def test_gosper_sum_AeqB_part2():
f2a = n**2*a**n
f2b = (n - r/2)*binomial(r, n)
f2c = factorial(n - 1)**2/(factorial(n - x)*factorial(n + x))
g2a = -a*(a + 1)/(a - 1)**3 + a**(
m + 1)*(a**2*m**2 - 2*a*m**2 + m**2 - 2*a*m + 2*m + a + 1)/(a - 1)**3
g2b = (m - r)*binomial(r, m)/2
ff = factorial(1 - x)*factorial(1 + x)
g2c = 1/ff*(
1 - 1/x**2) + factorial(m)**2/(x**2*factorial(m - x)*factorial(m + x))
g = gosper_sum(f2a, (n, 0, m))
assert g is not None and simplify(g - g2a) == 0
g = gosper_sum(f2b, (n, 0, m))
assert g is not None and simplify(g - g2b) == 0
g = gosper_sum(f2c, (n, 1, m))
assert g is not None and simplify(g - g2c) == 0
# delete these lines and unXFAIL the nan test below when it passes
f2d = n*(n + a + b)*a**n*b**n/(factorial(n + a)*factorial(n + b))
g2d = 1/(factorial(a - 1)*factorial(
b - 1)) - a**(m + 1)*b**(m + 1)/(factorial(a + m)*factorial(b + m))
assert simplify(
sum(f2d.subs(n, i) for i in range(3)) - g2d.subs(m, 2)) == 0
def test_gosper_sum_indefinite():
assert gosper_sum(k, k) == k * (k - 1) / 2
assert gosper_sum(k**2, k) == k * (k - 1) * (2 * k - 1) / 6
assert gosper_sum(1 / (k * (k + 1)), k) == -1 / k
assert gosper_sum(-(27*k**4 + 158*k**3 + 430*k**2 + 678*k + 445)*gamma(2*k + 4)/(3*(3*k + 7)*gamma(3*k + 6)), k) == \
(3*k + 5)*(k**2 + 2*k + 5)*gamma(2*k + 4)/gamma(3*k + 6)
def test_gosper_sum_AeqB_part2():
f2a = n**2 * a**n
f2b = (n - r / 2) * binomial(r, n)
f2c = factorial(n - 1)**2 / (factorial(n - x) * factorial(n + x))
g2a = -a * (a + 1) / (a - 1)**3 + a**(m + 1) * (
a**2 * m**2 - 2 * a * m**2 + m**2 - 2 * a * m + 2 * m + a + 1) / (a -
1)**3
g2b = (m - r) * binomial(r, m) / 2
ff = factorial(1 - x) * factorial(1 + x)
g2c = 1 / ff * (1 - 1 / x**2) + factorial(m)**2 / (
x**2 * factorial(m - x) * factorial(m + x))
g = gosper_sum(f2a, (n, 0, m))
assert g is not None and simplify(g - g2a) == 0
g = gosper_sum(f2b, (n, 0, m))
assert g is not None and simplify(g - g2b) == 0
g = gosper_sum(f2c, (n, 1, m))
assert g is not None and simplify(g - g2c) == 0
# delete these lines and unXFAIL the nan test below when it passes
f2d = n * (n + a + b) * a**n * b**n / (factorial(n + a) * factorial(n + b))
g2d = 1 / (factorial(a - 1) * factorial(b - 1)) - a**(m + 1) * b**(
m + 1) / (factorial(a + m) * factorial(b + m))
assert simplify(sum(f2d.subs(n, i)
for i in range(3)) - g2d.subs(m, 2)) == 0
def test_gosper_sum_indefinite():
assert gosper_sum(k, k) == k*(k - 1)/2
assert gosper_sum(k**2, k) == k*(k - 1)*(2*k - 1)/6
assert gosper_sum(1/(k*(k + 1)), k) == -1/k
assert gosper_sum(-(27*k**4 + 158*k**3 + 430*k**2 + 678*k + 445)*gamma(2*k + 4)/(3*(3*k + 7)*gamma(3*k + 6)), k) == \
(3*k + 5)*(k**2 + 2*k + 5)*gamma(2*k + 4)/gamma(3*k + 6)
def test_gosper_sum_iterated():
f1 = binomial(2*k, k)/4**k
f2 = (1 + 2*n)*binomial(2*n, n)/4**n
f3 = (1 + 2*n)*(3 + 2*n)*binomial(2*n, n)/(3*4**n)
f4 = (1 + 2*n)*(3 + 2*n)*(5 + 2*n)*binomial(2*n, n)/(15*4**n)
f5 = (1 + 2*n)*(3 + 2*n)*(5 + 2*n)*(7 + 2*n)*binomial(2*n, n)/(105*4**n)
assert gosper_sum(f1, (k, 0, n)) == f2
assert gosper_sum(f2, (n, 0, n)) == f3
assert gosper_sum(f3, (n, 0, n)) == f4
assert gosper_sum(f4, (n, 0, n)) == f5
def test_gosper_sum_iterated():
f1 = binomial(2*k, k)/4**k
f2 = (1 + 2*n)*binomial(2*n, n)/4**n
f3 = (1 + 2*n)*(3 + 2*n)*binomial(2*n, n)/(3*4**n)
f4 = (1 + 2*n)*(3 + 2*n)*(5 + 2*n)*binomial(2*n, n)/(15*4**n)
f5 = (1 + 2*n)*(3 + 2*n)*(5 + 2*n)*(7 + 2*n)*binomial(2*n, n)/(105*4**n)
assert gosper_sum(f1, (k, 0, n)) == f2
assert gosper_sum(f2, (n, 0, n)) == f3
assert gosper_sum(f3, (n, 0, n)) == f4
assert gosper_sum(f4, (n, 0, n)) == f5
def test_gosper_sum_AeqB_part2():
f2a = n**2*a**n
f2b = (n - r/2)*binomial(r, n)
f2c = factorial(n - 1)**2/(factorial(n - x)*factorial(n + x))
f2d = n*(n + a + b)*a**n*b**n/(factorial(n + a)*factorial(n + b))
g = gosper_sum(f2a, (n, 0, m))
assert g is not None # and simplify(g - g2a) == 0
g = gosper_sum(f2b, (n, 0, m))
assert g is not None # and simplify(g - g2b) == 0
g = gosper_sum(f2c, (n, 1, m))
assert g is not None # and simplify(g - g2c) == 0
g = gosper_sum(f2d, (n, 0, m))
assert g is not None # and simplify(g - g2d) == 0
def test_gosper_sum_AeqB_part1():
f1a = n**4
f1b = n**3*2**n
f1c = 1/(n**2 + sqrt(5)*n - 1)
f1d = n**4*4**n/binomial(2*n, n)
f1e = factorial(3*n)/(factorial(n)*factorial(n + 1)*factorial(n + 2)*27**n)
f1f = binomial(2*n, n)**2/((n + 1)*4**(2*n))
f1g = (4*n - 1)*binomial(2*n, n)**2/((2*n - 1)**2*4**(2*n))
f1h = n*factorial(n - S(1)/2)**2/factorial(n + 1)**2
g1a = m*(m + 1)*(2*m + 1)*(3*m**2 + 3*m - 1)/30
g1b = 26 + 2**(m + 1)*(m**3 - 3*m**2 + 9*m - 13)
g = gosper_sum(f1a, (n, 0, m))
assert g is not None and simplify(g - g1a) == 0
g = gosper_sum(f1b, (n, 0, m))
assert g is not None and simplify(g - g1b) == 0
g = gosper_sum(f1c, (n, 0, m))
