QEtaQSeriesTools(C, xiord, CX, xi)ΒΆ
qetaqseriestool.spad line 118 [edit on github]
xiord: PositiveInteger
CX: Algebra C
xi: CX
QEtaQSeriesTools provides functions for the transformation of $q
$-series.
- laurent: (QEtaPuiseuxSeries CX, Fraction Integer) -> QEtaLaurentSeries CX
Necessary input condition for laurent(
x
,w
) is thatw>0
. Ifp
(q
)=s
(z
) is a Puiseux seriesp
expressed as a Laurent seriess
in the variable z=q^r, then laurent(x
,w
) returns the laurent seriesl
such thatl
(x
)=s
(z
)=p
(q
) where x=q^(1/w) in caser*w
is an integer. It might happen thatr*w
is not an integer. That is even to be expected since non-modular eta-quotients involve a factor in terms ofq^
(1/24). If r=s/t, then we take only everyt
-th term and check that the intermediate terms come indeed with a zero coefficient.
- laurent: (QEtaPuiseuxSeries CX, PositiveInteger) -> QEtaLaurentSeries CX
laurent(p, w)
returns laurent(p
,w/1
)
- substitute: (QEtaLaurentSeries CX, Fraction Integer, Fraction Integer) -> QEtaPuiseuxSeries CX
If
s
(q
) is a series inq
, then substitute(s
,u
,v
) returns a seriest
(q
) such thatt
(q
)=s
(q^u*xi^v). Note that we assume that the seriess
has onlyq
powers that are divisible by denom(v
). Otherwise the function will abort with an error message if during the expansion oft
the unity rootxi
would have to be raised to a fractional power.
- substitute: (QEtaLaurentSeries CX, Fraction Integer, NonNegativeInteger) -> QEtaPuiseuxSeries CX
If
s
(q
) is a series inq
, then substitute(s
,u
,v
) returns a seriest
(q
) such thatt
(q
)=s
(q^u*xi^v).