QEtaCofactorSpace¶

undocumented

alphaInfinity: (QEtaSpecification, QEtaSpecification, PositiveInteger, List NonNegativeInteger) -> Integer: alphaInfinity(sspec,rspec,m,orb) implements the definition eqref{eq:alphabarInfinity} and eqref{eq:alphaInfinity}, i.e. the definition of Radu in cite{Radu:RamanujanKolberg:2015}, DOI=10.1016/j.jsc.2017.02.001, http://www.risc.jku.at/publications/download/risc_5338/DancingSambaRamanujan.pdf and can also be extracted from from formula (10.4) of cite{Chen+Du+Zhao:FindingModularFunctionsRamanujan:2019} when the respective cofactor part is taken into account. Note that it does not agree with alpha(t) as defined in cite{Chen+Du+Zhao:FindingModularFunctionsRamanujan:2019}.

etaCofactorSpace0: (PositiveInteger, QEtaSpecification, PositiveInteger, NonNegativeInteger) -> Record(indices: List List Integer, particular: Union(Vector Integer, failed), basis: List Vector Integer): etaCofactorSpace0(nn,rspec,m,t) returns a vector v and the basis of a space such that #v=#divisors(nn) and modularGamma0?(nn,members(s),rspec,m,t) is true for any s = v + reduce(_+, [z.i * basis.i for i in 1..#basis]) and any sufficiently long list z of integers. The function fails, if there is no such solution. The indices part of the result is a list of divisors of nn where each divisor is represented as a one-element list.

etaCofactorSpace0System: (PositiveInteger, QEtaSpecification, PositiveInteger, NonNegativeInteger) -> Record(comatrix: Matrix Integer, rhs: Vector Integer): etaCofactorSpace0System(nn, rsoec, m, t) returns a matrix mat and a vector v such that for the integer solutions s of the equation mat*s=v it holds modularGamma0?(ssoec,rspec,m,t) where sspec results from the list of the first #divisors(nn) entries of s.

etaCofactorSpace1: (PositiveInteger, QEtaSpecification, PositiveInteger, NonNegativeInteger) -> Record(indices: List List Integer, particular: Union(Vector Integer, failed), basis: List Vector Integer): etaCofactorSpace1(nn,rspec,m,t) returns etaCofactorSpace1(nn,rspec,m,t,idxs) for idxs=generalizedEtaFunctionIndices(nn)$QETAAUX.

etaCofactorSpace1: (PositiveInteger, QEtaSpecification, PositiveInteger, NonNegativeInteger, List List Integer) -> Record(indices: List List Integer, particular: Union(Vector Integer, failed), basis: List Vector Integer): etaCofactorSpace1(nn,rspec,m,t,idxs) returns a vector v and the basis of a space such that #v=#idxs and modularGamma1?(ssepc,rspec,m,t) is true for any sbar = v + reduce(_+, [z.i * basis.i for i in 1..#basis]) and any sufficiently long list z of integers. The function fails, if there is no such solution. The indices part of the result is equal to idxs and corresponds to the entries of the solution vectors. If idxs=[] then the computation is done for idxs=generalizedEtaFunctionIndices(nn)$QETAAUX..

etaCofactorSpace1System: (PositiveInteger, QEtaSpecification, PositiveInteger, NonNegativeInteger) -> Record(comatrix: Matrix Integer, rhs: Vector Integer): etaCofactorSpace1System(nn, rspec, m, t) returns a matrix mat and a vector v such that for the integer solutions s of the equation mat*s=v it holds modularGamma1?(sspec,rspec,m,t) where sbar is the translation of s into a generalized eta-quotient specification, see generalizedEtaQuotientSpecification.