Dancing samba with Ramanujan partition congruences

Hemmecke, Ralf

doi:10.1016/j.jsc.2017.02.001

Cited by 8 publications

(9 citation statements)

References 4 publications

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“…One can use the reduction process from [3] (see, in particular, Theorem 3.2) to algorithmically determine polynomials c 0 , . .…”

Section: A Basis For the Eta-quotients In M ∞mentioning

confidence: 99%

“…For this article, we use an implementation of the algorithm samba from [3] in our QEta package. Given modular functions m 1 = t, m 2 .…”

Section: A Basis For the Eta-quotients In M ∞mentioning

confidence: 99%

“…Another kind of witness identity is presented in [4, Thm. 1.1]: Suppose + 11 4 (−5 • 11 12 + 2 • 5 • 11 8 • 17 t − 2 2 • 3 • 11 3 • 1289 t 2 + 3 • 41 t 3 )f + 11 5 (11 4 + t)(11 11 − 3 • 7 • 11 7 t + 11 2 • 1321 t 2 + t 3 ).…”

Section: Introductionmentioning

confidence: 99%

“…For example, when using modular functions to prove Ramanujan's congruences for powers of 11 (i.e. 11 2 | p(11 2 n + 116), 11 3 | p(11 3 n + 721), etc.) one needs to work with a C[t]-module representation of M ∞ (11).…”

Section: Introductionmentioning

confidence: 99%

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Construction of modular function bases for Γ₀(121) related to p(11n+6)

Hemmecke

Paule

Radu

2021

Integral Transforms and Special Functions

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Motivated by arithmetic properties of partition numbers p(n), our goal is to find algorithmically a Ramanujan type identity of the form ∞ n=0 p(11n + 6)q n = R, where R is a polynomial in products of the form e α := ∞ n=1 (1 − q 11 α n ) with α = 0, 1, 2. To this end we multiply the left side by an appropriate factor such the result is a modular function for 0 (121) having only poles at infinity. It turns out that polynomials in the e α do not generate the full space of such functions, so we were led to modify our goal. More concretely, we give three different ways to construct the space of modular functions for 0 (121) having only poles at infinity. This in turn leads to three different representations of R not solely in terms of the e α but, for example, by using as generators also other functions like the modular invariant j.

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“…One can use the reduction process from [3] (see, in particular, Theorem 3.2) to algorithmically determine polynomials c 0 , . .…”

Section: A Basis For the Eta-quotients In M ∞mentioning

confidence: 99%

“…For this article, we use an implementation of the algorithm samba from [3] in our QEta package. Given modular functions m 1 = t, m 2 .…”

Section: A Basis For the Eta-quotients In M ∞mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Construction of modular function bases for Γ₀(121) related to p(11n+6)

Hemmecke

Paule

Radu

2021

Integral Transforms and Special Functions

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“…Radu [35] obtained a witness identity for p(11n + 6) by using the Ramanujan-Kolberg algorithm. Hemmecke [20] generalized Radu's algorithm and derived a witness identity for p(11n + 6). Paule and Radu [31] found a polynomial relation on the generating function of p(11n + 6), which can also be viewed as a witness identity.…”

Section: Introductionmentioning

confidence: 99%

Finding Modular Functions for Ramanujan-Type Identities

Chen

Zhao

2019

Ann. Comb.

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This paper is concerned with a class of partition functions a(n) introduced by Radu and defined in terms of eta-quotients. By utilizing the transformation laws of Newman, Schoeneberg and Robins, and Radu's algorithms, we present an algorithm to find Ramanujan-type identities for a(mn + t). While this algorithm is not guaranteed to succeed, it applies to many cases. For example, we deduce a witness identity for p(11n + 6) with integer coefficients. Our algorithm also leads to Ramanujan-type identities for the overpartition functions p(5n + 2) and p(5n + 3) and Andrews-Paule's broken 2-diamond partition functions △ 2 (25n + 14) and △ 2 (25n + 24). It can also be extended to derive Ramanujan-type identities on a more general class of partition functions. For example, it yields the Ramanujan-type identities on Andrews' singular overpartition functions Q 3,1 (9n + 3) and Q 3,1 (9n + 6) due to Shen, the 2-dissection formulas of Ramanujan and the 8-dissection formulas due to Hirschhorn.Atkin and Swinnerton-Dyer [5] have shown that g t can always be expressed by certain infinite products for m > 3. Then the left hand side of (1.3) can be expressed in terms of certain infinite products. Kolberg pointed out that when m > 5, this becomes much more complicated. For m = 11, 13, Bilgici and Ekin [7,8] used the method of Kolberg to compute the generating function ∞ n=0 p(mn + t)q mn+t 1. M |N .

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