def fdiff(self, argindex=4):
if argindex == 1:
# Diff wrt n
raise ArgumentIndexError(self, argindex)
elif argindex == 2:
# Diff wrt a
n, a, b, x = self.args
k = C.Dummy("k")
f1 = 1 / (a + b + n + k + 1)
f2 = ((a + b + 2 * k + 1) * C.RisingFactorial(b + k + 1, n - k) /
((n - k) * C.RisingFactorial(a + b + k + 1, n - k)))
return C.Sum(f1 * (jacobi(n, a, b, x) + f2 * jacobi(k, a, b, x)),
(k, 0, n - 1))
elif argindex == 3:
# Diff wrt b
n, a, b, x = self.args
k = C.Dummy("k")
f1 = 1 / (a + b + n + k + 1)
f2 = (-1)**(n - k) * (
(a + b + 2 * k + 1) * C.RisingFactorial(a + k + 1, n - k) /
((n - k) * C.RisingFactorial(a + b + k + 1, n - k)))
return C.Sum(f1 * (jacobi(n, a, b, x) + f2 * jacobi(k, a, b, x)),
(k, 0, n - 1))
elif argindex == 4:
# Diff wrt x
n, a, b, x = self.args
return S.Half * (a + b + n + 1) * jacobi(n - 1, a + 1, b + 1, x)
else:
raise ArgumentIndexError(self, argindex)
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, a, b, x):
# TODO: Make sure n \in N
k = C.Dummy("k")
kern = (C.RisingFactorial(-n, k) *
C.RisingFactorial(a + b + n + 1, k) *
C.RisingFactorial(a + k + 1, n - k) / C.factorial(k) *
((1 - x) / 2)**k)
return 1 / C.factorial(n) * C.Sum(kern, (k, 0, n))
def fdiff(self, argindex=3):
if argindex == 1:
# Diff wrt n
raise ArgumentIndexError(self, argindex)
elif argindex == 2:
# Diff wrt alpha
n, alpha, x = self.args
k = C.Dummy("k")
return C.Sum(assoc_laguerre(k, alpha, x) / (n - alpha), (k, 0, n - 1))
elif argindex == 3:
# Diff wrt x
n, alpha, x = self.args
return -assoc_laguerre(n - 1, alpha + 1, x)
else:
raise ArgumentIndexError(self, argindex)
def fdiff(self, argindex=3):
if argindex == 1:
# Diff wrt n
raise ArgumentIndexError(self, argindex)
elif argindex == 2:
# Diff wrt a
n, a, x = self.args
k = C.Dummy("k")
factor1 = 2 * (1 + (-1)**(n - k)) * (k + a) / ((k +
n + 2*a) * (n - k))
factor2 = 2*(k + 1) / ((k + 2*a) * (2*k + 2*a + 1)) + \
2 / (k + n + 2*a)
kern = factor1*gegenbauer(k, a, x) + factor2*gegenbauer(n, a, x)
return C.Sum(kern, (k, 0, n - 1))
elif argindex == 3:
# Diff wrt x
n, a, x = self.args
return 2*a*gegenbauer(n - 1, a + 1, x)
else:
raise ArgumentIndexError(self, argindex)
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):
# TODO: Should make sure n is in N_0
k = C.Dummy("k")
kern = C.RisingFactorial(
-n, k) / (C.gamma(k + alpha + 1) * C.factorial(k)) * x**k
return C.gamma(n + alpha + 1) / C.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 = C.Dummy("k")
kern = C.RisingFactorial(-n, k) / C.factorial(k)**2 * x**k
return C.Sum(kern, (k, 0, n))
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 = (-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 = 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)))
