def eval(cls, n, a, b, x):
# Simplify to other polynomials
# P^{a, a}_n(x)
if a == b:
if a == -S.Half:
return RisingFactorial(S.Half, n) / factorial(n) * chebyshevt(n, x)
elif a == S.Zero:
return legendre(n, x)
elif a == S.Half:
return RisingFactorial(3*S.Half, n) / factorial(n + 1) * chebyshevu(n, x)
else:
return RisingFactorial(a + 1, n) / RisingFactorial(2*a + 1, n) * gegenbauer(n, a + S.Half, x)
elif b == -a:
# P^{a, -a}_n(x)
return gamma(n + a + 1) / gamma(n + 1) * (1 + x)**(a/2) / (1 - x)**(a/2) * assoc_legendre(n, -a, x)
if not n.is_Number:
# Symbolic result P^{a,b}_n(x)
# P^{a,b}_n(-x) ---> (-1)**n * P^{b,a}_n(-x)
if x.could_extract_minus_sign():
return S.NegativeOne**n * jacobi(n, b, a, -x)
# We can evaluate for some special values of x
if x == S.Zero:
return (2**(-n) * gamma(a + n + 1) / (gamma(a + 1) * factorial(n)) *
hyper([-b - n, -n], [a + 1], -1))
if x == S.One:
return RisingFactorial(a + 1, n) / factorial(n)
elif x == S.Infinity:
if n.is_positive:
# Make sure a+b+2*n \notin Z
if (a + b + 2*n).is_integer:
raise ValueError("a + b + 2*n should not be an integer.")
return RisingFactorial(a + b + n + 1, n) * S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
return jacobi_poly(n, a, b, x)
def _eval_rewrite_as_hyper(self, z):
return (pi / 2) * hyper((S.Half, S.Half), (S.One, ), z)
def _eval_rewrite_as_hyper(self, *args):
if len(args) == 1:
z = args[0]
return (pi / 2) * hyper((-S.Half, S.Half), (S.One, ), z)
def _eval_rewrite_as_hyper(self, z):
pf1 = S.One / (3**Rational(2, 3) * gamma(Rational(2, 3)))
pf2 = z / (root(3, 3) * gamma(Rational(1, 3)))
return pf1 * hyper([], [Rational(2, 3)], z**3 / 9) - pf2 * hyper(
[], [Rational(4, 3)], z**3 / 9)
def _eval_rewrite_as_hyper(self, z):
pf1 = z**2 / (2 * root(3, 6) * gamma(Rational(2, 3)))
pf2 = root(3, 6) / gamma(Rational(1, 3))
return pf1 * hyper([], [Rational(5, 3)], z**3 / 9) + pf2 * hyper(
[], [Rational(1, 3)], z**3 / 9)
def _eval_rewrite_as_hyper(self, z):
pf1 = z**2 / (2 * 3**Rational(2, 3) * gamma(Rational(2, 3)))
pf2 = 1 / (root(3, 3) * gamma(Rational(1, 3)))
return pf1 * hyper([], [Rational(5, 3)], z**3 / 9) - pf2 * hyper(
[], [Rational(1, 3)], z**3 / 9)