The hyperoperation $a[n]b$ is addition when $n=1$, multiplication when $n=2$, exponentiation when $n=3$, tetration when $n=4$, and so on.

What happens when $n$ is noninteger?

Can we evaluate, e.g. $a[2.5]b$, $a[\pi]$b$, etc?

How about $\frac{\mathrm{d}}{\mathrm{d}x}a[2.5]x$, $\frac{\mathrm{d}}{\mathrm{d}x}x[2.5]b$, or $\frac{\mathrm{d}}{\mathrm{d}x}a[x]b$? Integrals?

Is this a meaningful concept?

  • $\begingroup$ The question is "can we assign some values to these operations which would work for any $a$ and $b$ in a consistent way and reduce to the usual operations for integer order". There may be several ways to do so (or no way, which is an interesting theorem to prove) $\endgroup$ – Yuriy S Nov 17 '18 at 12:06
  • $\begingroup$ There is an answer of mine giving one example see math.stackexchange.com/a/1272791/1714 . Note, this is not "the only" way to do that but it was focused to the question in the sense of binary operations $\endgroup$ – Gottfried Helms Nov 18 '18 at 23:22

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