def taylor_term(n, x, *previous_terms):
if n == 0:
return S.Pi * S.ImaginaryUnit / 2
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) >= 2 and n > 2:
p = previous_terms[-2]
return p * (n - 2)**2 / (n * (n - 1)) * x**2
else:
k = (n - 1) // 2
R = RisingFactorial(S.Half, k)
F = factorial(k)
return -R / F * S.ImaginaryUnit * x**n / n
def expansion_term(n, x, *previous_terms):
if n == 0:
return log(2 / x)
elif n < 0 or n % 2 == 1:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2 and n > 2:
p = previous_terms[-2]
return p * (n - 1)**2 // (n // 2)**2 * x**2 / 4
else:
k = n // 2
R = RisingFactorial(S.Half, k) * n
F = factorial(k) * n // 2 * n // 2
return -1 * R / F * x**n / 4
def eval(cls, n, a, x):
# For negative n the polynomials vanish
# See http://functions.wolfram.com/Polynomials/GegenbauerC3/03/01/03/0012/
if n.is_negative:
return S.Zero
# Some special values for fixed a
if a == S.Half:
return legendre(n, x)
elif a == S.One:
return chebyshevu(n, x)
elif a == S.NegativeOne:
return S.Zero
if not n.is_Number:
# Handle this before the general sign extraction rule
if x == S.NegativeOne:
if (re(a) > S.Half) == True:
return S.ComplexInfinity
else:
return (
cos(S.Pi * (a + n))
* sec(S.Pi * a)
* gamma(2 * a + n)
/ (gamma(2 * a) * gamma(n + 1))
)
# Symbolic result C^a_n(x)
# C^a_n(-x) ---> (-1)**n * C^a_n(x)
if x.could_extract_minus_sign():
return S.NegativeOne ** n * gegenbauer(n, a, -x)
# We can evaluate for some special values of x
if x.is_zero:
return (
2 ** n
* sqrt(S.Pi)
* gamma(a + S.Half * n)
/ (gamma((1 - n) / 2) * gamma(n + 1) * gamma(a))
)
if x == S.One:
return gamma(2 * a + n) / (gamma(2 * a) * gamma(n + 1))
elif x is S.Infinity:
if n.is_positive:
return RisingFactorial(a, n) * S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
return gegenbauer_poly(n, a, x)
def check(self, f):
# Levit, Irreducibility of Polynomials with Low Absolute Values, Theorem 2
n = f.degree()
N = (n + 1) // 2
suitable = []
rhs = RisingFactorial((n//2)/2, N) / (2 ** (N-1))
for a in range(-20, 20 + 1):
val = abs(f.eval(a))
if val == 0:
# clearly this is reducible...
return REDUCIBLE, None
if val < rhs:
suitable.append(a)
if len(suitable) >= n:
return IRREDUCIBLE, {'a': suitable}
return UNKNOWN, None
def eval(cls, n, a, b, x):
# Simplify to other polynomials
# P^{a, a}_n(x)
if a == b:
if a == Rational(-1, 2):
return RisingFactorial(S.Half, n) / factorial(n) * chebyshevt(n, x)
elif a.is_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.is_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 is S.Infinity:
if n.is_positive:
# Make sure a+b+2*n \notin Z
if (a + b + 2 * n).is_integer:
raise ValueError("Error. 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 test_assoc_laguerre():
n = Symbol("n")
m = Symbol("m")
alpha = Symbol("alpha")
# generalized Laguerre polynomials:
assert assoc_laguerre(0, alpha, x) == 1