assert g is not None # and simplify(g - g1c) == 0
g = gosper_sum(f1d, (n, 0, m))
assert g is not None # and simplify(g - g1d) == 0
g = gosper_sum(f1e, (n, 0, m))
assert g is not None # and simplify(g - g1e) == 0
g = gosper_sum(f1f, (n, 0, m))
assert g is not None # and simplify(g - g1f) == 0
g = gosper_sum(f1g, (n, 0, m))
assert g is not None # and simplify(g - g1g) == 0
g = gosper_sum(f1h, (n, 0, m))
assert g is not None # and simplify(g - g1h) == 0
def test_gosper_sum_AeqB_part1():
f1a = n**4
f1b = n**3*2**n
f1c = 1/(n**2 + sqrt(5)*n - 1)
f1d = n**4*4**n/binomial(2*n, n)
f1e = factorial(3*n)/(factorial(n)*factorial(n + 1)*factorial(n + 2)*27**n)
f1f = binomial(2*n, n)**2/((n + 1)*4**(2*n))
f1g = (4*n - 1)*binomial(2*n, n)**2/((2*n - 1)**2*4**(2*n))
f1h = n*factorial(n - S(1)/2)**2/factorial(n + 1)**2
g1a = m*(m + 1)*(2*m + 1)*(3*m**2 + 3*m - 1)/30
g1b = 26 + 2**(m + 1)*(m**3 - 3*m**2 + 9*m - 13)
g1c = (m + 1)*(m*(m**2 - 7*m + 3)*sqrt(5) - (3*m**3 - 7*m**2 + 19*m - 6))/(2*m**3*sqrt(5) + m**4 + m**2 - 1)/6
g1d = -S(2)/231 + 24**m*(m + 1)*(63*m**4 + 112*m**3 + 18*m**2 - 22*m + 3)/(693*binomial(2*m, m))
g1e = -S(9)/2 + (81*m**2 + 261*m + 200)*factorial(3*m + 2)/(40*27**m*factorial(m)*factorial(m + 1)*factorial(m + 2))
g1f = (2*m + 1)**2*binomial(2*m, m)**2/(4**(2*m)*(m + 1))
g1g = -binomial(2*m, m)**2/4**(2*m)
g1h = -(2*m + 1)**2*(3*m + 4)*factorial(m - S(1)/2)**2/factorial(m + 1)**2
g = gosper_sum(f1a, (n, 0, m))
assert g is not None and simplify(g - g1a) == 0
g = gosper_sum(f1b, (n, 0, m))
assert g is not None and simplify(g - g1b) == 0
g = gosper_sum(f1c, (n, 0, m))
assert g is not None # and simplify(g - g1c) == 0
g = gosper_sum(f1d, (n, 0, m))
assert g is not None # and simplify(g - g1d) == 0
g = gosper_sum(f1e, (n, 0, m))
assert g is not None # and simplify(g - g1e) == 0
g = gosper_sum(f1f, (n, 0, m))
assert g is not None # and simplify(g - g1f) == 0
g = gosper_sum(f1g, (n, 0, m))
assert g is not None # and simplify(g - g1g) == 0
g = gosper_sum(f1h, (n, 0, m))
assert g is not None # and simplify(g - g1h) == 0
def test_gosper_sum_AeqB_part3():
f3a = 1/n**4
f3b = (6*n + 3)/(4*n**4 + 8*n**3 + 8*n**2 + 4*n + 3)
f3c = 2**n*(n**2 - 2*n - 1)/(n**2*(n + 1)**2)
f3d = n**2*4**n/((n + 1)*(n + 2))
f3e = 2**n/(n + 1)
f3f = 4*(n - 1)*(n**2 - 2*n - 1)/(n**2*(n + 1)**2*(n - 2)**2*(n - 3)**2)
f3g = (n**4 - 14*n**2 - 24*n - 9)*2**n/(n**2*(n + 1)**2*(n + 2)**2*
(n + 3)**2)
# g3a -> no closed form
g3b = m*(m + 2)/(2*m**2 + 4*m + 3)
g3c = 2**m/m**2 - 2
g3d = S(2)/3 + 4**(m + 1)*(m - 1)/(m + 2)/3
# g3e -> no closed form
g3f = -(-S(1)/16 + 1/((m - 2)**2*(m + 1)**2)) # the AeqB key is wrong
g3g = -S(2)/9 + 2**(m + 1)/((m + 1)**2*(m + 3)**2)
g = gosper_sum(f3a, (n, 1, m))
assert g is None
g = gosper_sum(f3b, (n, 1, m))
assert g is not None and simplify(g - g3b) == 0
g = gosper_sum(f3c, (n, 1, m - 1))
assert g is not None and simplify(g - g3c) == 0
g = gosper_sum(f3d, (n, 1, m))
assert g is not None and simplify(g - g3d) == 0
g = gosper_sum(f3e, (n, 0, m - 1))
assert g is None
g = gosper_sum(f3f, (n, 4, m))
assert g is not None and simplify(g - g3f) == 0
g = gosper_sum(f3g, (n, 1, m))
assert g is not None and simplify(g - g3g) == 0
def test_gosper_sum_AeqB_part3():
f3a = 1 / n**4
f3b = (6 * n + 3) / (4 * n**4 + 8 * n**3 + 8 * n**2 + 4 * n + 3)
f3c = 2**n * (n**2 - 2 * n - 1) / (n**2 * (n + 1)**2)
f3d = n**2 * 4**n / ((n + 1) * (n + 2))
f3e = 2**n / (n + 1)
f3f = 4 * (n - 1) * (n**2 - 2 * n - 1) / (n**2 * (n + 1)**2 * (n - 2)**2 *
(n - 3)**2)
f3g = (n**4 - 14 * n**2 - 24 * n - 9) * 2**n / (n**2 * (n + 1)**2 *
(n + 2)**2 * (n + 3)**2)
# g3a -> no closed form
g3b = m * (m + 2) / (2 * m**2 + 4 * m + 3)
g3c = 2**m / m**2 - 2
g3d = S(2) / 3 + 4**(m + 1) * (m - 1) / (m + 2) / 3
# g3e -> no closed form
g3f = -(-S(1) / 16 + 1 / ((m - 2)**2 *
(m + 1)**2)) # the AeqB key is wrong
g3g = -S(2) / 9 + 2**(m + 1) / ((m + 1)**2 * (m + 3)**2)
g = gosper_sum(f3a, (n, 1, m))
assert g is None
g = gosper_sum(f3b, (n, 1, m))
assert g is not None and simplify(g - g3b) == 0
g = gosper_sum(f3c, (n, 1, m - 1))
assert g is not None and simplify(g - g3c) == 0
g = gosper_sum(f3d, (n, 1, m))
assert g is not None and simplify(g - g3d) == 0
g = gosper_sum(f3e, (n, 0, m - 1))
assert g is None
g = gosper_sum(f3f, (n, 4, m))
assert g is not None and simplify(g - g3f) == 0
g = gosper_sum(f3g, (n, 1, m))
assert g is not None and simplify(g - g3g) == 0
def test_gosper_sum_AeqB_part2():
f2a = n**2*a**n
f2b = (n - r/2)*binomial(r, n)
f2c = factorial(n - 1)**2/(factorial(n - x)*factorial(n + x))
g2a = -a*(a + 1)/(a - 1)**3 + a**(
m + 1)*(a**2*m**2 - 2*a*m**2 + m**2 - 2*a*m + 2*m + a + 1)/(a - 1)**3
g2b = (m - r)*binomial(r, m)/2
ff = factorial(1 - x)*factorial(1 + x)
g2c = 1/ff*(
