 Can we prove these coefficients must be constant? - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Can we prove these coefficients must be constant? (/showthread.php?tid=739) Can we prove these coefficients must be constant? - JmsNxn - 06/03/2012 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.