Can e^{e^{e^{x}}} Be Simplified or Related to Hypergeometric Functions?

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Discussion Overview

The discussion revolves around the potential simplification or relation of the expression e^{e^{e^{x}}} to hypergeometric functions, as well as the simpler case of e^{e^{x}}. Participants explore whether these expressions can be clarified or simplified in any meaningful way.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants inquire about the possibility of simplifying e^{e^{e^{x}}} and its relation to hypergeometric functions.
  • One participant asserts that the expressions do not simplify, assuming right-to-left association.
  • Another participant points out the ambiguity in the expression without parentheses and confirms that e^{e^{e^{x}}} does not simplify.
  • A later reply suggests a series expansion for e^{e^{e^{x}}} in the form of a power series, indicating a potential method to express it.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether the expressions can be simplified or related to hypergeometric functions. There are competing views on the nature of the expressions and their simplification.

Contextual Notes

The discussion highlights the ambiguity in notation and the assumptions regarding the order of operations, which may affect interpretations of the expressions.

PlasticOh-No
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Is there any way to simplify or clarify e^e^e^x ?
Some sort of equivalence to a hypergeometric?
How about e^e^x?
Thanks
 
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Assuming that these associate the usual way (right-to-left), they don't really simplify.
 


What you have written, without parentheses,is ambiguous. That is why CRGreathouse said "Assuming that these associate the usual way (right-to-left)".

e^{e^{e^x}}
does not simplify.

((e^e)^e)^x= (e^e)^{ex}= e^{e^2x}
 


PlasticOh-No said:
Is there any way to simplify or clarify e^e^e^x ?
Some sort of equivalence to a hypergeometric?
How about e^e^x?
Thanks

Assume it's the way I think you want it to be:

e^{e^{e^{x}}}=\sum_{n=0}^{\infty} a_n x^n

and you know how to figure out what each a_n is.
 


Thanks everyone for your help. Yes, I meant

e^{e^{e^{x}}}
 
Last edited:

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