1 - 1/x**2) + factorial(m)**2/(x**2*factorial(m - x)*factorial(m + x))
g = gosper_sum(f2a, (n, 0, m))
assert g is not None and simplify(g - g2a) == 0
g = gosper_sum(f2b, (n, 0, m))
assert g is not None and simplify(g - g2b) == 0
g = gosper_sum(f2c, (n, 1, m))
assert g is not None and simplify(g - g2c) == 0
def test_gosper_sum_AeqB_part2():
f2a = n**2*a**n
f2b = (n - r/2)*binomial(r, n)
f2c = factorial(n - 1)**2/(factorial(n - x)*factorial(n + x))
g2a = -a*(a + 1)/(a - 1)**3 + a**(
m + 1)*(a**2*m**2 - 2*a*m**2 + m**2 - 2*a*m + 2*m + a + 1)/(a - 1)**3
g2b = (m - r)*binomial(r, m)/2
ff = factorial(1 - x)*factorial(1 + x)
g2c = 1/ff*(
1 - 1/x**2) + factorial(m)**2/(x**2*factorial(m - x)*factorial(m + x))
g = gosper_sum(f2a, (n, 0, m))
assert g is not None and simplify(g - g2a) == 0
g = gosper_sum(f2b, (n, 0, m))
assert g is not None and simplify(g - g2b) == 0
g = gosper_sum(f2c, (n, 1, m))
assert g is not None and simplify(g - g2c) == 0
def test_gosper_sum_AeqB_part2():
f2a = n**2*a**n
f2b = (n - r/2)*binomial(r, n)
f2c = factorial(n - 1)**2/(factorial(n - x)*factorial(n + x))
f2d = n*(n + a + b)*a**n*b**n/(factorial(n + a)*factorial(n + b))
g2a = -a*(a + 1)/(a - 1)**3 + a**(m + 1)*(a**2*m**2 - 2*a*m*22 + m**2 - 2*a*m + 2*m + a + 1)/(a - 1)**3
g2b = -((-m + r)*binomial(r, m))
g2c = 1/(factorial(1 - x)*factorial(1 + x)) - 1/(x**2*factorial(1 - x)*factorial(1 + x)) + factorial(m)**2/(x**2*factorial(1 - x)*factorial(1 + x))
g2d = 1/(factorial(a - 1)*factorial(b - 1)) - a**(m + 1)*b**(m + 1)/(factorial(a + m)*factorial(b + m))
g = gosper_sum(f2a, (n, 0, m))
assert g is not None # and simplify(g - g2a) == 0
g = gosper_sum(f2b, (n, 0, m))
assert g is not None # and simplify(g - g2b) == 0
g = gosper_sum(f2c, (n, 1, m))
assert g is not None # and simplify(g - g2c) == 0
g = gosper_sum(f2d, (n, 0, m))
assert g is not None # and simplify(g - g2d) == 0
def test_gosper_nan():
a = Symbol('a', positive=True)
b = Symbol('b', positive=True)
n = Symbol('n', integer=True)
m = Symbol('m', integer=True)
f2d = n*(n + a + b)*a**n*b**n/(factorial(n + a)*factorial(n + b))
g2d = 1/(factorial(a - 1)*factorial(
b - 1)) - a**(m + 1)*b**(m + 1)/(factorial(a + m)*factorial(b + m))
g = gosper_sum(f2d, (n, 0, m))
assert simplify(g - g2d) == 0
def test_gosper_nan():
a = Symbol('a', positive=True)
b = Symbol('b', positive=True)
n = Symbol('n', integer=True)
m = Symbol('m', integer=True)
f2d = n * (n + a + b) * a**n * b**n / (factorial(n + a) * factorial(n + b))
g2d = 1 / (factorial(a - 1) * factorial(b - 1)) - a**(m + 1) * b**(
m + 1) / (factorial(a + m) * factorial(b + m))
g = gosper_sum(f2d, (n, 0, m))
assert simplify(g - g2d) == 0
def test_gosper_sum_AeqB_part1():
f1a = n**4
f1b = n**3 * 2**n
f1c = 1 / (n**2 + sqrt(5) * n - 1)
f1d = n**4 * 4**n / binomial(2 * n, n)
f1e = factorial(
3 * n) / (factorial(n) * factorial(n + 1) * factorial(n + 2) * 27**n)
f1f = binomial(2 * n, n)**2 / ((n + 1) * 4**(2 * n))
f1g = (4 * n - 1) * binomial(2 * n, n)**2 / ((2 * n - 1)**2 * 4**(2 * n))
f1h = n * factorial(n - S.Half)**2 / factorial(n + 1)**2
g1a = m * (m + 1) * (2 * m + 1) * (3 * m**2 + 3 * m - 1) / 30
g1b = 26 + 2**(m + 1) * (m**3 - 3 * m**2 + 9 * m - 13)
g1c = ((m + 1) * (m * (m**2 - 7 * m + 3) * sqrt(5) -
(3 * m**3 - 7 * m**2 + 19 * m - 6)) /
(2 * m**3 * sqrt(5) + m**4 + 5 * m**2 - 1) / 6)
g1d = Rational(
-2,
231) + 2 * 4**m * (m + 1) * (63 * m**4 + 112 * m**3 + 18 * m**2 -
22 * m + 3) / (693 * binomial(2 * m, m))
g1e = Rational(
-9, 2) + (81 * m**2 + 261 * m + 200) * factorial(3 * m + 2) / (
40 * 27**m * factorial(m) * factorial(m + 1) * factorial(m + 2))
g1f = (2 * m + 1)**2 * binomial(2 * m, m)**2 / (4**(2 * m) * (m + 1))
g1g = -binomial(2 * m, m)**2 / 4**(2 * m)
g1h = (4 * pi - (2 * m + 1)**2 *
(3 * m + 4) * factorial(m - S.Half)**2 / factorial(m + 1)**2)
g = gosper_sum(f1a, (n, 0, m))
assert g is not None and simplify(g - g1a) == 0
g = gosper_sum(f1b, (n, 0, m))
assert g is not None and simplify(g - g1b) == 0
g = gosper_sum(f1c, (n, 0, m))
assert g is not None and simplify(g - g1c) == 0
g = gosper_sum(f1d, (n, 0, m))
assert g is not None and simplify(g - g1d) == 0
g = gosper_sum(f1e, (n, 0, m))
assert g is not None and simplify(g - g1e) == 0
g = gosper_sum(f1f, (n, 0, m))
assert g is not None and simplify(g - g1f) == 0
g = gosper_sum(f1g, (n, 0, m))
assert g is not None and simplify(g - g1g) == 0
g = gosper_sum(f1h, (n, 0, m))
# need to call rewrite(gamma) here because we have terms involving
# factorial(1/2)
assert g is not None and simplify(g - g1h).rewrite(gamma) == 0
def test_gosper_sum_AeqB_part2():
f2a = n**2 * a**n
f2b = (n - r / 2) * binomial(r, n)
f2c = factorial(n - 1)**2 / (factorial(n - x) * factorial(n + x))
f2d = n * (n + a + b) * a**n * b**n / (factorial(n + a) * factorial(n + b))
