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Ultra power of higher orders and ultra exponential functional sequences. (English) Zbl 1384.33009

Summary: The word tetration (was coined by Reuben Louis Goodstein) is the next hyper operator after exponentiation, and is defined as iterated exponentiation. The first uniqueness theorem for approximation of tetration was proved by the author [Integral Transforms Spec. Funct. 17, No. 8, 549–558 (2006; Zbl 1121.39023)]. As a consequence of the researches the ultra power \((a^{\underline{x}})\), ultra exponential \((\operatorname{uxp}_a(x))\) and infra logarithm \((\log_a(x))\) functions have been uniquely introduced and studied by many recent papers. They are unique solutions of the functional equations \(f(x)=a^{f(x-1)}\) and the Ablel functional equation \(f(a^x)=f(x)+1\), under some conditions. In this paper we want to uniquely introduce ultra exponential functional sequence, give all the next hyper operators and obtain the serial binary operations: multiplication, power, ultra power, ultra power of order 2, and so on. Finally, we study many important mathematical analysis properties of ultra powers and ultra exponential functional sequences.

MSC:

33B99 Elementary classical functions
33E30 Other functions coming from differential, difference and integral equations
39B22 Functional equations for real functions

Citations:

Zbl 1121.39023
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References:

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