$. In fact, he shows the existence of a Dyson-type crank that explains a mod $5$ congruence in this subset. At the end of his paper, one of the problems he poses is to undertake a more extensive investigation on the properties of the subset of partitions. Since then there have been several investigations in various ways, including works of Jennings-Shaffer and Bringmann (Ann. Comb. 2019), Barman and Ray (2019), and Uncu (2019). In this paper, we study certain congruences satisfied by the above set of partitions (and the subset above) along with a certain subset of partitions (of Andrews' partitions above) studied by Uncu and also establish a connection between one of Andrews' partition function above with $p(n)$, the number of unrestricted partitions of $n$. Besides, we provide a combinatorial description of Uncu's partition function. },

number = {20-06},

year = {2019},

sponsor = {First author: SFB F50-06 of the Austrian Science Fund (FWF)},

length = {15},

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)}

}

[Hemmecke]

### Construction of all Polynomial Relations among Dedekind Eta Functions of Level $N$

#### Ralf Hemmecke, Silviu Radu

Journal of Symbolic Compuation 95, pp. 39-52. 2019. ISSN 0747-7171. Also available as RISC Report 18-03 http://www.risc.jku.at/publications/download/risc_5561/etarelations.pdf. [doi]@

author = {Ralf Hemmecke and Silviu Radu},

title = {{Construction of all Polynomial Relations among Dedekind Eta Functions of Level $N$}},

language = {english},

abstract = {We describe an algorithm that, given a positive integer $N$,computes a Gr\"obner basis of the ideal of polynomial relations among Dedekind$\eta$-functions of level $N$, i.e., among the elements of$\{\eta(\delta_1\tau),\ldots,\eta(\delta_n\tau)\}$ where$1=\delta_1<\delta_2\dots<\delta_n=N$ are the positive divisors of$N$.More precisely, we find a finite generating set (which is also aGr\"obner basis of the ideal $\ker\phi$ where\begin{gather*} \phi:Q[E_1,\ldots,E_n] \to Q[\eta(\delta_1\tau),\ldots,\eta(\delta_n\tau)], \quad E_k\mapsto \eta(\delta_k\tau), \quad k=1,\ldots,n.\end{gather*}},

journal = {Journal of Symbolic Compuation},

volume = {95},

pages = {39--52},

isbn_issn = {ISSN 0747-7171},

year = {2019},

note = {Also available as RISC Report 18-03 http://www.risc.jku.at/publications/download/risc_5561/etarelations.pdf},

refereed = {yes},

keywords = {Dedekind $\eta$ function, modular functions, modular equations, ideal of relations, Groebner basis},

length = {14},

url = {https://doi.org/10.1016/j.jsc.2018.10.001}

}

**article**{RISC5703,author = {Ralf Hemmecke and Silviu Radu},

title = {{Construction of all Polynomial Relations among Dedekind Eta Functions of Level $N$}},

language = {english},

abstract = {We describe an algorithm that, given a positive integer $N$,computes a Gr\"obner basis of the ideal of polynomial relations among Dedekind$\eta$-functions of level $N$, i.e., among the elements of$\{\eta(\delta_1\tau),\ldots,\eta(\delta_n\tau)\}$ where$1=\delta_1<\delta_2\dots<\delta_n=N$ are the positive divisors of$N$.More precisely, we find a finite generating set (which is also aGr\"obner basis of the ideal $\ker\phi$ where\begin{gather*} \phi:Q[E_1,\ldots,E_n] \to Q[\eta(\delta_1\tau),\ldots,\eta(\delta_n\tau)], \quad E_k\mapsto \eta(\delta_k\tau), \quad k=1,\ldots,n.\end{gather*}},

journal = {Journal of Symbolic Compuation},

volume = {95},

pages = {39--52},

isbn_issn = {ISSN 0747-7171},

year = {2019},

note = {Also available as RISC Report 18-03 http://www.risc.jku.at/publications/download/risc_5561/etarelations.pdf},

refereed = {yes},

keywords = {Dedekind $\eta$ function, modular functions, modular equations, ideal of relations, Groebner basis},

length = {14},

url = {https://doi.org/10.1016/j.jsc.2018.10.001}

}

[Hemmecke]

### The Generators of all Polynomial Relations among Jacobi Theta Functions

#### Ralf Hemmecke, Silviu Radu, Liangjie Ye

In: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, Johannes Blümlein and Carsten Schneider and Peter Paule (ed.), Texts & Monographs in Symbolic Computation 18-09, pp. 259-268. 2019. Springer International Publishing, Cham, 978-3-030-04479-4. Also available as RISC Report 18-09 http://www.risc.jku.at/publications/download/risc_5719/thetarelations.pdf. [doi]@

author = {Ralf Hemmecke and Silviu Radu and Liangjie Ye},

title = {{The Generators of all Polynomial Relations among Jacobi Theta Functions}},

booktitle = {{Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory}},

language = {english},

abstract = {In this article, we consider the classical Jacobi theta functions$\theta_i(z)$, $i=1,2,3,4$ and show that the ideal of all polynomialrelations among them with coefficients in$K :=\setQ(\theta_2(0|\tau),\theta_3(0|\tau),\theta_4(0|\tau))$ isgenerated by just two polynomials, that correspond to well knownidentities among Jacobi theta functions.},

series = {Texts & Monographs in Symbolic Computation},

number = {18-09},

pages = {259--268},

publisher = {Springer International Publishing},

address = {Cham},

isbn_issn = {978-3-030-04479-4},

year = {2019},

note = {Also available as RISC Report 18-09 http://www.risc.jku.at/publications/download/risc_5719/thetarelations.pdf},

