QEtaRamanujanKolberg(C, MODG)¶

C: EuclideanDomain
MODG: QEtaModularGammaCategory

QEtaRamanujanKolberg provides functions to compute Ramanujan-Kolberg identities by an algorithm that was developed by Radu in cite{Radu_RamanujanKolberg_2015} (https://doi.org/10.1016/j.jsc.2014.09.018), and turned into the Samba algorithm by Hemmecke in cite{Hemmecke_DancingSambaRamanujan_2018} (https://doi.org/10.1016/j.jsc.2017.02.001). See also at the top of the qetark.spad file where this package is implemented. This package also builds on the work of cite{ChenDuZhao_FindingModularFunctionsRamanujan_2019} (https://doi.org/10.1007/s00026-019-00457-4) and finds identities with generalized eta-quotients.

findIdentity: (List List Integer, QEtaSpecification, PositiveInteger, NonNegativeInteger, List List Integer) -> QEtaRamanujanKolbergIdentity C: findIdentity(sidxs, rspec, m, k, bidxs) returns findIdentity(sidxs,dissect,bidxs) where dissect:=choose(m,t)(generatingFunction(rspec)).

findIdentity: (List List Integer, QEtaSpecification, PositiveInteger, NonNegativeInteger, List QEtaSpecification) -> QEtaRamanujanKolbergIdentity C: findIdentity(idxs, rspec, m, k, mgens) returns findIdentity(idxs,dissect,mgens) where dissect:=choose(m,t)(generatingFunction(rspec)).

findIdentity: (List List Integer, QEtaSpecification, PositiveInteger, NonNegativeInteger, QEtaRamanujanKolbergIdentity C) -> QEtaRamanujanKolbergIdentity C: findIdentity(sidxs,rspec,m,k,id) returns findIdentity(sidxs,dissect,id) where dissect:=choose(m,t)(generatingFunction(rspec)).

findIdentity: (List List Integer, QGeneratingFunctionVariable, List List Integer) -> QEtaRamanujanKolbergIdentity C: findIdentity(sidxs,dissect,bidxs) returns all data for a Ramanujan-Kolberg identity where the indices for the cofactor and the generalized eta-quotients on the right-hand side are given by sidxs and bidxs, respectively.

findIdentity: (List List Integer, QGeneratingFunctionVariable, List QEtaSpecification) -> QEtaRamanujanKolbergIdentity C: findIdentity(idxs,dissect,mgens) returns all data for a Ramanujan-Kolberg identity where the cofactor may only involve generalized eta-functions given through the indices idxs. Note that elements from mspecs that specify constant generalized eta-quotients will be removed.

findIdentity: (List List Integer, QGeneratingFunctionVariable, QEtaRamanujanKolbergIdentity C) -> QEtaRamanujanKolbergIdentity C: findIdentity(sidxs,dissect,id) returns all data for a Ramanujan-Kolberg identity. The basis elements are taken from a previously computed identity, if possible.

findIdentity: (QEtaSpecification, PositiveInteger, NonNegativeInteger, List List Integer) -> QEtaRamanujanKolbergIdentity C: findIdentity(rspec,m,k,idxs) returns findIdentity(idxs,rspec,m,k,idxs).

findIdentity: (QEtaSpecification, QEtaSpecification, PositiveInteger, NonNegativeInteger, List List Integer) -> QEtaRamanujanKolbergIdentity C: findIdentity(sspec,rspec,m,k,idxs) returns findIdentity(sspec,dissect,idxs) where dissect:=choose(m,t)(generatingFunction(rspec)).

findIdentity: (QEtaSpecification, QEtaSpecification, PositiveInteger, NonNegativeInteger, List QEtaSpecification) -> QEtaRamanujanKolbergIdentity C: findIdentity(sspec, rspec, m, k, mspecs) returns findIdentity(sspec,dissect,mspecs) where dissect:=choose(m,t)(generatingFunction(rspec)).

findIdentity: (QEtaSpecification, QEtaSpecification, PositiveInteger, NonNegativeInteger, List QEtaSpecification, QEtaAlgebraBasis QEtaExtendedAlgebra(C, ModularFunctionQSeriesInfinity C, QEtaLazyAlgebra(C, Polynomial C))) -> QEtaRamanujanKolbergIdentity C: findIdentity(sspec,rspec,m,k,mspecs, xab) returns findIdentity(sspec,dissect,mspecs,xab) where dissect:=choose(m,t)(generatingFunction(rspec)).

findIdentity: (QEtaSpecification, QEtaSpecification, PositiveInteger, NonNegativeInteger, QEtaRamanujanKolbergIdentity C) -> QEtaRamanujanKolbergIdentity C: findIdentity(sspec, rspec, m, k, id) returns findIdentity(sspec,dissect,id) where dissect:=choose(m,t)(generatingFunction(rspec)).

findIdentity: (QEtaSpecification, QGeneratingFunctionVariable, List List Integer) -> QEtaRamanujanKolbergIdentity C: findIdentity(sspec,dissect,idxs) returns all data for a Ramanujan-Kolberg identity where the right-hand side may only involve generalized eta-quotients that are given through the indices idxs.

findIdentity: (QEtaSpecification, QGeneratingFunctionVariable, List QEtaSpecification) -> QEtaRamanujanKolbergIdentity C: findIdentity(sspec,dissect,mspecs) returns all data for a Ramanujan-Kolberg identity. Note that elements from mspecs that specify constant generalized eta-quotients will be removed.

findIdentity: (QEtaSpecification, QGeneratingFunctionVariable, List QEtaSpecification, QEtaAlgebraBasis QEtaExtendedAlgebra(C, ModularFunctionQSeriesInfinity C, QEtaLazyAlgebra(C, Polynomial C))) -> QEtaRamanujanKolbergIdentity C: findIdentity(sspec,dissect,mspecs,xab) returns all data for a Ramanujan-Kolberg identity. It is assumed that mspecs describe the Mi variables in xab in the same order.

findIdentity: (QEtaSpecification, QGeneratingFunctionVariable, QEtaRamanujanKolbergIdentity C) -> QEtaRamanujanKolbergIdentity C: findIdentity(sspec,dissect,id) returns all data for a Ramanujan-Kolberg identity. The basis elements are taken from a previously computed identity, if possible.

findIdentity: (QGeneratingFunctionVariable, List List Integer) -> QEtaRamanujanKolbergIdentity C: findIdentity(dissect,idxs) returns findIdentity(idxs,rspec,m,k,idxs).