g2a = -a * (a + 1) / (a - 1)**3 + a**(
m + 1) * (a**2 * m**2 - 2 * a * m * 22 + m**2 - 2 * a * m + 2 * m + a +
1) / (a - 1)**3
g2b = -((-m + r) * binomial(r, m))
g2c = 1 / (factorial(1 - x) * factorial(1 + x)) - 1 / (
x**2 * factorial(1 - x) * factorial(1 + x)) + factorial(m)**2 / (
x**2 * factorial(1 - x) * factorial(1 + x))
g2d = 1 / (factorial(a - 1) * factorial(b - 1)) - a**(m + 1) * b**(
m + 1) / (factorial(a + m) * factorial(b + m))
g = gosper_sum(f2a, (n, 0, m))
assert g is not None # and simplify(g - g2a) == 0
g = gosper_sum(f2b, (n, 0, m))
assert g is not None # and simplify(g - g2b) == 0
g = gosper_sum(f2c, (n, 1, m))
assert g is not None # and simplify(g - g2c) == 0
g = gosper_sum(f2d, (n, 0, m))
assert g is not None # and simplify(g - g2d) == 0
def test_gosper_sum_AeqB_part1():
f1a = n**4
f1b = n**3 * 2**n
f1c = 1 / (n**2 + sqrt(5) * n - 1)
f1d = n**4 * 4**n / binomial(2 * n, n)
f1e = factorial(
3 * n) / (factorial(n) * factorial(n + 1) * factorial(n + 2) * 27**n)
f1f = binomial(2 * n, n)**2 / ((n + 1) * 4**(2 * n))
f1g = (4 * n - 1) * binomial(2 * n, n)**2 / ((2 * n - 1)**2 * 4**(2 * n))
f1h = n * factorial(n - S(1) / 2)**2 / factorial(n + 1)**2
g1a = m * (m + 1) * (2 * m + 1) * (3 * m**2 + 3 * m - 1) / 30
g1b = 26 + 2**(m + 1) * (m**3 - 3 * m**2 + 9 * m - 13)
g1c = (m + 1) * (m * (m**2 - 7 * m + 3) * sqrt(5) -
(3 * m**3 - 7 * m**2 + 19 * m - 6)) / (
2 * m**3 * sqrt(5) + m**4 + 5 * m**2 - 1) / 6
g1d = -S(2) / 231 + 2 * 4**m * (m + 1) * (63 * m**4 + 112 * m**3 +
18 * m**2 - 22 * m +
3) / (693 * binomial(2 * m, m))
g1e = -S(9) / 2 + (81 * m**2 + 261 * m + 200) * factorial(3 * m + 2) / (
40 * 27**m * factorial(m) * factorial(m + 1) * factorial(m + 2))
g1f = (2 * m + 1)**2 * binomial(2 * m, m)**2 / (4**(2 * m) * (m + 1))
g1g = -binomial(2 * m, m)**2 / 4**(2 * m)
g1h = -(2 * m + 1)**2 * (
3 * m + 4) * factorial(m - S(1) / 2)**2 / factorial(m + 1)**2
g = gosper_sum(f1a, (n, 0, m))
assert g is not None and simplify(g - g1a) == 0
g = gosper_sum(f1b, (n, 0, m))
assert g is not None and simplify(g - g1b) == 0
g = gosper_sum(f1c, (n, 0, m))
assert g is not None and simplify(g - g1c) == 0
g = gosper_sum(f1d, (n, 0, m))
assert g is not None and simplify(g - g1d) == 0
g = gosper_sum(f1e, (n, 0, m))
assert g is not None and simplify(g - g1e) == 0
g = gosper_sum(f1f, (n, 0, m))
assert g is not None and simplify(g - g1f) == 0
g = gosper_sum(f1g, (n, 0, m))
assert g is not None and simplify(g - g1g) == 0
g = gosper_sum(f1h, (n, 0, m))
assert g is not None and simplify(g - g1h) == 0
def test_gosper_sum_AeqB_part3():
f3a = 1/n**4
f3b = (6*n + 3)/(4*n**4 + 8*n**3 + 8*n**2 + 4*n + 3)
f3c = 2**n*(n**2 - 2*n - 1)/(n**2*(n + 1)**2)
f3d = n**2*4**n/((n + 1)*(n + 2))
f3e = 2**n/(n + 1)
f3f = 4*(n - 1)*(n**2 - 2*n - 1)/(n**2*(n + 1)**2*(n - 2)**2*(n - 3)**2)
f3g = (n**4 - 14*n**2 - 24*n - 9)*2**n/(n**2*(n + 1)**2*(n + 2)**2*(n + 3)**2)
g = gosper_sum(f3a, (n, 1, m))
assert g is None
g = gosper_sum(f3b, (n, 1, m))
assert g is not None # and simplify(g - g3b) == 0
g = gosper_sum(f3c, (n, 1, m-1))
assert g is not None # and simplify(g - g3c) == 0
g = gosper_sum(f3d, (n, 1, m))
assert g is not None # and simplify(g - g3d) == 0
g = gosper_sum(f3e, (n, 0, m-1))
assert g is None
g = gosper_sum(f3f, (n, 4, m))
assert g is not None # and simplify(g - g3f) == 0
g = gosper_sum(f3g, (n, 1, m))
assert g is not None # and simplify(g - g3g) == 0
def eval_sum_symbolic(f, limits):
from sympy.functions import harmonic, bernoulli
f_orig = f
(i, a, b) = limits
if not f.has(i):
return f*(b - a + 1)
# Linearity
if f.is_Mul:
L, R = f.as_two_terms()
if not L.has(i):
sR = eval_sum_symbolic(R, (i, a, b))
if sR:
return L*sR
if not R.has(i):
sL = eval_sum_symbolic(L, (i, a, b))
if sL:
return R*sL
try:
f = apart(f, i) # see if it becomes an Add
except PolynomialError:
pass
if f.is_Add:
L, R = f.as_two_terms()
lrsum = telescopic(L, R, (i, a, b))
if lrsum:
return lrsum
lsum = eval_sum_symbolic(L, (i, a, b))
rsum = eval_sum_symbolic(R, (i, a, b))
if None not in (lsum, rsum):
r = lsum + rsum
if not r is S.NaN:
return r
# Polynomial terms with Faulhaber's formula
n = Wild('n')
result = f.match(i**n)
if result is not None:
n = result[n]
if n.is_Integer:
if n >= 0:
if (b is S.Infinity and not a is S.NegativeInfinity) or \
(a is S.NegativeInfinity and not b is S.Infinity):
return S.Infinity
return ((bernoulli(n + 1, b + 1) - bernoulli(n + 1, a))/(n + 1)).expand()
elif a.is_Integer and a >= 1:
if n == -1:
return harmonic(b) - harmonic(a - 1)
else:
return harmonic(b, abs(n)) - harmonic(a - 1, abs(n))
if not (a.has(S.Infinity, S.NegativeInfinity) or
b.has(S.Infinity, S.NegativeInfinity)):
# Geometric terms
c1 = Wild('c1', exclude=[i])
c2 = Wild('c2', exclude=[i])
c3 = Wild('c3', exclude=[i])
wexp = Wild('wexp')
