QEtaRamanujanKolbergIdentity C

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QEtaRamanujanKolbergIdentity is a container that holds all relevant data describing a Ramanujan-Kolberg identity together with the data from which it was computed.

algebraBasis: % -> QEtaAlgebraBasis QEtaExtendedAlgebra(C, ModularFunctionQSeriesInfinity C, QEtaLazyAlgebra(C, Polynomial C))

algebraBasis(x) returns the algebra basis that was given at creation time.

alphaInfinity: % -> Integer

alphaInfinity(x) returns the q exponent that corresponds to the orbit product and the respective co-exponent in order to make the whole product a modular function. It is defined already in the abstract of cite{Radu_RamanujanKolberg_2015}.

alphaOrbitInfinity: % -> Fraction Integer

alphaOrbitInfinity(x)=alphaInfinity(x)-rhoInfinity(coSpecification(x)). This corresponds to $beta$ in eqref{eq:beta} in qeta.tex.

coefficient: % -> C

coefficient(x) is the coefficient of F in identityPolynomial(x).

coerce: % -> OutputForm

from CoercibleTo OutputForm

coSpecification: % -> QEtaSpecification

coSpecification(x) returns the specification for the cofactor (generalized) eta-quotient that must be muliplied to the orbit product in order to obtain a modular function. It corresponds to $g_s(tau)$ of eqref{eq:F_s-r-m-t(tau)} in qeta.tex.

definingDissection: % -> QGeneratingFunctionVariable

definingDissection(x) returns the dissection specification of sum_{n=0}^infty a(m*n+t)q^n the generating series for which an eta-quotient representation had to be found. It is basically the initial input for all the data.

definingSpecification: % -> QEtaSpecification

definingSpecification(x) returns the specification of (generalized) eta-quotient that forms the generating series of the coefficients a(n) from definingDissection(x).

etaEquation: % -> Equation QEtaSpecificationExpression C

etaEquation(x) returns an equation with the expression for the orbit product of the dissection (together with the cofactor to make it modular), i.e. f(x) on the left-hand side and etaQuotientRepresentation(x) on the right-hand side

etaIdentityPolynomial: % -> Polynomial C

etaIdentityPolynomial(x) gives a representation of the identity (similar to identityPolynomial(x)) as a polynomial in the dissection(s) and eta-functions where ed_g stands for $eta_{d,g}(tau)$ and am_k stands for the dissection $q^r sum_{n=0}^infty a(m n + k)$ where . For c:=coefficient(etaIdentityPolynomial(x),F,1) it holds unitCanonical(c)=c.

etaQuotientRepresentation: % -> QEtaSpecificationExpression C

etaQuotientRepresentation(x) returns a C-linear combination of eta-quotients. It is replacing any Mi of coefficient(-identityPolynomial(x),'F,0) by the respective eta-specification.

f: % -> QEtaSpecificationExpressionMonomial

f(x) returns an expression for $F_{s,r,m,t}(tau)=g_s cdot P_{r,m,t}$ as specified in qeta.tex eqref{eq:F_r-r-m-t(tau)} and eqref{eq:Ramanujan-Kolberg-Identity}. It is equal to the product of the eta-quotient given by coSpecification(x) and orbitProduct(x).

identity: (QGeneratingFunctionVariable, QEtaSpecificationExpressionMonomial, List QEtaSpecification, Polynomial C, QEtaAlgebraBasis QEtaExtendedAlgebra(C, ModularFunctionQSeriesInfinity C, QEtaLazyAlgebra(C, Polynomial C))) -> %

identity(dissect, f, bspecs, rel, xab) returns a structure that represents all information about the q-expansion of $F_{s,r,m,t}$ at infinity, see Definition of $F_{s,r,m,t}$ in qeta.tex, Equation~eqref{eq:F_s-r-m-t(tau)}. rel is a polynomial in F and M1,…,Mn. The actual relation then is c*F=sum_{j}mon(j)) where F corresponds to the modular function $F_{s,r,m,t}$ in question. t is an element of orb, c=-coefficient(rel,F,1) and mon(j) is the j-th term in coefficient(rel,F,0) where each Mi is replaced by bspecs(i). At the same time this data structure serves as a container for a generalized eta-quotient identity for $F_{bar{s},bar{r},m,t}(tau)$.

identityPolynomial: % -> Polynomial C

identityPolynomial(x) gives the representation of the identity as a polynomial (which equates to zero). This polynomial is in terms of the variables F and Mi where F corresponds to the orbit product together with a prefactor to turn this orbit product into a modular function and the Mi correspond to eta-quotients given by monoidSpecifications(x). For c:=coefficient(identityPolynomial(x),F,1) it holds unitCanonical(c)=c.

moduleCoefficients: % -> List SparseUnivariatePolynomial C

moduleCoefficients(x) gives the list of coefficients of the 1 and the M2,…Mn in coefficient(-identityPolynomial(x),'F,0). That correspond to the (generalized) eta-quotients given by monoidSpecifications(x) and form the relation that that was computed. Note that the first entry of nonConstantMonoidSpecifications(x) must be replaced by 1 in order to make this relation a true relation.

moduleRepresentation: % -> QEtaSpecificationExpression SparseUnivariatePolynomial C

moduleRepresentation(x) returns a C[t]-linear combination of eta-quotients. It is replacing in coefficient(-identityPolynomial(x),'F,0) M1 by and indeterminat t and the M2,…Mk by the respective eta-quotients of the module basis.

monoidSpecifications: % -> List QEtaSpecification

monoidSpecifications(x) returns the specifications of the (generalized) eta-quotients that form the combination of the identity. The first non-constant element of this basis is the special element t so that the relation is actually formed by the module over C[t] generated by 1 and rest monoidSpecifications(x).

multiplier: % -> PositiveInteger

multiplier(x) returns factor m to select a certain subsequence a(m*n+k) of the defining series (where k is one number out of orbit(x).

nonConstantMonoidSpecifications: % -> List QEtaSpecification

monoidSpecifications(x) returns specifications for modular (generalized) eta-quotients with a pole only at infinity. nonConstantMonoidSpecifications(x) returns those elements of monoidSpecifications(x) whose pole order at infinity is positive.

nonConstantMonoidSpecifications: List QEtaSpecification -> List QEtaSpecification

nonConstantMonoidSpecifications(specs) returns a subset of specs containing only those specifications whose pole order at infinity is positive.

orbit: % -> List NonNegativeInteger

orbit(x) returns the orbit, i.e. only a product of the generating series for $a(m*n+k)$ for all k in the orbit can be turned into a modular function (for Gamma0) by multiplying it with an eta-quotient.

orbitProduct: % -> QEtaSpecificationExpression C

orbitProduct(x) returns an expression for $P_{r,m,t}(tau)=q^beta prod_{k in O} sum_{n=0}^infty a(m n + k) q^n$ (see eqref{eq:P_r-m-t(tau)} in qeta.tex) where k runs over the orbit that corresponds to m=multiplier(x) and t=offset(definingDissection(x)) and $beta$ is given by formula eqref{eq:beta} in qeta.tex. It corresponds to $P_{r,m,t}(tau)$ of eqref{eq:F_s-r-m-t(tau)} in qeta.tex.

qEquation: % -> Equation QEtaSpecificationExpression C

qEquation(x) returns an equation with the orbit product of the dissection on the left-hand side and the q-Pochhammer symbol quotients on the right-hand side.

CoercibleTo OutputForm