QEtaRepresentation¶

QEtaRepresentation implements functions to represent a $q$-series as a (generalized) eta-quotient.

etaQuotientSpecification: (List List Integer, QEtaLaurentSeries Integer, PositiveInteger) -> Record(fspec: QEtaSpecification, fexp: Fraction Integer, fser: QEtaLaurentSeries Integer): etaSpecification(idxs,s,n) assumes that s=q^a*t and t is a Taylor series with constant term 1 and returns [spec,e,r], i.e. an eta-quotient specification together with a series r such that if u=etaQuotientInfintiy(spec), then t and eulerExpansion(u) have the same coefficients up to the coefficient of q^n, a=e+eulerExponent(u), and t=r*eulerExpansion(u). The specification will contain only indices from idxs. The function basically fails if order(eulerExpansion(r),n)<n.

etaQuotientSpecification: (PositiveInteger, QEtaLaurentSeries Integer, PositiveInteger) -> Record(fspec: QEtaSpecification, fexp: Fraction Integer, fser: QEtaLaurentSeries Integer): etaSpecification(nn,s,n) assumes that s=q^a*t and t is a Taylor series with constant term 1 and returns [spec,e,r], i.e. an eta-quotient specification together with a series r such that if u=etaQuotientInfintiy(spec), then t and eulerExpansion(u) have the same coefficients up to the coefficient of q^n, a=e+eulerExponent(u), and t=r*eulerExpansion(u). The specification may contain generalized eta-quotients and will be of level nn. In fact, it is equivalent to etaQuotientSpecification(idxs, s, n) for idxs:=concat(eidxs,pidxs) where pidxs=[[nn,g] for g in 1..floor((nn-1)/2)] and eidx=[nn/2,nn] if nn is even and eidxs=[nn] for odd nn. The function basically fails if order(eulerExpansion(r),n)<n.

etaQuotientSpecification: (QEtaLaurentSeries Integer, PositiveInteger) -> Record(fspec: QEtaSpecification, fexp: Fraction Integer, fser: QEtaLaurentSeries Integer): etaSpecification(s,n) assumes that s=q^a*t and t is a Taylor series with constant term 1 and returns [spec,e,r], i.e. an eta-quotient specification together with a series r such that if u=etaQuotientInfintiy(spec), then t and eulerExpansion(u) have the same coefficients up to the coefficient of q^n, a=e+eulerExponent(u), and t=r*eulerExpansion(u). Only pure eta-quotients are returned. In fact, the call etaSpecification(s,n) is equivalent to etaSpecification(idxs,s,n) for idxs=[[d] for d in 1..n]. The function basically fails if order(eulerExpansion(r),n)<n.