def test_laguerre():
alpha = Symbol("alpha")
# generalized Laguerre polynomials:
assert laguerre_l(0, alpha, x) == 1
assert laguerre_l(1, alpha, x) == -x + alpha + 1
assert laguerre_l(2, alpha,
x).expand() == (x**2 / 2 - (alpha + 2) * x +
(alpha + 2) * (alpha + 1) / 2).expand()
assert laguerre_l(
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()
# Laguerre polynomials:
assert laguerre_l(0, 0, x) == 1
assert laguerre_l(1, 0, x) == 1 - x
assert laguerre_l(2, 0, x).expand() == 1 - 2 * x + x**2 / 2
assert laguerre_l(3, 0, x).expand() == 1 - 3 * x + 3 * x**2 / 2 - x**3 / 6
# Test the lowest 10 polynomials with laguerre_poly, to make sure that it
# works:
for i in range(10):
assert laguerre_l(i, 0, x).expand() == laguerre_poly(i, 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)
assert diff(assoc_laguerre(n, alpha, x), x) == \
-assoc_laguerre(n - 1, alpha + 1, x)
assert conjugate(assoc_laguerre(n, alpha, x)) == \
assoc_laguerre(n, conjugate(alpha), conjugate(x))
raises(ValueError, lambda: assoc_laguerre(-2.1, alpha, 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)
assert diff(assoc_laguerre(n, alpha, x), x) == \
-assoc_laguerre(n - 1, alpha + 1, x)
assert conjugate(assoc_laguerre(n, alpha, x)) == \
assoc_laguerre(n, conjugate(alpha), conjugate(x))
raises(ValueError, lambda: assoc_laguerre(-2.1, alpha, 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 test_laguerre():
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()
# Laguerre polynomials:
assert assoc_laguerre(0, 0, x) == 1
assert assoc_laguerre(1, 0, x) == 1 - x
assert assoc_laguerre(2, 0, x).expand() == 1 - 2 * x + x**2 / 2
assert assoc_laguerre(3, 0,
x).expand() == 1 - 3 * x + 3 * x**2 / 2 - x**3 / 6
# Test the lowest 10 polynomials with laguerre_poly, to make sure that it
# works:
for i in range(10):
assert assoc_laguerre(i, 0, x).expand() == laguerre_poly(i, x)
n = Symbol("n")
X = laguerre(n, x)
assert isinstance(X, laguerre)
assert laguerre(n, 0) == 1
assert diff(laguerre(n, x), x) == -assoc_laguerre(n - 1, 1, x)
m = Symbol("m")
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)
assert diff(assoc_laguerre(n, alpha, x),
x) == -assoc_laguerre(n - 1, alpha + 1, x)
def test_laguerre():
alpha = Symbol("alpha")
# generalized Laguerre polynomials:
assert laguerre_l(0, alpha, x) == 1
assert laguerre_l(1, alpha, x) == -x + alpha + 1
assert laguerre_l(2, alpha, x).expand() == (x**2/2 - (alpha+2)*x + (alpha+2)*(alpha+1)/2).expand()
assert laguerre_l(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()
# Laguerre polynomials:
assert laguerre_l(0, 0, x) == 1
assert laguerre_l(1, 0, x) == 1 - x
assert laguerre_l(2, 0, x).expand() == 1 - 2*x + x**2/2
assert laguerre_l(3, 0, x).expand() == 1 - 3*x + 3*x**2/2 - x**3/6
# Test the lowest 10 polynomials with laguerre_poly, to make sure that it
# works:
for i in range(10):
assert laguerre_l(i, 0, x).expand() == laguerre_poly(i, x)
def test_laguerre():
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()
# Laguerre polynomials:
assert assoc_laguerre(0, 0, x) == 1
assert assoc_laguerre(1, 0, x) == 1 - x
assert assoc_laguerre(2, 0, x).expand() == 1 - 2*x + x**2/2
assert assoc_laguerre(3, 0, x).expand() == 1 - 3*x + 3*x**2/2 - x**3/6
# Test the lowest 10 polynomials with laguerre_poly, to make sure that it
# works:
for i in range(10):
assert assoc_laguerre(i, 0, x).expand() == laguerre_poly(i, x)
n = Symbol("n")
X = laguerre(n, x)
assert isinstance(X, laguerre)
assert laguerre(n, 0) == 1
assert diff(laguerre(n, x), x) == -assoc_laguerre(n - 1, 1, x)
m = Symbol("m")
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)
assert diff(assoc_laguerre(n, alpha, x), x) == \
-assoc_laguerre(n - 1, alpha + 1, x)
