1"""Tests for Gosper's algorithm for hypergeometric summation. """ 2 3from sympy import binomial, factorial, gamma, Poly, S, simplify, sqrt, exp, \ 4 log, Symbol, pi, Rational 5from sympy.abc import a, b, j, k, m, n, r, x 6from sympy.concrete.gosper import gosper_normal, gosper_sum, gosper_term 7 8 9def test_gosper_normal(): 10 eq = 4*n + 5, 2*(4*n + 1)*(2*n + 3), n 11 assert gosper_normal(*eq) == \ 12 (Poly(Rational(1, 4), n), Poly(n + Rational(3, 2)), Poly(n + Rational(1, 4))) 13 assert gosper_normal(*eq, polys=False) == \ 14 (Rational(1, 4), n + Rational(3, 2), n + Rational(1, 4)) 15 16 17def test_gosper_term(): 18 assert gosper_term((4*k + 1)*factorial( 19 k)/factorial(2*k + 1), k) == (-k - S.Half)/(k + Rational(1, 4)) 20 21 22def test_gosper_sum(): 23 assert gosper_sum(1, (k, 0, n)) == 1 + n 24 assert gosper_sum(k, (k, 0, n)) == n*(1 + n)/2 25 assert gosper_sum(k**2, (k, 0, n)) == n*(1 + n)*(1 + 2*n)/6 26 assert gosper_sum(k**3, (k, 0, n)) == n**2*(1 + n)**2/4 27 28 assert gosper_sum(2**k, (k, 0, n)) == 2*2**n - 1 29 30 assert gosper_sum(factorial(k), (k, 0, n)) is None 31 assert gosper_sum(binomial(n, k), (k, 0, n)) is None 32 33 assert gosper_sum(factorial(k)/k**2, (k, 0, n)) is None 34 assert gosper_sum((k - 3)*factorial(k), (k, 0, n)) is None 35 36 assert gosper_sum(k*factorial(k), k) == factorial(k) 37 assert gosper_sum( 38 k*factorial(k), (k, 0, n)) == n*factorial(n) + factorial(n) - 1 39 40 assert gosper_sum((-1)**k*binomial(n, k), (k, 0, n)) == 0 41 assert gosper_sum(( 42 -1)**k*binomial(n, k), (k, 0, m)) == -(-1)**m*(m - n)*binomial(n, m)/n 43 44 assert gosper_sum((4*k + 1)*factorial(k)/factorial(2*k + 1), (k, 0, n)) == \ 45 (2*factorial(2*n + 1) - factorial(n))/factorial(2*n + 1) 46 47 # issue 6033: 48 assert gosper_sum( 49 n*(n + a + b)*a**n*b**n/(factorial(n + a)*factorial(n + b)), \ 50 (n, 0, m)).simplify() == -exp(m*log(a) + m*log(b))*gamma(a + 1) \ 51 *gamma(b + 1)/(gamma(a)*gamma(b)*gamma(a + m + 1)*gamma(b + m + 1)) \ 52 + 1/(gamma(a)*gamma(b)) 53 54 55def test_gosper_sum_indefinite(): 56 assert gosper_sum(k, k) == k*(k - 1)/2 57 assert gosper_sum(k**2, k) == k*(k - 1)*(2*k - 1)/6 58 59 assert gosper_sum(1/(k*(k + 1)), k) == -1/k 60 assert gosper_sum(-(27*k**4 + 158*k**3 + 430*k**2 + 678*k + 445)*gamma(2*k 61 + 4)/(3*(3*k + 7)*gamma(3*k + 6)), k) == \ 62 (3*k + 5)*(k**2 + 2*k + 5)*gamma(2*k + 4)/gamma(3*k + 6) 63 64 65def test_gosper_sum_parametric(): 66 assert gosper_sum(binomial(S.Half, m - j + 1)*binomial(S.Half, m + j), (j, 1, n)) == \ 67 n*(1 + m - n)*(-1 + 