Monday, 24 March 2014

Possible extension of the hyperations to real numbers

A hyperation is a hyperoperation from the hyperoperation sequence. The fisrt hyperation is addition, the second one multiplication and the third one exponentation. All the hyperations after multiplication can be denoted using knuth's up arrow notation. The third hyperation on a and b, sometimes denoted as ab can be denoted a ↑ b, the fourth one, tetration, sometimes denoted as ba, can be denoted a ↑↑ b. In fact for all integers x > 2, the xth hyperation is denoted as a then x - 2 up arrows then b or a ↑x - 2 b.

The hyperations after exponentation acting on pairs of numbers where the first number is a positive integer and the second one is a nonnegative integer are defined as follows: a ↑x b = 1 and a ↑x + 1 (b + 1) = a ↑x (a ↑x + 1 b). For any real number a > 0, just like for positive integers, we also define tetration to the base a and integral exponent to satisfy the 2 formulae:
  • a ↑↑ 0 = 1
  • a ↑↑ (b + 1) = a ↑ (a ↑↑ b)
so a ↑↑ -1 = 0 and a ↑↑ b is not defined for integers b < -1. When a < 1, a ↑↑ b is an alternating sequence with respect to b so a ↑↑ b is only defined when b is an integer. When a > 1, a ↑↑ b for integer exponents be is an increasing function with respect to b and there exists a real number c > 1 such that for all a, a ↑↑ b has an upper limit iff 1 ≤ a ≤ c. When 1 < a < c, we define the function of b, a ↑↑ b, to be the unique function that satisfies the following properties:
  • a ↑↑ 0 = 1
  • For all real numbers b, if a ↑↑ b exists, then a ↑↑ (b + 1) = a ↑ (a ↑↑ b)
  • a ↑↑ b approaches the upper limit exponentially.
  • There exists no other function satisfying the previous properties that has a number not in the domain of this function in its domain.
For any b such that there exists a number 1 < a < c such that a ↑↑ b exists, we define the function of a, a ↑↑ b to be the analytic continuation a ↑↑ b defined for 1 < a < c. When a = 1, a ↑↑ b is defined only for b > -1 and is defined to equal 1. For any a > 1, a ↑↑ b exists iff b > -2. Pentation is similarly defined in terms of tetration as follows:
  • a ↑↑↑ b is only defined for a ≥ 1
  • When a = 1, a ↑↑↑ b is defined for b > -1 and is equal to 1
  • There exists a real number d which is the largest real number such that d ↑↑↑ b has an upper limit for integer exponents b. When 1 < a < d, a ↑↑↑ b is defined on R such that a ↑↑↑ (b + 1) = a ↑↑ (a ↑↑↑ b). and a ↑↑↑ b approaches a constant exponentally as b approaches ∞.
  • When a ≥ d, a ↑↑↑ b is defined to be the analytic continuation of a ↑↑↑ b defined for 1 < a < d with respect to a.
In general, for all hyperations starting from exponentation, for any x, when a = 1, the x-ation a ↑x - 2 b is not defined for a < 1, when a = 1, it's defined for b > -1 and is equal to 1. There exists a largest real number r such that for integers b, a ↑x - 2 b has an upper limit. When 1 < a < r and x > 3, a ↑x - 2 b is defined such that a ↑x - 2 (b + 1) = a ↑x - 3 (a ↑x - 2 b) if a ↑x - 2 b exists and a ↑x - 2 b approaches a constant exponentially. When a ≥ r, a ↑x - 2 b is defined to be the analytic continuation with respect to a of a ↑x - 2 b defined for 1 < a < r. It turns out that for any x-ation where x ≥ 3, for any a > 1, a ↑x - 2 b is defined for all real numbers b with a lower limit on a ↑x - 2 b if x is odd and there exists a real number above which b must be in order for a ↑x - 2 b to be defined with no lower limit on a ↑x - 2 b if x is even.