assert assoc_laguerre(1, alpha, x) == -x + alpha + 1
assert assoc_laguerre(2, alpha, x).expand() == \
(x**2/2 - (alpha + 2)*x + (alpha + 2)*(alpha + 1)/2).expand()
assert assoc_laguerre(3, alpha, x).expand() == \
(-x**3/6 + (alpha + 3)*x**2/2 - (alpha + 2)*(alpha + 3)*x/2 +
(alpha + 1)*(alpha + 2)*(alpha + 3)/6).expand()
# Test the lowest 10 polynomials with laguerre_poly, to make sure it works:
for i in range(10):
assert assoc_laguerre(i, 0, x).expand() == laguerre_poly(i, x)
X = assoc_laguerre(n, m, x)
assert isinstance(X, assoc_laguerre)
assert assoc_laguerre(n, 0, x) == laguerre(n, x)
assert assoc_laguerre(n, alpha, 0) == binomial(alpha + n, alpha)
p = Symbol("p", positive=True)
assert assoc_laguerre(p, alpha, oo) == (-1)**p * oo
assert assoc_laguerre(p, alpha, -oo) is oo
assert diff(assoc_laguerre(n, alpha, x), x) == \
-assoc_laguerre(n - 1, alpha + 1, x)
_k = Dummy('k')
assert diff(assoc_laguerre(n, alpha, x), alpha).dummy_eq(
Sum(assoc_laguerre(_k, alpha, x) / (-alpha + n), (_k, 0, n - 1)))
assert conjugate(assoc_laguerre(n, alpha, x)) == \
assoc_laguerre(n, conjugate(alpha), conjugate(x))
assert assoc_laguerre(n, alpha, x).rewrite('polynomial').dummy_eq(
gamma(alpha + n + 1) * Sum(
x**_k * RisingFactorial(-n, _k) /
(factorial(_k) * gamma(_k + alpha + 1)),
(_k, 0, n)) / factorial(n))
raises(ValueError, lambda: assoc_laguerre(-2.1, alpha, x))
raises(ArgumentIndexError, lambda: assoc_laguerre(n, alpha, x).fdiff(1))
raises(ArgumentIndexError, lambda: assoc_laguerre(n, alpha, x).fdiff(4))
def _eval_expand_func(self, **hints):
arg = self.args[0]
if arg.is_Rational:
if abs(arg.p) > arg.q:
x = Dummy('x')
n = arg.p // arg.q
p = arg.p - n*arg.q
return gamma(x + n)._eval_expand_func().subs(x, Rational(p, arg.q))
if arg.is_Add:
coeff, tail = arg.as_coeff_add()
if coeff and coeff.q != 1:
intpart = floor(coeff)
tail = (coeff - intpart,) + tail
coeff = intpart
tail = arg._new_rawargs(*tail, reeval=False)
return gamma(tail)*RisingFactorial(tail, coeff)
return self.func(*self.args)
def test_guess():
i0, i1 = symbols('i0 i1')
assert guess([1, 2, 6, 24, 120],
evaluate=False) == [Product(i1 + 1, (i1, 1, i0 - 1))]
assert guess([1, 2, 6, 24, 120]) == [RisingFactorial(2, i0 - 1)]
assert guess([1, 2, 7, 42, 429, 7436, 218348, 10850216], niter=4) == [
2**(i0 - 1) * (Rational(27, 16))**(i0**2 / 2 - 3 * i0 / 2 + 1) *
Product(
RisingFactorial(Rational(5, 3), i1 - 1) *
RisingFactorial(Rational(7, 3), i1 - 1) /
(RisingFactorial(Rational(3, 2), i1 - 1) *
RisingFactorial(Rational(5, 2), i1 - 1)), (i1, 1, i0 - 1))
]
assert guess([1, 0, 2]) == []
x, y = symbols('x y')
assert guess([1, 2, 6, 24, 120],
variables=[x, y]) == [RisingFactorial(2, x - 1)]
def _eval_rewrite_as_polynomial(self, n, a, x):
from sympy import Sum
k = Dummy("k")
kern = ((-1)**k * RisingFactorial(a, n - k) * (2 * x)**(n - 2 * k) /
(factorial(k) * factorial(n - 2 * k)))
return Sum(kern, (k, 0, floor(n / 2)))
