SymbolicEtaQuotientGammaΒΆ
qetasymb.spad line 544 [edit on github]
SymbolicEtaQuotientGamma holds data to compute an eta quotient expansion of $p_{r,m,k}(gamma tau)$. See eqref{eq:p_r-m-k(gamma*tau)}.
- coerce: % -> OutputForm
from CoercibleTo OutputForm
- definingSpecification: % -> QEtaSpecification
If x=etaQuotient(
spec,m,k,gamma), then definingSpecification(x) returns purify(spec).
- elt: (%, NonNegativeInteger) -> SymbolicEtaQuotientLambdaGamma
x.lambda returns the data corresponding to the respective lambda.
- etaQuotient: (QEtaSpecification, Matrix Integer) -> %
etaQuotient(rspec,gamma)returns etaQuotient(rspec,1,0,gamma).
- etaQuotient: (QEtaSpecification, PositiveInteger, NonNegativeInteger, Matrix Integer) -> %
etaQuotient(rspec,m,k,gamma)represents the expansion of $p_{r,m,k}(gamma tau)$ where $r$ is given byrspec.
- hash: % -> SingleInteger
from Hashable
- hashUpdate!: (HashState, %) -> HashState
from Hashable
- latex: % -> String
from SetCategory
- minimalRootOfUnity: % -> PositiveInteger
minimalRootOfUnity(x)returns the smallest positive integernsuch that the expansion of the function $p_{r,m,k}(gamma tau)$ corresponding to x=etaQuotient(spec,m,k,gamma) (neglecting the (c*tau+d) factor) lives inQ[w][[z]] wherewis ann-th root of unity andza fractionalqpower. See eqref{p_r-m-k(gamma*tau)}.
- multiplier: % -> PositiveInteger
If x=etaQuotient(
spec,m,k,gamma), then multiplier(x) returnsm.
- offset: % -> NonNegativeInteger
If x=etaQuotient(
spec,m,k,gamma), then offset(x) returnsk. offset(x) returns the subsequence offset.
- one?: % -> Boolean
one?(x)returnstrueif the eta-quotient corresponding toxrepresents 1. This is the case if one?(definingSpecification(x)), i.e. if the representation is trivial.
- qExponentMin: % -> Fraction Integer
qExponentMin(x)returns the order of theq-expansion in terms of the originalq. Note that this exponent is only a lower bound for theq-expansion. The coefficient corresponding to thisq-power may be zero. See eqref{p_r-m-k(gamma*tau)}. It corresponds to $p(gamma)$ as defined in equation (51) of cite{Radu_AlgorithmicApproachRamanujanCongruences_2009}.
- transformationMatrix: % -> Matrix Integer
If x=etaQuotient(spec,
m,k,gamma), then transformationMatrix(x) returns gamma.