# Here we first attempt powsimp on f for easier matching with the
# exponential pattern, and attempt expansion on the exponent for easier
# matching with the linear pattern.
e = f.powsimp().match(c1 ** wexp)
if e is not None:
e_exp = e.pop(wexp).expand().match(c2*i + c3)
if e_exp is not None:
e.update(e_exp)
if e is not None:
p = (c1**c3).subs(e)
q = (c1**c2).subs(e)
r = p*(q**a - q**(b + 1))/(1 - q)
l = p*(b - a + 1)
return Piecewise((l, Eq(q, S.One)), (r, True))
r = gosper_sum(f, (i, a, b))
if not r in (None, S.NaN):
return r
return eval_sum_hyper(f_orig, (i, a, b))
def eval_sum_symbolic(f, limits):
from sympy.functions import harmonic, bernoulli
f_orig = f
(i, a, b) = limits
if not f.has(i):
return f * (b - a + 1)
# Linearity
if f.is_Mul:
L, R = f.as_two_terms()
if not L.has(i):
sR = eval_sum_symbolic(R, (i, a, b))
if sR:
return L * sR
if not R.has(i):
sL = eval_sum_symbolic(L, (i, a, b))
if sL:
return R * sL
try:
f = apart(f, i) # see if it becomes an Add
except PolynomialError:
pass
if f.is_Add:
L, R = f.as_two_terms()
lrsum = telescopic(L, R, (i, a, b))
if lrsum:
return lrsum
lsum = eval_sum_symbolic(L, (i, a, b))
rsum = eval_sum_symbolic(R, (i, a, b))
if None not in (lsum, rsum):
r = lsum + rsum
if not r is S.NaN:
return r
# Polynomial terms with Faulhaber's formula
n = Wild('n')
result = f.match(i**n)
if result is not None:
n = result[n]
if n.is_Integer:
if n >= 0:
if (b is S.Infinity and not a is S.NegativeInfinity) or \
(a is S.NegativeInfinity and not b is S.Infinity):
return S.Infinity
return ((bernoulli(n + 1, b + 1) - bernoulli(n + 1, a)) /
(n + 1)).expand()
elif a.is_Integer and a >= 1:
if n == -1:
return harmonic(b) - harmonic(a - 1)
else:
return harmonic(b, abs(n)) - harmonic(a - 1, abs(n))
if not (a.has(S.Infinity, S.NegativeInfinity)
or b.has(S.Infinity, S.NegativeInfinity)):
# Geometric terms
c1 = Wild('c1', exclude=[i])
c2 = Wild('c2', exclude=[i])
c3 = Wild('c3', exclude=[i])
wexp = Wild('wexp')
# Here we first attempt powsimp on f for easier matching with the
# exponential pattern, and attempt expansion on the exponent for easier
# matching with the linear pattern.
e = f.powsimp().match(c1**wexp)
if e is not None:
e_exp = e.pop(wexp).expand().match(c2 * i + c3)
if e_exp is not None:
e.update(e_exp)
if e is not None:
p = (c1**c3).subs(e)
q = (c1**c2).subs(e)
r = p * (q**a - q**(b + 1)) / (1 - q)
l = p * (b - a + 1)
return Piecewise((l, Eq(q, S.One)), (r, True))
r = gosper_sum(f, (i, a, b))
if not r in (None, S.NaN):
return r
return eval_sum_hyper(f_orig, (i, a, b))
def eval_sum_symbolic(f, limits):
(i, a, b) = limits
if not f.has(i):
return f*(b-a+1)
# Linearity
if f.is_Mul:
L, R = f.as_two_terms()
if not L.has(i):
sR = eval_sum_symbolic(R, (i, a, b))
if sR: return L*sR
if not R.has(i):
sL = eval_sum_symbolic(L, (i, a, b))
if sL: return R*sL
try:
f = apart(f, i) # see if it becomes an Add
except PolynomialError:
pass
if f.is_Add:
L, R = f.as_two_terms()
lrsum = telescopic(L, R, (i, a, b))
if lrsum:
return lrsum
lsum = eval_sum_symbolic(L, (i, a, b))
rsum = eval_sum_symbolic(R, (i, a, b))
if None not in (lsum, rsum):
return lsum + rsum
# Polynomial terms with Faulhaber's formula
n = Wild('n')
result = f.match(i**n)
if result is not None:
n = result[n]
if n.is_Integer:
if n >= 0:
return ((C.bernoulli(n+1, b+1) - C.bernoulli(n+1, a))/(n+1)).expand()
elif a.is_Integer and a >= 1:
if n == -1:
return C.harmonic(b) - C.harmonic(a - 1)
else:
return C.harmonic(b, abs(n)) - C.harmonic(a - 1, abs(n))
# Geometric terms
c1 = C.Wild('c1', exclude=[i])
c2 = C.Wild('c2', exclude=[i])
c3 = C.Wild('c3', exclude=[i])
e = f.match(c1**(c2*i+c3))
if e is not None:
c1 = c1.subs(e)
c2 = c2.subs(e)
c3 = c3.subs(e)
# TODO: more general limit handling
return c1**c3 * (c1**(a*c2) - c1**(c2+b*c2)) / (1 - c1**c2)
if not (a.has(S.Infinity, S.NegativeInfinity) or \
b.has(S.Infinity, S.NegativeInfinity)):
r = gosper_sum(f, (i, a, b))
if not r in (None, S.NaN):
return r
return eval_sum_hyper(f, (i, a, b))
def eval_sum_symbolic(f, limits):
f_orig = f
(i, a, b) = limits
if not f.has(i):
return f*(b - a + 1)
# Linearity
if f.is_Mul:
L, R = f.as_two_terms()
if not L.has(i):
sR = eval_sum_symbolic(R, (i, a, b))
if sR:
return L*sR
if not R.has(i):
sL = eval_sum_symbolic(L, (i, a, b))
if sL:
return R*sL
try:
f = apart(f, i) # see if it becomes an Add
except PolynomialError:
pass
if f.is_Add:
L, R = f.as_two_terms()
lrsum = telescopic(L, R, (i, a, b))
if lrsum:
return lrsum
lsum = eval_sum_symbolic(L, (i, a, b))
rsum = eval_sum_symbolic(R, (i, a, b))
if None not in (lsum, rsum):
r = lsum + rsum
if not r is S.NaN:
return r
# Polynomial terms with Faulhaber's formula
n = Wild('n')
result = f.match(i**n)
if result is not None:
n = result[n]
if n.is_Integer:
if n >= 0:
if (b is S.Infinity and not a is S.NegativeInfinity) or \
(a is S.NegativeInfinity and not b is S.Infinity):
return S.Infinity
return ((C.bernoulli(n + 1, b + 1) - C.bernoulli(n + 1, a))/(n + 1)).expand()
elif a.is_Integer and a >= 1:
if n == -1:
return C.harmonic(b) - C.harmonic(a - 1)
else:
return C.harmonic(b, abs(n)) - C.harmonic(a - 1, abs(n))
if not (a.has(S.Infinity, S.NegativeInfinity) or
b.has(S.Infinity, S.NegativeInfinity)):
# Geometric terms
c1 = C.Wild('c1', exclude=[i])
c2 = C.Wild('c2', exclude=[i])
c3 = C.Wild('c3', exclude=[i])
e = f.match(c1**(c2*i + c3))
if e is not None:
p = (c1**c3).subs(e)
q = (c1**c2).subs(e)
r = p*(q**a - q**(b + 1))/(1 - q)
l = p*(b - a + 1)
