QGeneratingFunctionVariable¶

qgenfunspec.spad line 122 [edit on github]

QGeneratingFunctionVariable formally specifices dissections of generating functions. It is the variable domain for QGeneratingFunctionSpecification.

<=: (%, %) -> Boolean: from PartialOrder

<: (%, %) -> Boolean: from PartialOrder

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

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

>: (%, %) -> Boolean: from PartialOrder

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

choose: (PositiveInteger, NonNegativeInteger) -> % -> %: choose(m,t)(x) returns a specification that represents $sum_{n=0}^infty a(m n + t) q^n$, if x is a specification that represents $sum_{n=0}^infty a(n) q^n$.

coefficientSymbol: % -> Symbol: coefficientSymbol(x) returns the symbol that is used to denote the coefficient of q^n in the series expansion.

coerce: % -> OutputForm: from CoercibleTo OutputForm

construct: (QEtaSpecification, PositiveInteger, NonNegativeInteger, Symbol, Symbol) -> %: [spec,m,t,a,n] returns a specification that represents $sum_{n=0}^infty a(m n + t) q^n$ where $sum_{n=0}^infty a(n) q^n$ is given by the eta-quotient as specified by spec divided by $q^rho$ with rho=rhoInfinity(spec).

construct: (QPochhammerSpecification, PositiveInteger, NonNegativeInteger, Symbol, Symbol) -> %: [qspec, m,t,a,n] returns construct(spec,m,t,a,n) if spec:=lift(qspec) and empty?(properGeneralizedParts(missingSpecificationForEta(qspec))). The function fails, if this condition is not fulfilled.

definingSpecification: % -> QEtaSpecification: definingSpecification(x) returns the eta-quotient specification of the generating function. Although the returned value denotes an eta-quotient, it actually specifies the product of the corresponding q-Pochhammer symbols, i.e. the corresponding q-series expansion starts with constant term 1.

dilate: (%, PositiveInteger) -> %: dilate(x,n) is the respective operation of replacing q by q^n. At the same time the level, the multiplier, and the offset are multiplied by n.

generatingFunction: (QEtaSpecification, Symbol) -> %: genratingFunction(spec,a) returns generatingFunction(spec,a,”n”::Symbol).

generatingFunction: (QEtaSpecification, Symbol, Symbol) -> %: genratingFunction(spec,a,n) returns construct(spec,1,0,a,n).

generatingFunction: (QPochhammerSpecification, Symbol) -> %: genratingFunction(qspec,a) returns generatingFunction(qspec,a,”n”::Symbol).

generatingFunction: (QPochhammerSpecification, Symbol, Symbol) -> %: genratingFunction(qspec,a,n) returns construct(qspec,1,0,a,n).

generatingFunction: QEtaSpecification -> %: genratingFunction(spec) returns generatingFunction(qspec,”a”::Symbol).

generatingFunction: QPochhammerSpecification -> %: genratingFunction(qspec) returns generatingFunction(qspec,”a”::Symbol).

hash: % -> SingleInteger: from Hashable

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

latex: % -> String: from SetCategory

level: % -> PositiveInteger: level(x) returns level(definingSpecification(x)).

max: (%, %) -> %: from OrderedSet

min: (%, %) -> %: from OrderedSet

multiplier: % -> PositiveInteger: mulltiplier(x) returns the dissection multiplier of the series.

offset: % -> NonNegativeInteger: offset(x) returns the shift in the dissection.

offsetEquivalent: (%, %) -> Boolean: offsetEquivalent(x,y) is true if x and y are equal, except for the value of offset(x) and offset(y), which is counted as irrelevant.

qPochhammerSpecification: % -> QPochhammerSpecification: qPochhammerSpecification(x) returns the q-Pochhammer specification of the generating series. The result is always a quotient of q-Pochhammer symbols. There is no separate q power.

rename: (%, Symbol) -> %: rename(x,a) changes the coefficient symbol to a.

rename: (%, Symbol, Symbol) -> %: rename(x,a,n) changes the coefficient symbol to a and the running variable to n.

rhoInfinity: % -> Fraction Integer: rhoInfinity(x) returns rhoInfinity(definingSpecification(x),multiplier(x),offset(x)).

runningVariable: % -> Symbol: runningVariable(x) returns the symbol that is used to denote the running variable in the series expansion.

smaller?: (%, %) -> Boolean: from Comparable

symbol: % -> Symbol: symbol(x) creates the symbol indexedSymbol(string(coefficientSymbol(x)),[multiplier(x),offset(x)]).

CoercibleTo OutputForm