2*m + 2*n)*binomial(S.Half, 1 + m - n)* \ 68 binomial(S.Half, m + n)/(m*(1 + 2*m)) 69 70 71def test_gosper_sum_algebraic(): 72 assert gosper_sum( 73 n**2 + sqrt(2), (n, 0, m)) == (m + 1)*(2*m**2 + m + 6*sqrt(2))/6 74 75 76def test_gosper_sum_iterated(): 77 f1 = binomial(2*k, k)/4**k 78 f2 = (1 + 2*n)*binomial(2*n, n)/4**n 79 f3 = (1 + 2*n)*(3 + 2*n)*binomial(2*n, n)/(3*4**n) 80 f4 = (1 + 2*n)*(3 + 2*n)*(5 + 2*n)*binomial(2*n, n)/(15*4**n) 81 f5 = (1 + 2*n)*(3 + 2*n)*(5 + 2*n)*(7 + 2*n)*binomial(2*n, n)/(105*4**n) 82 83 assert gosper_sum(f1, (k, 0, n)) == f2 84 assert gosper_sum(f2, (n, 0, n)) == f3 85 assert gosper_sum(f3, (n, 0, n)) == f4 86 assert gosper_sum(f4, (n, 0, n)) == f5 87 88# the AeqB tests test expressions given in 89# www.math.upenn.edu/~wilf/AeqB.pdf 90 91 92def test_gosper_sum_AeqB_part1(): 93 f1a = n**4 94 f1b = n**3*2**n 95 f1c = 1/(n**2 + sqrt(5)*n - 1) 96 f1d = n**4*4**n/binomial(2*n, n) 97 f1e = factorial(3*n)/(factorial(n)*factorial(n + 1)*factorial(n + 2)*27**n) 98 f1f = binomial(2*n, n)**2/((n + 1)*4**(2*n)) 99 f1g = (4*n - 1)*binomial(2*n, n)**2/((2*n - 1)**2*4**(2*n)) 100 f1h = n*factorial(n - S.Half)**2/factorial(n + 1)**2 101 102 g1a = m*(m + 1)*(2*m + 1)*(3*m**2 + 3*m - 1)/30 103 g1b = 26 + 2**(m + 1)*(m**3 - 3*m**2 + 9*m - 13) 104 g1c = (m + 1)*(m*(m**2 - 7*m + 3)*sqrt(5) - ( 105 3*m**3 - 7*m**2 + 19*m - 6))/(2*m**3*sqrt(5) + m**4 + 5*m**2 - 1)/6 106 g1d = Rational(-2, 231) + 2*4**m*(m + 1)*(63*m**4 + 112*m**3 + 18*m**2 - 107 22*m + 3)/(693*binomial(2*m, m)) 108 g1e = Rational(-9, 2) + (81*m**2 + 261*m + 200)*factorial( 109 3*m + 2)/(40*27**m*factorial(m)*factorial(m + 1)*factorial(m + 2)) 110 g1f = (2*m + 1)**2*binomial(2*m, m)**2/(4**(2*m)*(m + 1)) 111 g1g = -binomial(2*m, m)**2/4**(2*m) 112 g1h = 4*pi -(2*m + 1)**2*(3*m + 4)*factorial(m - S.Half)**2/factorial(m + 1)**2 113 114 g = gosper_sum(f1a, (n, 0, m)) 115 assert g is not None and simplify(g - g1a) == 0 116 g = gosper_sum(f1b, (n, 0, m)) 117 assert g is not None and simplify(g - g1b) == 0 118 g = gosper_sum(f1c, (n, 0, m)) 119 assert g is not None and simplify(g - g1c) == 0 120 g = gosper_sum(f1d, (n, 0, m)) 121 assert g is not None and simplify(g - g1d) == 0 122 g = gosper_sum(f1e, (n, 0, m)) 123 assert g is not None and simplify(g - g1e) == 0 124 g = gosper_sum(f1f, (n, 0, m)) 125 assert g is not None and simplify(g - g1f) == 0 126 g = gosper_sum(f1g, (n, 0, m)) 127 assert g is not None and simplify(g - g1g) == 0 128 g = gosper_sum(f1h, (n, 0, m)) 129 # need to call rewrite(gamma) here because we