SymbolicEtaQuotientGamma¶

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)}.

=: (%, %) -> Boolean: from BasicType

~=: (%, %) -> Boolean: from BasicType

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 by rspec.

hash: % -> SingleInteger: from Hashable

hashUpdate!: (HashState, %) -> HashState: from Hashable

latex: % -> String: from SetCategory

minimalRootOfUnity: % -> PositiveInteger: minimalRootOfUnity(x) returns the smallest positive integer n such 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 in Q[w][[z]] where w is an n-th root of unity and z a fractional q power. See eqref{p_r-m-k(gamma*tau)}.

multiplier: % -> PositiveInteger: If x=etaQuotient(spec,m,k,gamma), then multiplier(x) returns m.

offset: % -> NonNegativeInteger: If x=etaQuotient(spec,m,k,gamma), then offset(x) returns k. offset(x) returns the subsequence offset.

one?: % -> Boolean: one?(x) returns true if the eta-quotient corresponding to x represents 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 the q-expansion in terms of the original q. Note that this exponent is only a lower bound for the q-expansion. The coefficient corresponding to this q-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.

unityExponent: % -> Fraction Integer: If x=etaQuotient(spec,m,k,gamma), then unityExponent(x) returns the fractional part of -(k+rho_infinity(r))/m.

CoercibleTo OutputForm