def J_inv_ZZ(self):
r"""
Return the rational Fourier expansion of ``J_inv``,
where the parameter ``d`` is replaced by ``1``.
This is the main function used to determine all Fourier expansions!
.. NOTE:
The Fourier expansion of ``J_inv`` for ``d!=1``
is given by ``J_inv_ZZ(q/d)``.
.. TODO:
The functions that are used in this implementation are
products of hypergeometric series with other, elementary,
functions. Implement them and clean up this representation.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(prec=3).J_inv_ZZ()
q^-1 + 31/72 + 1823/27648*q + O(q^2)
sage: MFSeriesConstructor(group=5, prec=3).J_inv_ZZ()
q^-1 + 79/200 + 42877/640000*q + O(q^2)
sage: MFSeriesConstructor(group=5, prec=3).J_inv_ZZ().parent()
Laurent Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=infinity, prec=3).J_inv_ZZ()
q^-1 + 3/8 + 69/1024*q + O(q^2)
"""
F1 = lambda a, b: self._series_ring([ZZ(0)] + [
rising_factorial(a, k) * rising_factorial(b, k) /
(ZZ(k).factorial())**2 * sum(
ZZ(1) / (a + j) + ZZ(1) / (b + j) - ZZ(2) / ZZ(1 + j)
for j in range(ZZ(0), ZZ(k)))
for k in range(ZZ(1), ZZ(self._prec + 1))
], ZZ(self._prec + 1))
F = lambda a, b, c: self._series_ring([
rising_factorial(a, k) * rising_factorial(b, k) / rising_factorial(
c, k) / ZZ(k).factorial()
for k in range(ZZ(0), ZZ(self._prec + 1))
], ZZ(self._prec + 1))
a = self._group.alpha()
b = self._group.beta()
Phi = F1(a, b) / F(a, b, ZZ(1))
q = self._series_ring.gen()
# the current implementation of power series reversion is slow
# J_inv_ZZ = ZZ(1) / ((q*Phi.exp()).reverse())
temp_f = (q * Phi.exp()).polynomial()
new_f = temp_f.revert_series(temp_f.degree() + 1)
J_inv_ZZ = ZZ(1) / (new_f + O(q**(temp_f.degree() + 1)))
return J_inv_ZZ
def _deflated(self, a, b, z):
"""
Private helper to return list of deflated terms.
EXAMPLES::
sage: x = hypergeometric([5], [4], 3)
sage: y = x.deflated()
sage: y
7/4*hypergeometric((), (), 3)
sage: x.n(); y.n()
35.1496896155784
35.1496896155784
"""
new = self.eliminate_parameters()
aa = new.operands()[0].operands()
bb = new.operands()[1].operands()
for i, aaa in enumerate(aa):
for j, bbb in enumerate(bb):
m = aaa - bbb
if m in ZZ and m > 0:
aaaa = aa[:i] + aa[i + 1:]
bbbb = bb[:j] + bb[j + 1:]
terms = []
for k in range(m + 1):
# TODO: could rewrite prefactors as recurrence
term = binomial(m, k)
for c in aaaa:
term *= rising_factorial(c, k)
for c in bbbb:
term /= rising_factorial(c, k)
term *= z**k
term /= rising_factorial(aaa - m, k)
F = hypergeometric([c + k for c in aaaa],
[c + k for c in bbbb], z)
unique = []
counts = []
for c, f in F._deflated():
if f in unique:
counts[unique.index(f)] += c
else:
unique.append(f)
counts.append(c)
Fterms = zip(counts, unique)
terms += [(term * termG, G)
for (termG, G) in Fterms]
return terms
return ((1, new), )
def _deflated(cls, self, a, b, z):
"""
Private helper to return list of deflated terms.
