QEtaRamanujanKolbergIdentity C¶
qetark.spad line 197 [edit on github]
QEtaRamanujanKolbergIdentity is a container that holds all relevant data describing a Ramanujan-Kolberg identity.
- 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- qexponent 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- Fin 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
- q^alphaOrbbitInfinity( - x) * etaQuotient(coSpecification(- x),”E”::Symbol).
- 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(x)returns equationX(- x,”a”::Symbol).
- equationX: (%, Symbol) -> Equation Expression C if C has Comparable
- equationX(x,a)returns the result stored in- xas an equation of expressions in “E” variables and with a (fractional) power of the- qvariable.
- etaQuotient: (QEtaSpecification, Symbol) -> Fraction Polynomial C
- etaQuotient(spec,v)returns the same as monomialQuotient(- spec,- v).
- etaQuotientRepresentation: % -> QEtaSpecificationRing C
- etaQuotientRepresentation(x)returns a- C-linear combination of eta-quotients. It is replacing any- Miof coefficient(identityPolynomial(- x),- 'F,0) by the respective eta-specification.
- etaQuotientX: (QEtaSpecification, Symbol) -> Expression C if C has Comparable
- etaQuotientX(spec,v)returns monomialQuotientX(- spec,- v).
- 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- rspecand coefficient- ciwhich is multiplied by q^e with variables given through- vand in a form given by- fmt.
- expand?: Integer -> Boolean
- expand?(fmt)is- trueif the respective bit in- fmtcorresponding to formatExpanded() is set.
- f: (%, Integer, (Integer, Integer) -> OutputForm, C, Fraction Integer, QEtaSpecification, OutputForm, OutputForm) -> OutputForm
- f(id,fmt.v,c,e,rspec,a,n)returns the dissection part of the equation, multiplied by the eta-quotient given by- rspecand c*q^e. The coefficient under the sum is named by a and the summation variable is denoted by- n.
- formatAsNonModular: (Integer)
- formatAsNonModular()is a number to be used as a format value in the function pretty. If set like prettyVariable(- x,…+formatAsNonModular()+…,- e) it means that the output is such that on the left hand side appears only the disected orbit sum without any factor. The corresponding test is modularFunctionIdentity?.
- formatExpanded: (Integer)
- formatExpanded()is a number to be used as a format value in the function pretty. If set like pretty(- x,…+formatExpanded()+…,- e) it means that the output is entirely in eta-quotients even the value for- M1(which is otherwise shown as a variable). The corresponding test is expand?.
- identity: (QEtaSpecification, QEtaSpecification, PositiveInteger, List NonNegativeInteger, List QEtaSpecification, Polynomial C, QEtaAlgebraBasis QEtaExtendedAlgebra(C, ModularFunctionQSeriesInfinity C, QEtaLazyAlgebra(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- Fcorresponds to the modular function $- F_{- s,- r,- m,- t}$ in question.- tis 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- Fand- Miwhere- Fcorresponds to the orbit product together with a prefactor to turn this orbit product into a modular function and the- Micorrespond to eta-quotients given by monoidSpecifications(- x).
- modularFunctionIdentity?: Integer -> Boolean
- modularFunctionIdentity?(fmt)is- trueif the respective bit in- fmtcorresponding to formatAsNonModular() is not set.
- 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- truerelation.
- moduleRepresentation: % -> QEtaSpecificationRing SparseUnivariatePolynomial C
- moduleRepresentation(x)returns a- C[- t]-linear combination of eta-quotients. It is replacing in coefficient(identityPolynomial(- x),- 'F,0)- M1by- tand the- M2,…- Mkby 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- tso that the relation is actually formed by the module over- C[- t] generated by 1 and rest monoidSpecifications(- x).
- monomial: (QEtaSpecification, Symbol) -> Polynomial C
- monomial(spec,v)assumes that all exponents in the specification are positive and returns a product of powers of specification parts. If [- d,- g,- e] is in parts(- spec), then the result is- v[- d,- g]^e where- v[- d,- g] is the symbol- vsubscripted by [- d,- g].
- monomialQuotient: (QEtaSpecification, Symbol) -> Fraction Polynomial C
- monomialQuotient(spec,v)returns monomial(numer(- spec),- v)/monomial(denom(- spec),- v).
