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| """Tests for Gosper's algorithm for hypergeometric summation. """ | |
| from sympy.core.numbers import (Rational, pi) | |
| from sympy.core.singleton import S | |
| from sympy.core.symbol import Symbol | |
| from sympy.functions.combinatorial.factorials import (binomial, factorial) | |
| from sympy.functions.elementary.exponential import (exp, log) | |
| from sympy.functions.elementary.miscellaneous import sqrt | |
| from sympy.functions.special.gamma_functions import gamma | |
| from sympy.polys.polytools import Poly | |
| from sympy.simplify.simplify import simplify | |
| from sympy.concrete.gosper import gosper_normal, gosper_sum, gosper_term | |
| from sympy.abc import a, b, j, k, m, n, r, x | |
| def test_gosper_normal(): | |
| eq = 4*n + 5, 2*(4*n + 1)*(2*n + 3), n | |
| assert gosper_normal(*eq) == \ | |
| (Poly(Rational(1, 4), n), Poly(n + Rational(3, 2)), Poly(n + Rational(1, 4))) | |
| assert gosper_normal(*eq, polys=False) == \ | |
| (Rational(1, 4), n + Rational(3, 2), n + Rational(1, 4)) | |
| def test_gosper_term(): | |
| assert gosper_term((4*k + 1)*factorial( | |
| k)/factorial(2*k + 1), k) == (-k - S.Half)/(k + Rational(1, 4)) | |
| 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_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_parametric(): | |
| assert gosper_sum(binomial(S.Half, m - j + 1)*binomial(S.Half, m + j), (j, 1, n)) == \ | |
| n*(1 + m - n)*(-1 + 2*m + 2*n)*binomial(S.Half, 1 + m - n)* \ | |
| binomial(S.Half, m + n)/(m*(1 + 2*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 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 | |
| # the AeqB tests test expressions given in | |
| # www.math.upenn.edu/~wilf/AeqB.pdf | |
| 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)) | |
| 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_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_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 = Rational(2, 3) + 4**(m + 1)*(m - 1)/(m + 2)/3 | |
| # g3e -> no closed form | |
| g3f = -(Rational(-1, 16) + 1/((m - 2)**2*(m + 1)**2)) # the AeqB key is wrong | |
| g3g = Rational(-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 | |