What's the relation between these two definition of Ackermann function?

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The definition 1 is from Introduction to Algorithm and the second is from wikipedia. [dd](!https://www.latexlive.com/#QV9rKGopJTIwPSUyMCU1Q2xlZnQlNUMlN0IlMjAlMEElNUNiZWdpbiU3QmFsaWduZWQlN0QlMEEmaisxJTIwJTIwJTVDcXVhZCUyMCZrPTAlNUMlNUMlMEEmQV8lN0JrLTElN0QlNUUlN0IoaisxKSU3RChqKSUyMCZrJTIwJTVDZ2UlMjAxJTIwJTBBJTVDZW5kJTdCYWxpZ25lZCU3RCUwQSU1Q3JpZ2h0Lg==)

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The second form is equivalent to formular enter image description here

if we let the first definiction be A and the second one be \bar{A}, then we have

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Proof:

  • first

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since enter image description here

  • second

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then

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so

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This is the only relationshift I can find.