def test_sympy__functions__combinatorial__factorials__RisingFactorial():
from sympy.functions.combinatorial.factorials import RisingFactorial
assert _test_args(RisingFactorial(2, x))
def test_jacobi():
n = Symbol("n")
a = Symbol("a")
b = Symbol("b")
assert jacobi(0, a, b, x) == 1
assert jacobi(1, a, b, x) == a / 2 - b / 2 + x * (a / 2 + b / 2 + 1)
assert jacobi(n, a, a, x) == RisingFactorial(a + 1, n) * gegenbauer(
n, a + S.Half, x) / RisingFactorial(2 * a + 1, n)
assert jacobi(n, a, -a,
x) == ((-1)**a * (-x + 1)**(-a / 2) * (x + 1)**(a / 2) *
assoc_legendre(n, a, x) * factorial(-a + n) *
gamma(a + n + 1) / (factorial(a + n) * gamma(n + 1)))
assert jacobi(n, -b, b, x) == ((-x + 1)**(b / 2) * (x + 1)**(-b / 2) *
assoc_legendre(n, b, x) *
gamma(-b + n + 1) / gamma(n + 1))
assert jacobi(n, 0, 0, x) == legendre(n, x)
assert jacobi(n, S.Half, S.Half, x) == RisingFactorial(Rational(
3, 2), n) * chebyshevu(n, x) / factorial(n + 1)
assert jacobi(
n, Rational(-1, 2), Rational(-1, 2),
x) == RisingFactorial(S.Half, n) * chebyshevt(n, x) / factorial(n)
X = jacobi(n, a, b, x)
assert isinstance(X, jacobi)
assert jacobi(n, a, b, -x) == (-1)**n * jacobi(n, b, a, x)
assert jacobi(n, a, b, 0) == 2**(-n) * gamma(a + n + 1) * hyper(
(-b - n, -n), (a + 1, ), -1) / (factorial(n) * gamma(a + 1))
assert jacobi(n, a, b, 1) == RisingFactorial(a + 1, n) / factorial(n)
m = Symbol("m", positive=True)
assert jacobi(m, a, b, oo) == oo * RisingFactorial(a + b + m + 1, m)
assert unchanged(jacobi, n, a, b, oo)
assert conjugate(jacobi(m, a, b, x)) == \
jacobi(m, conjugate(a), conjugate(b), conjugate(x))
_k = Dummy('k')
assert diff(jacobi(n, a, b, x), n) == Derivative(jacobi(n, a, b, x), n)
assert diff(jacobi(n, a, b, x), a).dummy_eq(
Sum((jacobi(n, a, b, x) + (2 * _k + a + b + 1) *
RisingFactorial(_k + b + 1, -_k + n) * jacobi(_k, a, b, x) /
((-_k + n) * RisingFactorial(_k + a + b + 1, -_k + n))) /
(_k + a + b + n + 1), (_k, 0, n - 1)))
assert diff(jacobi(n, a, b, x), b).dummy_eq(
Sum(((-1)**(-_k + n) * (2 * _k + a + b + 1) *
RisingFactorial(_k + a + 1, -_k + n) * jacobi(_k, a, b, x) /
((-_k + n) * RisingFactorial(_k + a + b + 1, -_k + n)) +
jacobi(n, a, b, x)) / (_k + a + b + n + 1), (_k, 0, n - 1)))
assert diff(jacobi(n, a, b, x), x) == \
(a/2 + b/2 + n/2 + S.Half)*jacobi(n - 1, a + 1, b + 1, x)
assert jacobi_normalized(n, a, b, x) == \
(jacobi(n, a, b, x)/sqrt(2**(a + b + 1)*gamma(a + n + 1)*gamma(b + n + 1)
/((a + b + 2*n + 1)*factorial(n)*gamma(a + b + n + 1))))
raises(ValueError, lambda: jacobi(-2.1, a, b, x))
raises(ValueError,
lambda: jacobi(Dummy(positive=True, integer=True), 1, 2, oo))
assert jacobi(n, a, b, x).rewrite("polynomial").dummy_eq(
Sum((S.Half - x / 2)**_k * RisingFactorial(-n, _k) *
RisingFactorial(_k + a + 1, -_k + n) *
RisingFactorial(a + b + n + 1, _k) / factorial(_k),
(_k, 0, n)) / factorial(n))
