def eval(cls, n, alpha, x):
# L_{n}^{0}(x) ---> L_{n}(x)
if alpha == S.Zero:
return laguerre(n, x)
if not n.is_Number:
# We can evaluate for some special values of x
if x == S.Zero:
return C.binomial(n+alpha, alpha)
elif x == S.Infinity and n > S.Zero:
return S.NegativeOne**n * S.Infinity
elif x == S.NegativeInfinity and n > S.Zero:
return S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
if n.is_negative:
raise ValueError("The index n must be nonnegative integer (got %r)" % n)
else:
return laguerre_poly(n, x, alpha)
def eval(cls, n, alpha, x):
# L_{n}^{0}(x) ---> L_{n}(x)
if alpha == S.Zero:
return laguerre(n, x)
if not n.is_Number:
# We can evaluate for some special values of x
if x == S.Zero:
return C.binomial(n + alpha, alpha)
elif x == S.Infinity and n > S.Zero:
return S.NegativeOne**n * S.Infinity
elif x == S.NegativeInfinity and n > S.Zero:
return S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
if n.is_negative:
raise ValueError(
"The index n must be nonnegative integer (got %r)" % n)
else:
return laguerre_poly(n, x, alpha)
def _eval_rewrite_as_polynomial(self, n, x):
k = C.Dummy("k")
kern = C.binomial(n, 2 * k) * (x**2 - 1)**k * x**(n - 2 * k)
return C.Sum(kern, (k, 0, C.floor(n / 2)))
def _eval_rewrite_as_polynomial(self, n, x):
k = C.Dummy("k")
kern = (-1)**k * C.binomial(n, k)**2 * ((1 + x) / 2)**(n - k) * (
(1 - x) / 2)**k
return C.Sum(kern, (k, 0, n))
def _eval_rewrite_as_polynomial(self, n, x):
k = C.Dummy("k")
kern = (-1)**k*C.binomial(n, k)**2*((1 + x)/2)**(n - k)*((1 - x)/2)**k
return C.Sum(kern, (k, 0, n))
def _eval_rewrite_as_polynomial(self, n, x):
k = C.Dummy("k")
kern = C.binomial(n, 2*k) * (x**2 - 1)**k * x**(n - 2*k)
return C.Sum(kern, (k, 0, C.floor(n/2)))