editor = {Johannes Blümlein and Carsten Schneider and Peter Paule},

refereed = {yes},

length = {9},

url = {https://doi.org/10.1007/978-3-030-04480-0_11}

}

**incollection**{RISC5913,author = {Ralf Hemmecke and Silviu Radu and Liangjie Ye},

title = {{The Generators of all Polynomial Relations among Jacobi Theta Functions}},

booktitle = {{Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory}},

language = {english},

abstract = {In this article, we consider the classical Jacobi theta functions$\theta_i(z)$, $i=1,2,3,4$ and show that the ideal of all polynomialrelations among them with coefficients in$K :=\setQ(\theta_2(0|\tau),\theta_3(0|\tau),\theta_4(0|\tau))$ isgenerated by just two polynomials, that correspond to well knownidentities among Jacobi theta functions.},

series = {Texts & Monographs in Symbolic Computation},

number = {18-09},

pages = {259--268},

publisher = {Springer International Publishing},

address = {Cham},

isbn_issn = {978-3-030-04479-4},

year = {2019},

note = {Also available as RISC Report 18-09 http://www.risc.jku.at/publications/download/risc_5719/thetarelations.pdf},

editor = {Johannes Blümlein and Carsten Schneider and Peter Paule},

refereed = {yes},

length = {9},

url = {https://doi.org/10.1007/978-3-030-04480-0_11}

}

### 2018

[Jiu]

### 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]@

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}

}

**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}

}

[Magnusson]

### The functional equation of Dedekind's $\eta$-function

#### Tobias Magnusson

Submitted to the RISC Report Series. June 15 2018. [pdf] [tex]@

author = {Tobias Magnusson},

title = {{The functional equation of Dedekind's $\eta$-function}},

language = {English},

year = {2018},

month = {June 15},

length = {14},

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**{RISC5701,author = {Tobias Magnusson},

title = {{The functional equation of Dedekind's $\eta$-function}},

language = {English},

year = {2018},

month = {June 15},

length = {14},

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)}

}

### 2017

[Jiu]

### An extension of the method of brackets. Part 1

#### Ivan Gonzalez, Karen Kohl, Lin Jiu, and Victor H. Moll

Open Mathematics (formerly Central European Journal of Mathematics) 15, pp. 1181-1211. 2017. 2391-5455. [url]@

author = {Ivan Gonzalez and Karen Kohl and Lin Jiu and and Victor H. Moll},

title = {{An extension of the method of brackets. Part 1}},

language = {english},

abstract = {The method of brackets is an efficient method for the evaluation of a large class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the concepts of null and divergent series. These are formal representations of functions, whose coefficients $a_n$ have meromorphic representations for $n\in\mathbb{C}$, but might vanish or blow up when $n\in\mathbb{N}$. These ideas are illustrated with the evaluation of a variety of entries from the classical table of integrals by Gradshteyn and Ryzhik.},

journal = {Open Mathematics (formerly Central European Journal of Mathematics)},

volume = {15},

pages = {1181--1211},

isbn_issn = {2391-5455},

year = {2017},

refereed = {no},

length = {31},

url = {https://www.degruyter.com/view/j/math.2017.15.issue-1/math-2017-0100/math-2017-0100.xml?format=INT}

}

**article**{RISC5483,author = {Ivan Gonzalez and Karen Kohl and Lin Jiu and and Victor H. Moll},

title = {{An extension of the method of brackets. Part 1}},

language = {english},

abstract = {The method of brackets is an efficient method for the evaluation of a large class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the concepts of null and divergent series. These are formal representations of functions, whose coefficients $a_n$ have meromorphic representations for $n\in\mathbb{C}$, but might vanish or blow up when $n\in\mathbb{N}$. These ideas are illustrated with the evaluation of a variety of entries from the classical table of integrals by Gradshteyn and Ryzhik.},

journal = {Open Mathematics (formerly Central European Journal of Mathematics)},

volume = {15},

pages = {1181--1211},

isbn_issn = {2391-5455},

year = {2017},

refereed = {no},

length = {31},

url = {https://www.degruyter.com/view/j/math.2017.15.issue-1/math-2017-0100/math-2017-0100.xml?format=INT}

}

[Jiu]

### Integral representations of equally positive integer-indexed harmonic sums at infinity

#### L. Jiu

Research in Number Theory 3(10), pp. 1-4. 2017. 2363-9555. [url]@

author = {L. Jiu},

title = {{Integral representations of equally positive integer-indexed harmonic sums at infinity}},

language = {English},

abstract = {We identify a partition-theoretic generalization of Riemann zeta function and the equally positive integer-indexed harmonic sums at infinity, to obtain the generating function and the integral representations of the latter. The special cases coincide with zeta values at positive integer arguments.},

journal = {Research in Number Theory},

volume = {3},

number = {10},

pages = {1--4},

isbn_issn = {2363-9555},

year = {2017},

refereed = {no},

length = {4},

url = {https://resnumtheor.springeropen.com/articles/10.1007/s40993-017-0074-x}

}

**article**{RISC5385,author = {L. Jiu},

title = {{Integral representations of equally positive integer-indexed harmonic sums at infinity}},

language = {English},

abstract = {We identify a partition-theoretic generalization of Riemann zeta function and the equally positive integer-indexed harmonic sums at infinity, to obtain the generating function and the integral representations of the latter. The special cases coincide with zeta values at positive integer arguments.},

journal = {Research in Number Theory},

volume = {3},

number = {10},

pages = {1--4},

isbn_issn = {2363-9555},

year = {2017},

refereed = {no},

length = {4},

url = {https://resnumtheor.springeropen.com/articles/10.1007/s40993-017-0074-x}

}