- monomialQuotientX: (QEtaSpecification, Symbol) -> Expression C if C has Comparable
- monomialQuotientX(spec,v)returns monomialQuotient(- spec,- v)::Expression(- C).
- monomialX: (QEtaSpecification, Symbol) -> Expression C if C has Comparable
- monomialX(spec,v)returns monomial(- spec,- v)::Expression(- C).
- multiplier: % -> PositiveInteger
- multiplier(x)returns factor- mto select a certain subsequence a(m*n+k) of the defining series (where- kis 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- kin the orbit can be turned into a modular function (for- Gamma0) by multiplying it with an eta-quotient.
- orbitProduct: % -> OutputForm
- orbitProduct(x)is the same as orbitProduct(- x, a,- n) where a and- nare the outputforms of the symbols “a” and- "n".
- orbitProduct: (%, OutputForm, OutputForm) -> OutputForm
- orbitProduct(x, a, n)returns the orbit product as an OutputForm where the series coefficients are printed as a(- n) and- nis the summation variable.
- orbitProductX: % -> Expression C if C has Comparable
- orbitProductX(x)returns orbitProductX(- x,”a”::Symbol).
- orbitProductX: (%, Symbol) -> Expression C if C has Comparable
- orbitProductX(x,a)returns the respective orbit product part of- xas an expression with summation variable- nand series variable a(- m- n+- k) where- mis the respective multiplier and- kis in orbit(- x).
- pretty: (%, Integer) -> Equation OutputForm
- pretty(x, fmt)returns the same as pretty(- x,- fmt,”a”::Symbol::OutputForm,”n”::Symbol::OutputForm).
- pretty: (%, Integer, (Integer, Integer) -> OutputForm) -> Equation OutputForm
- pretty(x, fmt, v)returns the same as pretty(- x,- fmt,- v,”a”::Symbol::OutputForm,”n”::Symbol::OutputForm).
- pretty: (%, Integer, (Integer, Integer) -> OutputForm, C, Fraction Integer, QEtaSpecification, OutputForm, OutputForm) -> Equation OutputForm
- pretty(x, fmt, v, c, e, sspec, a, n)outputs- xas computed in eta-quotients, but multiplied by c*q^e*etaQuotient(- sspec).
- pretty: (%, Integer, (Integer, Integer) -> OutputForm, OutputForm, OutputForm) -> Equation OutputForm
- pretty(x,fmt,v,a,n)outputs- xas computed with respect to a style given by the- fmtparameter. If the lowest bit is set in- fmtthen, the- qfactor is shown. If the bit 1 is set, then the- tparameter is replaced by the respective eta-quotient, so that the coefficients will just be elements of the coefficient domain. The summation index is- nand the coefficients under the sum are a(- n).
- pretty: (%, Integer, OutputForm, OutputForm) -> Equation OutputForm
- pretty(x, fmt, a, n)returns pretty(- x,- fmt,- v, a,- n) where- vis a variable indexing function chosen by bit 3 and 4 of- fmtaccording 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=1means subscripts. If- bit3=1, then expressions are used. If- bit0=0, then v=varEta. If- bit0=1, then v=varPochhammer. The summation index is- nand the coefficients under the sum are a(- n).
- qcofactor: % -> Fraction Polynomial C
- qcofactor(x)returns q^alphaInfinity(- x)*etaQuotient(coSpecification(- x),”u”::Symbol).
- qcofactorX: % -> Expression C if C has Comparable
- qcofactorX(x)returns qcofactor(- x)::Expression(- C).
- qequationX: % -> Equation Expression C if C has Comparable
- qequationX(x)returns qequationX(- x,”a”::Symbol).
- qequationX: (%, Symbol) -> Equation Expression C if C has Comparable
- qequationX(x,a)returns the result stored in- xas an equation of expressions in “u” variables and with a power of the- qvariable.
- sumX: (BasicOperator, Symbol, PositiveInteger, NonNegativeInteger) -> Expression C if C has Comparable
- sumX(a,n,m,t)returns the expression for $sum_{- n=0}^infty{a(- m{- n}- +t)- q^n}$.