QEtaSeriesExpansion(C, xiord, CX, xi, QMOD)¶

qetaquot.spad line 139 [edit on github]

QEtaSeriesExpansion computes an eta quotient expansion of $g_{r,m,lambda}(gamma tau)$, $p_{r,m,t}(gammatau)$, $F_{s,r,m,t}(gammatau)$. See eqref{eq:g_r-m-lambda(gamma*tau)} and eqref{eq:F_s-r-m-t(gamma*tau)}.

coerce: % -> OutputForm: from CoercibleTo OutputForm

laurent: (QEtaPuiseuxSeries CX, PositiveInteger) -> QEtaLaurentSeries CX: If p(q)=s(z) is a Puiseux series p expressed as a Laurent series s in the variable z=q^r, then laurent(x, w) returns the laurent series l such that l(x)=s(z)=p(q) where x=q^(1/w) in case r*w is an integer. It might happen that r*w is not an integer. That is even to be expected since non-modular eta-quotients involve a factor in terms of q^(1/24). If r=s/t, then we take only every t-th term and check that the intermediate terms come indeed with a zero coefficient.

laurentExpansion: SymbolicModularEtaQuotientGamma QMOD -> QEtaLaurentSeries CX: laurentExpansion(y) represents the q-expansion of $F_{s,r,m,t}(gammatau)$ given by y in the canonical variables given by the width of the cusp wrt. QMOD.

laurentExpansionInfinity: SymbolicModularEtaQuotientGamma QMOD -> QEtaLaurentSeries CX: laurentExpansionInfinity(y) represents the q-expansion of $F_{s,r,m,t}(tau)$ given by y wrt. QMOD. See eqref{eq:F_s-r-m-t(tau)}.

laurentExpansions: (SymbolicModularEtaQuotient QMOD, List Cusp) -> XHashTable(Cusp, QEtaLaurentSeries CX): laurentExpansions(y, cusps) represents the q-expansion of $F_{s,r,m,t}(gammatau)$ at the given cusps (which should be a subset of the cusps given by y) in the canonical variables given by the width of the cusp wrt. QMOD.

laurentExpansions: SymbolicModularEtaQuotient QMOD -> XHashTable(Cusp, QEtaLaurentSeries CX): laurentExpansions y represents the q-expansion of $F_{s,r,m,t}(gammatau)$ at all cusps given by y in the canonical variables given by the width of the cusp wrt. QMOD.

puiseuxExpansion: (SymbolicEtaQuotientLambdaGamma, Fraction Integer) -> QEtaPuiseuxSeries CX: puiseuxExpansion(y, r) computes the Puiseux expansion of $g_{r,m,lambda(gammatau)$ in terms of $q$ multiplied by $exp(2pi i r)$. The $(ctau+d)$ factor is missing. See eqref{eq:g_r-m-lambda(gamma*tau)}.

puiseuxExpansion: SymbolicEtaQuotientGamma -> QEtaPuiseuxSeries CX: puiseuxExpansion(y) computes the Puiseux expansion of the q-expansion of $p_{r,m,t}(gamma tau)$, see eqref{eq:p_r-m-t(gamma*tau)}. The $(ctau+d)$ factor is missing.

puiseuxExpansion: SymbolicModularEtaQuotientGamma QMOD -> QEtaPuiseuxSeries CX: puiseuxExpansion(y) represents the q-expansion of $F_{s,r,m,t}(gamma tau)$, see eqref{eq:F_s-r-m-t(gamma*tau)}.

substitute: (QEtaLaurentSeries CX, Fraction Integer, NonNegativeInteger) -> QEtaPuiseuxSeries CX: If s(q) is a series in q, then substitute(s, u, v) returns a series t(q) such that t(q)=s(q^u*xi^v).

CoercibleTo OutputForm