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Can we prove these coefficients must be constant?
I have boiled down the recursion of analytic hyper operators into a formula based on their coefficients. If we write these coefficients as follows:

We can write the recursive formula; without giving a proof for it (it just requires a few series rearrangement); as:

As you can see; this appears very off. can vary freely and the result on the L.H.S. doesn't change at all. However, it's being summed across an infinite series so that may compensate. But I wonder if declaring, that since takes on every value in between and ; at least; we can say over that interval

Since we can set this implies a strict contradiction:

This is a contradiction because it implies is constant and therefore constant for all b in . This would imply there is no analytic continuation of hyper operators! At least, not representable by its Taylor series.

I didn't write out the proof because I'm stuck and I'm curious if it's justifiable to do that last move, or if there is some other routine I can go about to prove the constancy of these coefficients.

If hyper operators aren't analytic; and hopefully I can prove not continuous; I have a separate way of defining them that admit a discrete solution with a more number theoretical algebraic approach.

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