raises(ArgumentIndexError, lambda: jacobi(n, a, b, x).fdiff(5))
def _eval_rewrite_as_polynomial(self, n, a, b, x):
# TODO: Make sure n \in N
k = Dummy("k")
kern = (RisingFactorial(-n, k) * RisingFactorial(a + b + n + 1, k) * RisingFactorial(a + k + 1, n - k) /
factorial(k) * ((1 - x)/2)**k)
return 1 / factorial(n) * C.Sum(kern, (k, 0, n))
def _eval_rewrite_as_polynomial(self, n, x):
# TODO: Should make sure n is in N_0
k = Dummy("k")
kern = RisingFactorial(
-n, k) / (gamma(k + alpha + 1) * factorial(k)) * x**k
return gamma(n + alpha + 1) / factorial(n) * C.Sum(kern, (k, 0, n))
def _eval_rewrite_as_polynomial(self, n, x):
# TODO: Should make sure n is in N_0
k = Dummy("k")
kern = RisingFactorial(-n, k) / factorial(k)**2 * x**k
return C.Sum(kern, (k, 0, n))
def _eval_product(self, term, limits):
(k, a, n) = limits
if k not in term.free_symbols:
if (term - 1).is_zero:
return S.One
return term**(n - a + 1)
if a == n:
return term.subs(k, a)
from .delta import deltaproduct, _has_simple_delta
if term.has(KroneckerDelta) and _has_simple_delta(term, limits[0]):
return deltaproduct(term, limits)
dif = n - a
definite = dif.is_Integer
if definite and (dif < 100):
return self._eval_product_direct(term, limits)
elif term.is_polynomial(k):
poly = term.as_poly(k)
A = B = Q = S.One
all_roots = roots(poly)
M = 0
for r, m in all_roots.items():
M += m
A *= RisingFactorial(a - r, n - a + 1)**m
Q *= (n - r)**m
if M < poly.degree():
arg = quo(poly, Q.as_poly(k))
B = self.func(arg, (k, a, n)).doit()
return poly.LC()**(n - a + 1) * A * B
elif term.is_Add:
factored = factor_terms(term, fraction=True)
if factored.is_Mul:
return self._eval_product(factored, (k, a, n))
elif term.is_Mul:
# Factor in part without the summation variable and part with
without_k, with_k = term.as_coeff_mul(k)
if len(with_k) >= 2:
# More than one term including k, so still a multiplication
exclude, include = [], []
for t in with_k:
p = self._eval_product(t, (k, a, n))
if p is not None:
exclude.append(p)
else:
include.append(t)
if not exclude:
return None
else:
arg = term._new_rawargs(*include)
A = Mul(*exclude)
B = self.func(arg, (k, a, n)).doit()
return without_k**(n - a + 1) * A * B
else:
# Just a single term
p = self._eval_product(with_k[0], (k, a, n))
if p is None:
p = self.func(with_k[0], (k, a, n)).doit()
return without_k**(n - a + 1) * p
elif term.is_Pow:
if not term.base.has(k):
s = summation(term.exp, (k, a, n))
return term.base**s
elif not term.exp.has(k):
p = self._eval_product(term.base, (k, a, n))
if p is not None:
return p**term.exp
elif isinstance(term, Product):
evaluated = term.doit()
f = self._eval_product(evaluated, limits)
if f is None:
return self.func(evaluated, limits)
else:
return f
if definite:
return self._eval_product_direct(term, limits)
def test_issue_6878():
n = symbols('n', integer=True)
assert combsimp(RisingFactorial(
-10, n)) == 3628800 * (-1)**n / factorial(10 - n)
