totally monotonic

[Gottfried]

This may be slightly irrelevent, but there was a large thread on here, I want to say 7 years ago about totally monotonic solutions, and whether that was a uniqueness criterion. It evolved into a thread on MO, in which a couple of things were revealed.
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I forget because it was so long ago, but I believe Sheldon and I and the original poster agreed on the answer. I cannot remember his name for the life of me, he used to be on here occasionally. It was a long russian name, I want to say Reshetnikov, or something close to that. Damn my memory. I'll try to find the answer. But I believe if you're willing to put the work in this answer was affirmed in the positive.

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I think it has been this thread (Jan. 2017) and his author's name is Vladimir Reshetnikov

The basic idea was around to take "q-binomials" as basis for the representation of the tetration as series. Some of the involved members of this forum have deleted their accounts, which measure unfortunately anonymized their contributions; even more V.Reshetnikov reduced his presence in MO shortly after that with no more answers and only few questions, and so this thread had no "children".

Funny thing...

Gottfried

p.s. perhaps it's worth to transfer the contents of the thread to the forum here in some format? I think it's well written material...