QEtaRamanujanKolbergIdentity C

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QEtaRamanujanKolbergIdentity is a container that holds all relevant data describing a Ramanujan-Kolberg identity.

algebraBasis: % -> QEtaAlgebraBasis(C, QEtaExtendedAlgebra(C, QEtaAlgebraCachedPower(C, Finite0Series C), QEtaAlgebraCachedPower(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-rhoInfinity(coSpecification(x))

coefficient: % -> C

coefficient(x) is the coefficient of F in identityPolynomial(x) that results from a reduction process in case it cannot be inverted. For a coefficient field, it will give 1.

coerce: % -> OutputForm

from CoercibleTo OutputForm

cofactorX: % -> Expression C if C has Comparable

coSpecification: % -> QEtaSpecification

coExponent(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.

definingSpecification: % -> QEtaSpecification

definingSpecification(x) returns the specification of (generalized) eta-quotient that forms the generating series of the coefficients a(n), i.e. sum_{n=0}^infty a(n)q^n =

equationX: % -> Equation Expression C if C has Comparable

equationX: (%, Symbol) -> Equation Expression C if C has Comparable

etaQuotient: (QEtaSpecification, (Integer, Integer) -> OutputForm) -> OutputForm

etaQuotient(rspec, v, e) returns the generalized eta-quotient given by rspec where each part is formatted by the function v.

etaQuotientX: (QEtaSpecification, Symbol) -> Expression C if C has Comparable

etaRelation: (List SparseUnivariatePolynomial C, List QEtaSpecification, Integer, (Integer, Integer) -> OutputForm, C, Fraction Integer, QEtaSpecification) -> OutputForm

etaRelation(cis, bas, fmt, v, c, e, sspec) returns the eta-relation part multiplied by c*q^e*etaQuotient(sspec).

etaRelationMonomial: (SparseUnivariatePolynomial C, QEtaSpecification, Integer, (Integer, Integer) -> OutputForm, Fraction Integer) -> OutputForm

etaRelationMonomial(ci, rspec, fmt, v, e) return an eta-quoatient given by rspec and coefficient ci which is multiplied by q^e with variables given through v and in a form given by fmt.

f: (%, Integer, (Integer, Integer) -> OutputForm, C, Fraction Integer, QEtaSpecification) -> OutputForm

identity: (QEtaSpecification, QEtaSpecification, PositiveInteger, List NonNegativeInteger, List QEtaSpecification, Polynomial C, QEtaAlgebraBasis(C, QEtaExtendedAlgebra(C, QEtaAlgebraCachedPower(C, Finite0Series C), QEtaAlgebraCachedPower(C, Polynomial C)))) -> %

identity(sspec, rspec, m, orb, bspecss, 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)}. The relations is then c*F=sum_{i}(rel(i)*etaQuotient(bspecs(i))) where F corresponds to the modular function $F_{s,r,m,t}$ in question. t is an element of orb. 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 etaQuotientMonoidSpecifications(x).

moduleCoefficients: % -> List SparseUnivariatePolynomial C

moduleCoefficients(x) gives the list of coefficients that correspond to the (generalized) eta-quotients given by nonConstantMonoidSpecifications(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.

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).

monomial: (QEtaSpecification, (Integer, Integer) -> OutputForm) -> OutputForm

monomialQuotient: (QEtaSpecification, (Integer, Integer) -> OutputForm) -> OutputForm

monomialQuotientX: (QEtaSpecification, Symbol) -> Expression C if C has Comparable

monomialX: (QEtaSpecification, Symbol) -> Expression C if C has Comparable

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

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.

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: % -> OutputForm

orbitProduct(x) returns the orbit product as an OutputForm.

orbitProductX: % -> Expression C if C has Comparable

orbitProductX: (%, Symbol) -> Expression C if C has Comparable

pretty: (%, Integer) -> Equation OutputForm

pretty(x, fmt) returns pretty(x, fmt, v) where v is a variable indexing function chosen by bit 3 and 4 of fmt according to the rules. If bit3=0, then variables are used. If bit0=0, then the variable is “E” otherwise it is “u”. If bit4=0, then no subscripts are uses, bit4=1 means subscripts. If bit3=1, then expressions are used. If bit0=0, then v=varEta. If bit0=1, then v=varPochhammer.

pretty: (%, Integer, (Integer, Integer) -> OutputForm) -> Equation OutputForm

pretty(x, fmt, v) outputs x as computed with respect to a style given by the fmt parameter. If the lowest bit is set in fmt then, the q factor is shown. If the bit 1 is set, then the t parameter is replaces by the respective eta-quotient, so that the coefficients will just be elements of the coefficient domain.

pretty: (%, Integer, (Integer, Integer) -> OutputForm, C, Fraction Integer, QEtaSpecification) -> Equation OutputForm

pretty(x, fmt, v, c, e, sspec) outputs x as computed in eta-quotients, but multiplied by c*q^e*etaQuotient(sspec).

qcofactorX: % -> Expression C if C has Comparable

qequationX: % -> Equation Expression C if C has Comparable

qequationX: (%, Symbol) -> Equation Expression C if C has Comparable

qEtaQuotient: (QEtaSpecification, (Integer, Integer) -> OutputForm) -> OutputForm

qEtauotient(rspec, v) returns qQuotient(rspec, v, e) with e=rhoInfinity(rspec).

qQuotient: (QEtaSpecification, (Integer, Integer) -> OutputForm, Fraction Integer) -> OutputForm

qQuotient(rspec, v, e) returns the generalized eta-quotient given by rspec where each part is formatted by the function v. The whole eta-quatient comes multiplied with a q power with exponent e.

sumX: (BasicOperator, Symbol, PositiveInteger, NonNegativeInteger) -> Expression C if C has Comparable

var: Symbol -> (Integer, Integer) -> OutputForm

varEta: (Integer, Integer) -> OutputForm

varPochhammer: (Integer, Integer) -> OutputForm

varPower: (List Integer, (Integer, Integer) -> OutputForm) -> OutputForm

varsub: Symbol -> (Integer, Integer) -> OutputForm

CoercibleTo OutputForm