def _eval_rewrite_as_polynomial(self, n, m, x):
k = C.Dummy("k")
kern = C.factorial(2 * n - 2 * k) / (
2**n * C.factorial(n - k) * C.factorial(k) *
C.factorial(n - 2 * k - m)) * (-1)**k * x**(n - m - 2 * k)
return (1 - x**2)**(m / 2) * C.Sum(kern,
(k, 0, C.floor((n - m) * S.Half)))
def _eval_rewrite_as_polynomial(self, n, m, x):
k = C.Dummy("k")
kern = (
C.factorial(2 * n - 2 * k)
/ (2 ** n * C.factorial(n - k) * C.factorial(k) * C.factorial(n - 2 * k - m))
* (-1) ** k
* x ** (n - m - 2 * k)
)
return (1 - x ** 2) ** (m / 2) * C.Sum(kern, (k, 0, C.floor((n - m) * S.Half)))
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, a, x):
k = C.Dummy("k")
kern = ((-1)**k * C.RisingFactorial(a, n - k) * (2 * x)**(n - 2 * k) /
(C.factorial(k) * C.factorial(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.factorial(k) *
C.factorial(n - 2 * k)) * (2 * x)**(n - 2 * k)
return C.factorial(n) * C.Sum(kern, (k, 0, C.floor(n / 2)))
def _eval_rewrite_as_polynomial(self, n, x):
k = C.Dummy("k")
kern = S.NegativeOne**k * C.factorial(n - k) * (2 * x)**(n - 2 * k) / (
C.factorial(k) * C.factorial(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 = S.NegativeOne**k * C.factorial(
n - k) * (2*x)**(n - 2*k) / (C.factorial(k) * C.factorial(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 = 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, a, x):
k = C.Dummy("k")
kern = ((-1)**k * C.RisingFactorial(a, n - k) * (2*x)**(n - 2*k) /
(C.factorial(k) * C.factorial(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.factorial(k)*C.factorial(n - 2*k)) * (2*x)**(n - 2*k)
return C.factorial(n)*C.Sum(kern, (k, 0, C.floor(n/2)))
