# Can we evaluate noninteger hyperoparations?

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? • 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) – Yuriy S Nov 17 '18 at 12:06
• 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 – Gottfried Helms Nov 18 '18 at 23:22