def coef(l1, l2, l, p, n, algorithm=None):
q = var('q')
if p >= 0:
result1 = q**(2*n*(l1+l2) + n)
qpoch1 = qPochhammerSymbol([q**(-4*l2), q**(-4*l1 + 2*p)], q**2, n)
qpoch2 = qPochhammerSymbol([q**2, q**(2*p + 2)], q**2, n)
result2 = sqrt(qpoch1.evaluate() / qpoch2.evaluate())
if p <= 2*l1 - 2*l2:
result3 = dual_q_Hahn_polynomials( \
n, l1 + l2 - l, q**(-4*l1 + 2*p - 2), \
q**(-4*l2 - 2*p - 2), 2*l2, q**2 \
)
else:
result3 = dual_q_Hahn_polynomials( \
n, l1 + l2 - n, q**(-4*l2 - 2), \
q**(-4*l1 - 2), 2**l1 - 2*p, q**2
)
else:
return coef(l2, l1, l, -p, n)
if algorithm == 'mathematica':
return mathematica(result1*result2*result3).FullSimplify().sage()
return result1*result2*result3
def dual_q_Hahn_polynomials(n, x, c, d, N, q):
if n != N:
l1 = [q**(-n), q**(-x), c*d*q**(x + 1)]
l2 = [c*q, q**(-N)]
result = BasicHypergeometricSeries(l1, l2, q, q).evaluate()
else:
# For n = N we must understand the dual q-Hahn polynomials as
# polynomials by continuity.
result = 0
for k in range(N + 1):
qpoch1 = qPochhammerSymbol([q**(-x), c*d*q**(x + 1)], q, k)
if qpoch1.evaluate() == 0:
break
qpoch2 = qPochhammerSymbol([c*q, q], q, k)
result += qpoch1.evaluate() / qpoch2.evaluate() * q**k
return result
def new_normalized_cgc(l1, l2, l, p, n):
q = var('q')
sign = (-1)**(l1+l2-l)
qpoch1 = qPochhammerSymbol(q**(-4*l2 - 2*p), q**2, 2*l2).evaluate()
qpoch2 = qPochhammerSymbol([q**(-4*l1 + 2*p), q**(-4*l1 - 4*l2 - 2), \
q**(-4*l2)], q**2, l1 + l2 - l).evaluate()
qpoch3 = qPochhammerSymbol(q**(-4*l1 - 4*l2), q**2, 2*l2).evaluate()
qpoch4 = qPochhammerSymbol([q**2, q**(-4*l1), q**(-4*l2 - 2*p)], q**2, \
l1 + l2 - l).evaluate()
line1 = sqrt(qpoch1 * qpoch2 / (qpoch3 * qpoch4))
line2 = sqrt((1 - q**(-4*l - 2)) / (1 - q**(-4*l1 - 4*l2 - 2))) \
* q**((l1+l2-l)*(2*l1+2*l2-p)+l2*(2*p-4*l1)-binomial(l1 + l2 - l, 2))
qpoch5 = qPochhammerSymbol([q**(-4*l2), q**(-4*l1 + 2*p)], \
q**2, n).evaluate()
qpoch6 = qPochhammerSymbol([q**2, q**(2*p + 2)], q**2, n).evaluate()
poly = dual_q_Hahn_polynomials(n, l1 + l2 - l, q**(-4*l1 + 2*p - 2), \
q**(-4*l2 - 2*p - 2), 2*l2, q**2)
line3 = q**((2*l1 + 2*l2 + 1)*n) * sqrt(qpoch5 / qpoch6) * poly
return line1 * line2 * line3
def qbinomial(n, k, q):
nominator = qPochhammerSymbol(q**n, q**(-1), k).evaluate()
denominator = qPochhammerSymbol(q, q, k).evaluate()
return nominator/denominator