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  1. from mpmath import nsum, inf, nprod
  2. import math
  3. def fgh(iter, num):
  4.     if iter == 0:
  5.         return num + 1
  6.     elif iter == 1:
  7.         return num * 2
  8.     elif iter == 2:
  9.         return 2 ** num
  10.     else:
  11.         result = fgh(iter - 1, num)
  12.         for i in range(num - 1):
  13.             result = fgh(iter - 1, result)
  14.         return result
  15.  
  16. def sgh(iter, num):
  17.     if iter == 0:
  18.         return 0
  19.     else:
  20.         return sgh(iter - 1, num) + 1
  21.  
  22. def mgh(iter, num):
  23.     if iter == 0:
  24.         return num + 1
  25.     else:
  26.         return mgh(iter - 1, mgh(iter - 1, num))
  27.  
  28. def hh(iter, num):
  29.     if iter == 0:
  30.         return num
  31.     else:
  32.         return hh(iter - 1, num + 1)
  33.  
  34. def fact(num):
  35.     if num == 1:
  36.         return 1
  37.     else:
  38.         return num * fact(num - 1)
  39.  
  40. def pochhammer_symbol(num, iter):
  41.     if num == 1 or iter == 0:
  42.         return num
  43.     else:
  44.         result = num
  45.         for it in range(1, iter):
  46.             result *= it + num
  47.         return result
  48.  
  49. def qfact(k, q):
  50.     result = 1
  51.     chapter = 1
  52.     for iter in range(1, k):
  53.         chapter += q ** iter
  54.         result *= chapter
  55.     return result
  56.  
  57. def qseries(a, q, n):
  58.     result = 1 - a
  59.     for _ in range(1, n):
  60.         result *= 1 - a * (q ** _)
  61.     return result
  62.  
  63. def qps(a, q, k=inf):
  64.     if k > 0:
  65.         return nprod(lambda j: 1 - a * (q ** j), [0, k - 1])
  66.     elif k == 0:
  67.         return 1
  68.     elif k < 0:
  69.         return nprod(lambda j: (1 - a * (q ** -j)) ** -1, [0, math.abs(k - 1)])
  70. def qbc(n, k):
  71.     if 0 <= k and k < n:
  72.         return fact(n) / (fact(k) * fact(n - k))
  73.     else:
  74.         return 0
  75.  
  76. def qhypgeom(asl, bsl, q, z):
  77.     return nsum(lambda x: (nprod(lambda a: qps(asl[int(a)], q, x), [0, len(asl) - 1]) / nprod(lambda b: qps(bsl[int(b)], q, x), [0, len(bsl) - 1]) * ((z ** x) / qps(q, q, x)) * ((-1 ** x) * (q ** (x * (x - 1)/2))) ** (1 + len(bsl) - len(asl))), [0, inf])
  78.  
  79. def qe(q, x):
  80.     return nsum(lambda x: (z ** x) / qps(q, q, x), [0, inf])
  81. def qE(q, x):
  82.     return qps(-x, q, inf)
  83. def gamma(z):
  84.     #Euler's limit
  85.     return (1.0 / z) * nprod(lambda x: ((1 + (1.0 / x)) ** z) * ((1 + (z / x)) ** -1), [1, inf])
  86. def qgamma(q, z):
  87.     return (qps(q, q, inf) / qps(q ** z, q, inf)) * ((1 - q) ** (1 - x))
  88. def beta(q, a, b):
  89.     return (qgamma(q, a) * qgamma(q, b)) / qgamma(a + b)
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