return Piecewise((l, Eq(q, S.One)), (r, True))
r = gosper_sum(f, (i, a, b))
if not r in (None, S.NaN):
return r
return eval_sum_hyper(f_orig, (i, a, b))
def test_gosper_sum():
assert gosper_sum(1, (k, 0, n)) == 1 + n
assert gosper_sum(k, (k, 0, n)) == n * (1 + n) / 2
assert gosper_sum(k**2, (k, 0, n)) == n * (1 + n) * (1 + 2 * n) / 6
assert gosper_sum(k**3, (k, 0, n)) == n**2 * (1 + n)**2 / 4
assert gosper_sum(2**k, (k, 0, n)) == 2 * 2**n - 1
assert gosper_sum(factorial(k), (k, 0, n)) is None
assert gosper_sum(binomial(n, k), (k, 0, n)) is None
assert gosper_sum(factorial(k) / k**2, (k, 0, n)) is None
assert gosper_sum((k - 3) * factorial(k), (k, 0, n)) is None
assert gosper_sum(k * factorial(k), k) == factorial(k)
assert gosper_sum(k * factorial(k),
(k, 0, n)) == n * factorial(n) + factorial(n) - 1
assert gosper_sum((-1)**k * binomial(n, k), (k, 0, n)) == 0
assert gosper_sum((-1)**k * binomial(n, k),
(k, 0, m)) == -(-1)**m * (m - n) * binomial(n, m) / n
assert gosper_sum((4*k + 1)*factorial(k)/factorial(2*k + 1), (k, 0, n)) == \
(2*factorial(2*n + 1) - factorial(n))/factorial(2*n + 1)
def test_gosper_sum_parametric():
assert gosper_sum(binomial(S(1)/2, m - j + 1)*binomial(S(1)/2, m + j), (j, 1, n)) == \
n*(1 + m - n)*(-1 + 2*m + 2*n)*binomial(S(1)/2, 1 + m - n)* \
binomial(S(1)/2, m + n)/(m*(1 + 2*m))
def test_gosper_sum():
assert gosper_sum(1, (k, 0, n)) == 1 + n
assert gosper_sum(k, (k, 0, n)) == n*(1 + n)/2
assert gosper_sum(k**2, (k, 0, n)) == n*(1 + n)*(1 + 2*n)/6
assert gosper_sum(k**3, (k, 0, n)) == n**2*(1 + n)**2/4
assert gosper_sum(2**k, (k, 0, n)) == 2*2**n - 1
assert gosper_sum(factorial(k), (k, 0, n)) is None
assert gosper_sum(binomial(n, k), (k, 0, n)) is None
assert gosper_sum(factorial(k)/k**2, (k, 0, n)) is None
assert gosper_sum((k - 3)*factorial(k), (k, 0, n)) is None
assert gosper_sum(k*factorial(k), k) == factorial(k)
assert gosper_sum(
k*factorial(k), (k, 0, n)) == n*factorial(n) + factorial(n) - 1
assert gosper_sum((-1)**k*binomial(n, k), (k, 0, n)) == 0
assert gosper_sum((
-1)**k*binomial(n, k), (k, 0, m)) == -(-1)**m*(m - n)*binomial(n, m)/n
assert gosper_sum((4*k + 1)*factorial(k)/factorial(2*k + 1), (k, 0, n)) == \
(2*factorial(2*n + 1) - factorial(n))/factorial(2*n + 1)
# issue 6033:
assert gosper_sum(
n*(n + a + b)*a**n*b**n/(factorial(n + a)*factorial(n + b)), \
(n, 0, m)) == -a*b*(exp(m*log(a))*exp(m*log(b))*factorial(a)* \
factorial(b) - factorial(a + m)*factorial(b + m))/(factorial(a)* \
factorial(b)*factorial(a + m)*factorial(b + m))
def test_gosper_sum_algebraic():
assert gosper_sum(
n**2 + sqrt(2), (n, 0, m)) == (m + 1)*(2*m**2 + m + 6*sqrt(2))/6
def eval_sum_symbolic(f, limits):
(i, a, b) = limits
if not f.has(i):
return f*(b - a + 1)
# Linearity
if f.is_Mul:
L, R = f.as_two_terms()
if not L.has(i):
sR = eval_sum_symbolic(R, (i, a, b))
if sR:
return L*sR
if not R.has(i):
sL = eval_sum_symbolic(L, (i, a, b))
if sL:
return R*sL
try:
f = apart(f, i) # see if it becomes an Add
except PolynomialError:
pass
if f.is_Add:
L, R = f.as_two_terms()
lrsum = telescopic(L, R, (i, a, b))
if lrsum:
return lrsum
lsum = eval_sum_symbolic(L, (i, a, b))
rsum = eval_sum_symbolic(R, (i, a, b))
if None not in (lsum, rsum):
return lsum + rsum
# Polynomial terms with Faulhaber's formula
n = Wild('n')
result = f.match(i**n)
if result is not None:
n = result[n]
if n.is_Integer:
if n >= 0:
if (b is S.Infinity and not a is S.NegativeInfinity) or \
(a is S.NegativeInfinity and not b is S.Infinity):
return S.Infinity
return ((C.bernoulli(n + 1, b + 1) - C.bernoulli(n + 1, a))/(n + 1)).expand()
elif a.is_Integer and a >= 1:
if n == -1:
return C.harmonic(b) - C.harmonic(a - 1)
else:
return C.harmonic(b, abs(n)) - C.harmonic(a - 1, abs(n))
if not (a.has(S.Infinity, S.NegativeInfinity) or
b.has(S.Infinity, S.NegativeInfinity)):
# Geometric terms
c1 = C.Wild('c1', exclude=[i])
c2 = C.Wild('c2', exclude=[i])
c3 = C.Wild('c3', exclude=[i])
e = f.match(c1**(c2*i + c3))
if e is not None:
p = (c1**c3).subs(e)
q = (c1**c2).subs(e)
r = p*(q**a - q**(b + 1))/(1 - q)
l = p*(b - a + 1)
return Piecewise((l, Eq(q, S.One)), (r, True))
r = gosper_sum(f, (i, a, b))
if not r in (None, S.NaN):
return r
return eval_sum_hyper(f, (i, a, b))
def test_gosper_sum():
assert gosper_sum(1, (k, 0, n)) == 1 + n
assert gosper_sum(k, (k, 0, n)) == n * (1 + n) / 2
assert gosper_sum(k**2, (k, 0, n)) == n * (1 + n) * (1 + 2 * n) / 6
assert gosper_sum(k**3, (k, 0, n)) == n**2 * (1 + n)**2 / 4
assert gosper_sum(2**k, (k, 0, n)) == 2 * 2**n - 1
assert gosper_sum(factorial(k), (k, 0, n)) is None
assert gosper_sum(binomial(n, k), (k, 0, n)) is None
assert gosper_sum(factorial(k) / k**2, (k, 0, n)) is None
assert gosper_sum((k - 3) * factorial(k), (k, 0, n)) is None
assert gosper_sum(k * factorial(k), k) == factorial(k)
assert gosper_sum(k * factorial(k),
(k, 0, n)) == n * factorial(n) + factorial(n) - 1
assert gosper_sum((-1)**k * binomial(n, k), (k, 0, n)) == 0
assert gosper_sum((-1)**k * binomial(n, k),
(k, 0, m)) == -(-1)**m * (m - n) * binomial(n, m) / n
assert gosper_sum((4*k + 1)*factorial(k)/factorial(2*k + 1), (k, 0, n)) == \
(2*factorial(2*n + 1) - factorial(n))/factorial(2*n + 1)
# issue 2934:
assert gosper_sum(
n*(n + a + b)*a**n*b**n/(factorial(n + a)*factorial(n + b)), \
(n, 0, m)) == -a*b*(exp(m*log(a))*exp(m*log(b))*factorial(a)* \
factorial(b) - factorial(a + m)*factorial(b + m))/(factorial(a)* \
factorial(b)*factorial(a + m)*factorial(b + m))
def test_gosper_sum():
