• DOI:10.1080/10652460500422247
• Corpus ID: 120431576

Ultra power and ultra exponential functions

@article{Hooshmand2006UltraPA,
  title={Ultra power and ultra exponential functions},
  author={M. H. Hooshmand},
  journal={Integral Transforms and Special Functions},
  year={2006},
  volume={17},
  pages={549 - 558}
}

• M. Hooshmand
• Published 1 August 2006
• Mathematics
• Integral Transforms and Special Functions

Supposing that a is a positive real number, then for each natural number n, the notation a n is called a to the ultra power of n , and we define by In the other words, , n times. In [Euler, L., De formulis exponentialibus replicatus, Acta Academiac Petropolitenae, 1, 38–60.] the necessary and sufficient condition for the convergence of lim n→∞ a n is proved by Euler and in [Baker, I.N. and Rippon, P.J., 1983, Convergence of infinite exponentials. Annales Academiac Scentiarium Fennicae Mathematika Series AI., 8, 179–186] it is studied for the case a∈ℂ. Also in Macdonnell, J. [1989, Some critical points on the hyperpower function n x = x x⋮ International Journal of Mathematical Education in Science and Technology, 20(2), 297–305.], by supposing a = x be variable, some property of the function f(x)=lim n→∞ x n is introduced. But, in this paper, we want to introduce and discuss a different topic, considering the following explanation: we can consider the serial operations as production, power and ultra power respectively. As we know, due to algebraic properties of power, it can be extended from natural to rational numbers (and then other numbers) but this cannot be done for ultra power, because it does not have any useful algebraic properties except . Perhaps this is the reason that ultra power is not extended so far. For removing this problem, we first introduce the following functional equation and then study some of its properties (If f be a function satisfying the equation, then f(n)=a n for all natural n). We state and prove a uniqueness theorem about the above functional equation and extend the definition of ultra power by it. Also, we introduce the new functions uxp a (ultra exponential functions) and study their properties.