Journal of Integer Sequences, Vol. 28 (2025), Article 25.2.8

Exponential versus Factorial Revisited


Karol Gryszka
Institute of Mathematics
University of the National Education Commission
Podchorążych 2
30-084 Kraków
Poland

Abstract:

We study the relationship between iterated factorials, the iterated Euler gamma function, and iterated exponentiation. We demonstrate that both the iterated factorial and the iterated gamma function can be represented as a power tower, where all the terms are identical except for the top term. Additionally, we show that the top terms of this representation form a converging sequence. Finally, we prove that this representation can be reversed.


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(Concerned with sequences A000142 A014221.)


Received March 3 2025; revised versions received March 4 2025; April 3 2025. Published in Journal of Integer Sequences, April 5 2025.


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