assert gosper_sum(1, (k, 0, n)) == 1 + n
assert gosper_sum(k, (k, 0, n)) == n*(1 + n)/2
assert gosper_sum(k**2, (k, 0, n)) == n*(1 + n)*(1 + 2*n)/6
assert gosper_sum(k**3, (k, 0, n)) == n**2*(1 + n)**2/4
assert gosper_sum(2**k, (k, 0, n)) == 2*2**n - 1
assert gosper_sum(factorial(k), (k, 0, n)) is None
assert gosper_sum(binomial(n, k), (k, 0, n)) is None
assert gosper_sum(factorial(k)/k**2, (k, 0, n)) is None
assert gosper_sum((k - 3)*factorial(k), (k, 0, n)) is None
assert gosper_sum(k*factorial(k), k) == factorial(k)
assert gosper_sum(
k*factorial(k), (k, 0, n)) == n*factorial(n) + factorial(n) - 1
assert gosper_sum((-1)**k*binomial(n, k), (k, 0, n)) == 0
assert gosper_sum((
-1)**k*binomial(n, k), (k, 0, m)) == -(-1)**m*(m - n)*binomial(n, m)/n
assert gosper_sum((4*k + 1)*factorial(k)/factorial(2*k + 1), (k, 0, n)) == \
(2*factorial(2*n + 1) - factorial(n))/factorial(2*n + 1)
def eval_sum_symbolic(f, limits):
from sympy.functions import harmonic, bernoulli
f_orig = f
(i, a, b) = limits
if not f.has(i):
return f*(b - a + 1)
# Linearity
if f.is_Mul:
L, R = f.as_two_terms()
if not L.has(i):
sR = eval_sum_symbolic(R, (i, a, b))
if sR:
return L*sR
if not R.has(i):
sL = eval_sum_symbolic(L, (i, a, b))
if sL:
return R*sL
try:
f = apart(f, i) # see if it becomes an Add
except PolynomialError:
pass
if f.is_Add:
L, R = f.as_two_terms()
lrsum = telescopic(L, R, (i, a, b))
if lrsum:
return lrsum
lsum = eval_sum_symbolic(L, (i, a, b))
rsum = eval_sum_symbolic(R, (i, a, b))
if None not in (lsum, rsum):
r = lsum + rsum
if not r is S.NaN:
return r
# Polynomial terms with Faulhaber's formula
n = Wild('n')
result = f.match(i**n)
if result is not None:
n = result[n]
if n.is_Integer:
if n >= 0:
if (b is S.Infinity and not a is S.NegativeInfinity) or \
(a is S.NegativeInfinity and not b is S.Infinity):
return S.Infinity
return ((bernoulli(n + 1, b + 1) - bernoulli(n + 1, a))/(n + 1)).expand()
elif a.is_Integer and a >= 1:
if n == -1:
return harmonic(b) - harmonic(a - 1)
else:
return harmonic(b, abs(n)) - harmonic(a - 1, abs(n))
if not (a.has(S.Infinity, S.NegativeInfinity) or
b.has(S.Infinity, S.NegativeInfinity)):
# Geometric terms
c1 = Wild('c1', exclude=[i])
c2 = Wild('c2', exclude=[i])
c3 = Wild('c3', exclude=[i])
wexp = Wild('wexp')
# Here we first attempt powsimp on f for easier matching with the
# exponential pattern, and attempt expansion on the exponent for easier
# matching with the linear pattern.
e = f.powsimp().match(c1 ** wexp)
if e is not None:
e_exp = e.pop(wexp).expand().match(c2*i + c3)
if e_exp is not None:
e.update(e_exp)
if e is not None:
p = (c1**c3).subs(e)
q = (c1**c2).subs(e)
r = p*(q**a - q**(b + 1))/(1 - q)
l = p*(b - a + 1)
return Piecewise((l, Eq(q, S.One)), (r, True))
r = gosper_sum(f, (i, a, b))
if isinstance(r, (Mul,Add)):
from sympy import ordered, Tuple
non_limit = r.free_symbols - Tuple(*limits[1:]).free_symbols
den = denom(together(r))
den_sym = non_limit & den.free_symbols
args = []
for v in ordered(den_sym):
try:
s = solve(den, v)
m = Eq(v, s[0]) if s else S.false
if m != False:
args.append((Sum(f_orig.subs(*m.args), limits).doit(), m))
break
except NotImplementedError:
continue
args.append((r, True))
return Piecewise(*args)
if not r in (None, S.NaN):
return r
h = eval_sum_hyper(f_orig, (i, a, b))
if h is not None:
return h
factored = f_orig.factor()
if factored != f_orig:
return eval_sum_symbolic(factored, (i, a, b))
if c.is_integer and c >= 0:
s = (B(c+1, b+1) - B(c+1, a))/(c+1)
return s.expand()
# Geometric terms
c1 = C.Wild('c1', exclude=[i])
c2 = C.Wild('c2', exclude=[i])
c3 = C.Wild('c3', exclude=[i])
e = f.match(c1**(c2*i+c3))
if e is not None:
c1 = c1.subs(e)
c2 = c2.subs(e)
c3 = c3.subs(e)
# TODO: more general limit handling
return c1**c3 * (c1**(a*c2) - c1**(c2+b*c2)) / (1 - c1**c2)
return gosper_sum(f, (i, a, b))
def eval_sum_direct(expr, (i, a, b)):
s = S.Zero
if i in expr.free_symbols:
for j in xrange(a, b+1):
s += expr.subs(i, j)
else:
for j in xrange(a, b+1):
s += expr
return s
def test_gosper_nan():
f2d = n * (n + a + b) * a**n * b**n / (factorial(n + a) * factorial(n + b))
g2d = 1 / (factorial(a - 1) * factorial(b - 1)) - a**(m + 1) * b**(
m + 1) / (factorial(a + m) * factorial(b + m))
g = gosper_sum(f2d, (n, 0, m))
assert g is not S.NaN and simplify(g - g2d) == 0
def eval_sum_symbolic(f, limits):
(i, a, b) = limits
if not f.has(i):
return f * (b - a + 1)
# Linearity
if f.is_Mul:
L, R = f.as_two_terms()
if not L.has(i):
sR = eval_sum_symbolic(R, (i, a, b))
if sR: return L * sR
if not R.has(i):
sL = eval_sum_symbolic(L, (i, a, b))
if sL: return R * sL
try:
f = apart(f, i) # see if it becomes an Add
except PolynomialError:
pass
if f.is_Add:
L, R = f.as_two_terms()
lrsum = telescopic(L, R, (i, a, b))
if lrsum:
return lrsum
lsum = eval_sum_symbolic(L, (i, a, b))
rsum = eval_sum_symbolic(R, (i, a, b))
if None not in (lsum, rsum):
return lsum + rsum
# Polynomial terms with Faulhaber's formula
n = Wild('n')
result = f.match(i**n)
if result is not None:
n = result[n]
if n.is_Integer:
if n >= 0:
return ((C.bernoulli(n + 1, b + 1) - C.bernoulli(n + 1, a)) /
(n + 1)).expand()
elif a.is_Integer and a >= 1:
if n == -1:
return C.harmonic(b) - C.harmonic(a - 1)
else:
return C.harmonic(b, abs(n)) - C.harmonic(a - 1, abs(n))
# Geometric terms
c1 = C.Wild('c1', exclude=[i])
c2 = C.Wild('c2', exclude=[i])
c3 = C.Wild('c3', exclude=[i])
e = f.match(c1**(c2 * i + c3))
if e is not None:
c1 = c1.subs(e)
c2 = c2.subs(e)
