An algorithmic approach to Ramanujan–Kolberg identities

Radu, Cristian-Silviu

doi:10.1016/j.jsc.2014.09.018

Cited by 48 publications

(64 citation statements)

References 7 publications

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“…Given an eta quotient f , its expansion at ∞ has integer coefficients, as does its inverse 1/f . Moreover, we have a precise formula for the order of f at any given cusp, as given in [18,Theorem 23], generally attributed to Ligozat:…”

Section: )mentioning

confidence: 99%

“…In that case, ω · L 1 ∈ E ∞ (20) Q . In order to give the exact expression of ω · L 1 ∈ E ∞ (20) Q , we take advantage of an algorithm given in [18] to produce the following algebra basis for E ∞ (20) Q…”

Section: Rogers-ramanujan Subpartitionsmentioning

confidence: 99%

“…We want to determine its membership in E ∞ (N ) Q = 1, g 1 , ..., g v Q[t] , in which we assume that the functions g j , t satisfy the conditions of Theorem 8. We will describe the membership check algorithm (MW) from [18].…”

Section: Rogers-ramanujan Subpartitionsmentioning

confidence: 99%

“…In Section 3 we discuss the generating function (1.1) and outline the key algorithmic steps to check Theorem 2 for a large number of α. We make use of an important theorem whose proof can be found in [18], which allows us to construct a useful algebra basis for the space of eta quotients over Γ 0 (N ). From here we show how the basis can be suitably modified to interact more carefully with the U 5 operator.…”

Section: Introductionmentioning

confidence: 99%

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A method of verifying partition congruences by symbolic computation

Radu¹,

Smoot²

2021

Journal of Symbolic Computation

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Conjectures involving infinite families of restricted partition congruences can be difficult to verify for a number of individual cases, even with a computer. We demonstrate how the machinery of Radu's algorithm may be modified and employed to efficiently check a very large number of cases of such conjectures. This allows substantial evidence to be collected for a given conjecture, before a complete proof is attempted."...for certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards because their investigation by the said method did not furnish an actual demonstration. But it is of course easier, when we have previously acquired... some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge."-Archimedes, The Method CRISTIAN-SILVIU RADU AND NICOLAS ALLEN SMOOT a(25n + 24) ≡ 0 (mod 5), and suggested additional congruences for a(n) to higher powers of 5. At first sight, the matter of specifying a family of congruences might seem easy enough. Certainly, one could directly compute a list of the numerical values of a(mn + j) for a fixed m, j ∈ Z ≥0 , as n varies over a large number of nonnegative integers. We could program a computer to check the greatest common divisor of this list.Yet more interesting, one of the authors has developed algorithms [18] that can take series of the form ∞ n=0 a(mn + j)q n and expand them into a finite, linear combination of eta quotients. By examining the coefficients of each term in such a finite combination, and knowing that each eta quotient expands into an integer power series, we can often determine whether a(mn + j) is divisible by a given power of a prime (in our case, 5) for all n ∈ Z ≥0 .However, for 24n ≡ 1 (mod 5 2α ), one can quickly show that

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Section: )mentioning

confidence: 99%

Section: Rogers-ramanujan Subpartitionsmentioning

confidence: 99%

Section: Rogers-ramanujan Subpartitionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A method of verifying partition congruences by symbolic computation

Radu¹,

Smoot²

2021

Journal of Symbolic Computation

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“…Given the intricacy of this subject, we can only provide a brief outline here. The interested reader is invited to consult [4], [6,Chapters 1,2], and [8] for an outline of the general theory, and [6, Chapters 3-8] [10] and [11] for specific applications of the theory.…”

Section: Initial Casesmentioning

confidence: 99%