def eval(cls, n, a, b, x):
# Simplify to other polynomials
# P^{a, a}_n(x)
if a == b:
if a == -S.Half:
return C.RisingFactorial(S.Half, n) / C.factorial(n) * chebyshevt(n, x)
elif a == S.Zero:
return legendre(n, x)
elif a == S.Half:
return C.RisingFactorial(3 * S.Half, n) / C.factorial(n + 1) * chebyshevu(n, x)
else:
return C.RisingFactorial(a + 1, n) / C.RisingFactorial(2 * a + 1, n) * gegenbauer(n, a + S.Half, x)
elif b == -a:
# P^{a, -a}_n(x)
return (
C.gamma(n + a + 1) / C.gamma(n + 1) * (1 + x) ** (a / 2) / (1 - x) ** (a / 2) * assoc_legendre(n, -a, x)
)
elif a == -b:
# P^{-b, b}_n(x)
return (
C.gamma(n - b + 1) / C.gamma(n + 1) * (1 - x) ** (b / 2) / (1 + x) ** (b / 2) * assoc_legendre(n, b, 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)
* C.gamma(a + n + 1)
/ (C.gamma(a + 1) * C.factorial(n))
* C.hyper([-b - n, -n], [a + 1], -1)
)
if x == S.One:
return C.RisingFactorial(a + 1, n) / C.factorial(n)
elif x == S.Infinity:
if n.is_positive:
# TODO: Make sure a+b+2*n \notin Z
return C.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(cls, n, a, b, x):
# Simplify to other polynomials
# P^{a, a}_n(x)
if a == b:
if a == -S.Half:
return C.RisingFactorial(
S.Half, n) / C.factorial(n) * chebyshevt(n, x)
elif a == S.Zero:
return legendre(n, x)
elif a == S.Half:
return C.RisingFactorial(
3 * S.Half, n) / C.factorial(n + 1) * chebyshevu(n, x)
else:
return C.RisingFactorial(a + 1, n) / C.RisingFactorial(
2 * a + 1, n) * gegenbauer(n, a + S.Half, x)
elif b == -a:
# P^{a, -a}_n(x)
return C.gamma(n + a + 1) / C.gamma(n + 1) * (1 + x)**(a / 2) / (
1 - x)**(a / 2) * assoc_legendre(n, -a, x)
elif a == -b:
# P^{-b, b}_n(x)
return C.gamma(n - b + 1) / C.gamma(n + 1) * (1 - x)**(b / 2) / (
1 + x)**(b / 2) * assoc_legendre(n, b, 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) * C.gamma(a + n + 1) /
(C.gamma(a + 1) * C.factorial(n)) *
C.hyper([-b - n, -n], [a + 1], -1))
if x == S.One:
return C.RisingFactorial(a + 1, n) / C.factorial(n)
elif x == S.Infinity:
if n.is_positive:
# TODO: Make sure a+b+2*n \notin Z
return C.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)