have terms involving 130 # factorial(1/2) 131 assert g is not None and simplify(g - g1h).rewrite(gamma) == 0 132 133 134def test_gosper_sum_AeqB_part2(): 135 f2a = n**2*a**n 136 f2b = (n - r/2)*binomial(r, n) 137 f2c = factorial(n - 1)**2/(factorial(n - x)*factorial(n + x)) 138 139 g2a = -a*(a + 1)/(a - 1)**3 + a**( 140 m + 1)*(a**2*m**2 - 2*a*m**2 + m**2 - 2*a*m + 2*m + a + 1)/(a - 1)**3 141 g2b = (m - r)*binomial(r, m)/2 142 ff = factorial(1 - x)*factorial(1 + x) 143 g2c = 1/ff*( 144 1 - 1/x**2) + factorial(m)**2/(x**2*factorial(m - x)*factorial(m + x)) 145 146 g = gosper_sum(f2a, (n, 0, m)) 147 assert g is not None and simplify(g - g2a) == 0 148 g = gosper_sum(f2b, (n, 0, m)) 149 assert g is not None and simplify(g - g2b) == 0 150 g = gosper_sum(f2c, (n, 1, m)) 151 assert g is not None and simplify(g - g2c) == 0 152 153 154def test_gosper_nan(): 155 a = Symbol('a', positive=True) 156 b = Symbol('b', positive=True) 157 n = Symbol('n', integer=True) 158 m = Symbol('m', integer=True) 159 f2d = n*(n + a + b)*a**n*b**n/(factorial(n + a)*factorial(n + b)) 160 g2d = 1/(factorial(a - 1)*factorial( 161 b - 1)) - a**(m + 1)*b**(m + 1)/(factorial(a + m)*factorial(b + m)) 162 g = gosper_sum(f2d, (n, 0, m)) 163 assert simplify(g - g2d) == 0 164 165 166def test_gosper_sum_AeqB_part3(): 167 f3a = 1/n**4 168 f3b = (6*n + 3)/(4*n**4 + 8*n**3 + 8*n**2 + 4*n + 3) 169 f3c = 2**n*(n**2 - 2*n - 1)/(n**2*(n + 1)**2) 170 f3d = n**2*4**n/((n + 1)*(n + 2)) 171 f3e = 2**n/(n + 1) 172 f3f = 4*(n - 1)*(n**2 - 2*n - 1)/(n**2*(n + 1)**2*(n - 2)**2*(n - 3)**2) 173 f3g = (n**4 - 14*n**2 - 24*n - 9)*2**n/(n**2*(n + 1)**2*(n + 2)**2* 174 (n + 3)**2) 175 176 # g3a -> no closed form 177 g3b = m*(m + 2)/(2*m**2 + 4*m + 3) 178 g3c = 2**m/m**2 - 2 179 g3d = Rational(2, 3) + 4**(m + 1)*(m - 1)/(m + 2)/3 180 # g3e -> no closed form 181 g3f = -(Rational(-1, 16) + 1/((m - 2)**2*(m + 1)**2)) # the AeqB key is wrong 182 g3g = Rational(-2, 9) + 2**(m + 1)/((m + 1)**2*(m + 3)**2) 183 184 g = gosper_sum(f3a, (n, 1, m)) 185 assert g is None 186 g = gosper_sum(f3b, (n, 1, m)) 187 assert g is not None and simplify(g - g3b) == 0 188 g = gosper_sum(f3c, (n, 1, m - 1)) 189 assert g is not None and simplify(g - g3c) == 0 190 g = gosper_sum(f3d, (n, 1, m)) 191 assert g is not None and simplify(g - g3d) == 0 192 g = gosper_sum(f3e, (n, 0, m - 1)) 193 assert g is None 194 g = gosper_sum(f3f, (n, 4, m)) 195 assert g is not None and simplify(g - g3f) == 0 196 g = gosper_sum(f3g, (n, 1, m)) 197 assert g is not None and simplify(g - g3g) == 0 198