QEtaRepresentationΒΆ
qetarep.spad line 94 [edit on github]
QEtaRepresentation implements functions to represent a $q
$-series as a (generalized) eta-quotient.
- guessEtaQuotientSpecification: (List List Integer, QEtaLaurentSeries Integer, PositiveInteger) -> Record(fspec: QEtaSpecification, fexp: Fraction Integer, fser: QEtaLaurentSeries Integer)
guessEtaQuotientSpecification(idxs,s,n)
assumes that s=q^a*t for some integer a andt
is a Taylor series with constant term 1 and returns [spec,e
,r
], i.e. an eta-quotient specification together with a (rational) exponente
and a seriesr
such that if u=etaQuotientInfintiy(spec), thent
and eulerExpansion(u
) have the same coefficients up to the coefficient ofq^n
, a=e+eulerExponent(u
), and t=r*eulerExpansion(u
). In other words, s=u*q^e*r whereu
is the eta-quotient given by spec. The specification will contain only indices fromidxs
. If order(eulerExpansion(r
)-1
,n
)<n
, then the function has not succeeded and only returns a specification it has found so far.
- guessEtaQuotientSpecification: (PositiveInteger, QEtaLaurentSeries Integer, PositiveInteger) -> Record(fspec: QEtaSpecification, fexp: Fraction Integer, fser: QEtaLaurentSeries Integer)
guessEtaQuotientSpecification(nn,s,n)
is equivalent to guessEtaQuotientSpecification(idxs,s
,n
) for idxs:=concat(eidxs,pidxs) where pidxs=[[nn
,g
] forg
in 1..floor((nn
-1)/2
)] and eidx=[nn/2
,nn
] ifnn
is even and eidxs=[nn
] for oddnn
. The specification may contain generalized eta-quotients and will be of levelnn
.
- guessEtaQuotientSpecification: (QEtaLaurentSeries Integer, PositiveInteger) -> Record(fspec: QEtaSpecification, fexp: Fraction Integer, fser: QEtaLaurentSeries Integer)
guessEtaQuotientSpecification(s,n)
is equivalent to guessEtaQuotientSpecification(idxs,s
,n
) for idxs=[[d
] ford
in 1..n
]. Only pure eta-quotients are returned.