EXAMPLES::
sage: x = hypergeometric([5], [4], 3)
sage: y = x.deflated()
sage: y
7/4*hypergeometric((), (), 3)
sage: x.n(); y.n()
35.1496896155784
35.1496896155784
"""
new = self.eliminate_parameters()
aa = new.operands()[0].operands()
bb = new.operands()[1].operands()
for i, aaa in enumerate(aa):
for j, bbb in enumerate(bb):
m = aaa - bbb
if m in ZZ and m > 0:
aaaa = aa[:i] + aa[i + 1:]
bbbb = bb[:j] + bb[j + 1:]
terms = []
for k in xrange(m + 1):
# TODO: could rewrite prefactors as recurrence
term = binomial(m, k)
for c in aaaa:
term *= rising_factorial(c, k)
for c in bbbb:
term /= rising_factorial(c, k)
term *= z ** k
term /= rising_factorial(aaa - m, k)
F = hypergeometric([c + k for c in aaaa],
[c + k for c in bbbb], z)
unique = []
counts = []
for c, f in F._deflated():
if f in unique:
counts[unique.index(f)] += c
else:
unique.append(f)
counts.append(c)
Fterms = zip(counts, unique)
terms += [(term * termG, G) for (termG, G) in
Fterms]
return terms
return ((1, new),)
def _sympysage_rf(self):
"""
EXAMPLES::
sage: from sympy import Symbol, rf
sage: _ = var('x, y')
sage: rfxy = rf(Symbol('x'), Symbol('y'))
sage: assert rising_factorial(x,y)._sympy_() == rfxy.rewrite('gamma')
sage: assert rising_factorial(x,y) == rfxy._sage_()
"""
from sage.arith.all import rising_factorial
return rising_factorial(self.args[0]._sage_(), self.args[1]._sage_())
def _sympysage_rf(self):
"""
EXAMPLES::
sage: from sympy import Symbol, rf
sage: _ = var('x, y')
sage: rfxy = rf(Symbol('x'), Symbol('y'))
sage: assert rising_factorial(x,y)._sympy_() == rfxy.rewrite('gamma')
sage: assert rising_factorial(x,y) == rfxy._sage_()
"""
from sage.arith.all import rising_factorial
return rising_factorial(self.args[0]._sage_(), self.args[1]._sage_())
def J_inv_ZZ(self):
r"""
Return the rational Fourier expansion of ``J_inv``,
where the parameter ``d`` is replaced by ``1``.
This is the main function used to determine all Fourier expansions!
.. NOTE::
The Fourier expansion of ``J_inv`` for ``d!=1``
is given by ``J_inv_ZZ(q/d)``.
.. TODO::
The functions that are used in this implementation are
products of hypergeometric series with other, elementary,
functions. Implement them and clean up this representation.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(prec=3).J_inv_ZZ()
q^-1 + 31/72 + 1823/27648*q + O(q^2)
sage: MFSeriesConstructor(group=5, prec=3).J_inv_ZZ()
q^-1 + 79/200 + 42877/640000*q + O(q^2)
sage: MFSeriesConstructor(group=5, prec=3).J_inv_ZZ().parent()
Laurent Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=infinity, prec=3).J_inv_ZZ()
q^-1 + 3/8 + 69/1024*q + O(q^2)
"""
F1 = lambda a,b: self._series_ring(
[ ZZ(0) ]
+ [
rising_factorial(a,k) * rising_factorial(b,k) / (ZZ(k).factorial())**2
* sum(ZZ(1)/(a+j) + ZZ(1)/(b+j) - ZZ(2)/ZZ(1+j)
for j in range(ZZ(0),ZZ(k))
)
for k in range(ZZ(1), ZZ(self._prec+1))
],
ZZ(self._prec+1)
)
F = lambda a,b,c: self._series_ring(
[
rising_factorial(a,k) * rising_factorial(b,k) / rising_factorial(c,k) / ZZ(k).factorial()
for k in range(ZZ(0), ZZ(self._prec+1))
],
ZZ(self._prec+1)
)
a = self._group.alpha()
b = self._group.beta()
Phi = F1(a,b) / F(a,b,ZZ(1))
q = self._series_ring.gen()
# the current implementation of power series reversion is slow
# J_inv_ZZ = ZZ(1) / ((q*Phi.exp()).reverse())
temp_f = (q*Phi.exp()).polynomial()
new_f = temp_f.revert_series(temp_f.degree()+1)
J_inv_ZZ = ZZ(1) / (new_f + O(q**(temp_f.degree()+1)))
return J_inv_ZZ