c3 = c3.subs(e)
# TODO: more general limit handling
return c1**c3 * (c1**(a * c2) - c1**(c2 + b * c2)) / (1 - c1**c2)
r = gosper_sum(f, (i, a, b))
if not r in (None, S.NaN):
return r
return eval_sum_hyper(f, (i, a, b))
def test_gosper_sum():
assert gosper_sum(1, (k, 0, n)) == 1 + n
assert gosper_sum(k, (k, 0, n)) == n * (1 + n) / 2
assert gosper_sum(k**2, (k, 0, n)) == n * (1 + n) * (1 + 2 * n) / 6
assert gosper_sum(k**3, (k, 0, n)) == n**2 * (1 + n)**2 / 4
assert gosper_sum(2**k, (k, 0, n)) == 2 * 2**n - 1
assert gosper_sum(factorial(k), (k, 0, n)) is None
assert gosper_sum(binomial(n, k), (k, 0, n)) is None
assert gosper_sum(factorial(k) / k**2, (k, 0, n)) is None
assert gosper_sum((k - 3) * factorial(k), (k, 0, n)) is None
assert gosper_sum(k * factorial(k), k) == factorial(k)
assert gosper_sum(k * factorial(k),
(k, 0, n)) == n * factorial(n) + factorial(n) - 1
assert gosper_sum((-1)**k * binomial(n, k), (k, 0, n)) == 0
assert gosper_sum((-1)**k * binomial(n, k),
(k, 0, m)) == -(-1)**m * (m - n) * binomial(n, m) / n
assert gosper_sum((4*k + 1)*factorial(k)/factorial(2*k + 1), (k, 0, n)) == \
(2*factorial(2*n + 1) - factorial(n))/factorial(2*n + 1)
# issue 6033:
assert gosper_sum(
n*(n + a + b)*a**n*b**n/(factorial(n + a)*factorial(n + b)), \
(n, 0, m)).simplify() == -exp(m*log(a) + m*log(b))*gamma(a + 1) \
*gamma(b + 1)/(gamma(a)*gamma(b)*gamma(a + m + 1)*gamma(b + m + 1)) \
+ 1/(gamma(a)*gamma(b))
def test_gosper_sum_parametric():
assert gosper_sum(binomial(S(1)/2, m - j + 1)*binomial(S(1)/2, m + j), (j, 1, n)) == \
n*(1 + m - n)*(-1 + 2*m + 2*n)*binomial(S(1)/2, 1 + m - n)* \
binomial(S(1)/2, m + n)/(m*(1 + 2*m))
else:
return C.harmonic(b, abs(n)) - C.harmonic(a - 1, abs(n))
# Geometric terms
c1 = C.Wild('c1', exclude=[i])
c2 = C.Wild('c2', exclude=[i])
c3 = C.Wild('c3', exclude=[i])
e = f.match(c1**(c2 * i + c3))
if e is not None:
c1 = c1.subs(e)
c2 = c2.subs(e)
c3 = c3.subs(e)
# TODO: more general limit handling
return c1**c3 * (c1**(a * c2) - c1**(c2 + b * c2)) / (1 - c1**c2)
return gosper_sum(f, (i, a, b))
def eval_sum_direct(expr, (i, a, b)):
s = S.Zero
if i in expr.free_symbols:
for j in xrange(a, b + 1):
s += expr.subs(i, j)
else:
for j in xrange(a, b + 1):
s += expr
return s
def test_gosper_sum_algebraic():
assert gosper_sum(n**2 + sqrt(2),
(n, 0, m)) == (m + 1) * (2 * m**2 + m + 6 * sqrt(2)) / 6
def test_gosper_nan():
f2d = n*(n + a + b)*a**n*b**n/(factorial(n + a)*factorial(n + b))
g2d = 1/(factorial(a - 1)*factorial(
b - 1)) - a**(m + 1)*b**(m + 1)/(factorial(a + m)*factorial(b + m))
g = gosper_sum(f2d, (n, 0, m))
assert g is not S.NaN and simplify(g - g2d) == 0
def eval_sum_symbolic(f, limits):
from sympy.functions import harmonic, bernoulli
f_orig = f
(i, a, b) = limits
if not f.has(i):
return f*(b - a + 1)
# Linearity
if f.is_Mul:
L, R = f.as_two_terms()
if not L.has(i):
sR = eval_sum_symbolic(R, (i, a, b))
if sR:
return L*sR
if not R.has(i):
sL = eval_sum_symbolic(L, (i, a, b))
if sL:
return R*sL
try:
f = apart(f, i) # see if it becomes an Add
except PolynomialError:
pass
if f.is_Add:
L, R = f.as_two_terms()
lrsum = telescopic(L, R, (i, a, b))
if lrsum:
return lrsum
lsum = eval_sum_symbolic(L, (i, a, b))
rsum = eval_sum_symbolic(R, (i, a, b))
if None not in (lsum, rsum):
r = lsum + rsum
if not r is S.NaN:
return r
# Polynomial terms with Faulhaber's formula
n = Wild('n')
result = f.match(i**n)
if result is not None:
n = result[n]
if n.is_Integer:
if n >= 0:
if (b is S.Infinity and not a is S.NegativeInfinity) or \
(a is S.NegativeInfinity and not b is S.Infinity):
return S.Infinity
return ((bernoulli(n + 1, b + 1) - bernoulli(n + 1, a))/(n + 1)).expand()
elif a.is_Integer and a >= 1:
if n == -1:
return harmonic(b) - harmonic(a - 1)
else:
return harmonic(b, abs(n)) - harmonic(a - 1, abs(n))
if not (a.has(S.Infinity, S.NegativeInfinity) or
b.has(S.Infinity, S.NegativeInfinity)):
# Geometric terms
c1 = Wild('c1', exclude=[i])
c2 = Wild('c2', exclude=[i])
c3 = Wild('c3', exclude=[i])
wexp = Wild('wexp')
# Here we first attempt powsimp on f for easier matching with the
# exponential pattern, and attempt expansion on the exponent for easier
# matching with the linear pattern.
e = f.powsimp().match(c1 ** wexp)
if e is not None:
e_exp = e.pop(wexp).expand().match(c2*i + c3)
if e_exp is not None:
e.update(e_exp)
if e is not None:
p = (c1**c3).subs(e)
q = (c1**c2).subs(e)
r = p*(q**a - q**(b + 1))/(1 - q)
l = p*(b - a + 1)
return Piecewise((l, Eq(q, S.One)), (r, True))
r = gosper_sum(f, (i, a, b))
if isinstance(r, (Mul,Add)):
from sympy import ordered, Tuple
non_limit = r.free_symbols - Tuple(*limits[1:]).free_symbols
den = denom(together(r))
den_sym = non_limit & den.free_symbols
args = []
for v in ordered(den_sym):
try:
s = solve(den, v)
m = Eq(v, s[0]) if s else S.false
if m != False:
args.append((Sum(f_orig.subs(*m.args), limits).doit(), m))
break
except NotImplementedError:
continue
args.append((r, True))
return Piecewise(*args)
if not r in (None, S.NaN):
return r
h = eval_sum_hyper(f_orig, (i, a, b))
if h is not None:
return h
factored = f_orig.factor()
if factored != f_orig:
return eval_sum_symbolic(factored, (i, a, b))
from sympy import S from sympy import * from sympy.concrete.gosper import gosper_sum from sympy.functions import factorial from sympy.abc import n, z f1 = 3**n * z**(-n) F1_z = gosper_sum(f1, (n, 0, S.Infinity)) f2 = cos((pi / 2)**n) F2_z = gosper_sum(f1, (n, 0, S.Infinity)) print(F1_z) print(F2_z)
