"""
q-expansions of Theta Series
AUTHOR:
-- William Stein
"""
from sage.rings.all import Integer, ZZ, PowerSeriesRing
from math import sqrt
def theta2_qexp(prec=10, var='q', K=ZZ, sparse=False):
r"""
Return the $q$-expansion of the series
$$
\theta_2 = \sum_{n odd} q^{n^2}.
$$
INPUT:
prec -- integer; the absolute precision of the output
var -- (default: 'q') variable name
K -- (default: ZZ) base ring of answer
OUTPUT:
a power series over K
EXAMPLES:
sage: theta2_qexp(18)
q + q^9 + O(q^18)
sage: theta2_qexp(49)
q + q^9 + q^25 + O(q^49)
sage: theta2_qexp(100, 'q', QQ)
q + q^9 + q^25 + q^49 + q^81 + O(q^100)
sage: f = theta2_qexp(100, 't', GF(3)); f
t + t^9 + t^25 + t^49 + t^81 + O(t^100)
sage: parent(f)
Power Series Ring in t over Finite Field of size 3
sage: theta2_qexp(200)
q + q^9 + q^25 + q^49 + q^81 + q^121 + q^169 + O(q^200)
sage: f = theta2_qexp(20,sparse=True); f
q + q^9 + O(q^20)
sage: parent(f)
Sparse Power Series Ring in q over Integer Ring
"""
prec = Integer(prec)
if prec <= 0:
raise ValueError, "prec must be positive"
if sparse:
v = {}
else:
v = [Integer(0)] * prec
one = Integer(1)
n = int(sqrt(prec))
if n*n < prec:
n += 1
for m in xrange(1, n, 2):
v[m*m] = one
R = PowerSeriesRing(K, sparse=sparse, names=var)
return R(v, prec=prec)
def theta_qexp(prec=10, var='q', K=ZZ, sparse=False):
r"""
Return the $q$-expansion of the standard $\theta$ series
$$
\theta = 1 + 2\sum_{n=1}{^\infty} q^{n^2}.
$$
INPUT:
prec -- integer; the absolute precision of the output
var -- (default: 'q') variable name
K -- (default: ZZ) base ring of answer
OUTPUT:
a power series over K
EXAMPLES:
sage: theta_qexp(25)
1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + O(q^25)
sage: theta_qexp(10)
1 + 2*q + 2*q^4 + 2*q^9 + O(q^10)
sage: theta_qexp(100)
1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + 2*q^25 + 2*q^36 + 2*q^49 + 2*q^64 + 2*q^81 + O(q^100)
sage: theta_qexp(100, 't')
1 + 2*t + 2*t^4 + 2*t^9 + 2*t^16 + 2*t^25 + 2*t^36 + 2*t^49 + 2*t^64 + 2*t^81 + O(t^100)
sage: theta_qexp(100, 't', GF(2))
1 + O(t^100)
sage: f = theta_qexp(20,sparse=True); f
1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + O(q^20)
sage: parent(f)
Sparse Power Series Ring in q over Integer Ring
"""
prec = Integer(prec)
if prec <= 0:
raise ValueError, "prec must be positive"
if sparse:
v = {}
else:
v = [Integer(0)] * prec
v[0] = Integer(1)
two = Integer(2)
n = int(sqrt(prec))
if n*n != prec:
n += 1
for m in xrange(1, n):
v[m*m] = two
R = PowerSeriesRing(K, sparse=sparse, names=var)
return R(v, prec=prec)
