01/14/2017, 09:09 PM
(This post was last modified: 01/15/2017, 11:38 PM by Vladimir Reshetnikov.)
By the way, in the well-known representation of the tetration as an exponential series
the coefficients have a q-binomial representation
\( {^z a} = \sum_{m=0}^{\infty} \; c_m q^{mz}, \)
the coefficients have a q-binomial representation
\( c_m=\sum_{n=m}^\infty \sum_{k=0}^{n}(-1)^{m+n+k} \; q^{\binom{n-k}{2} + \binom{m}{2}-m(n-1)}\;\frac{\binom{n}{m}_q\;\binom{n}{k}_q}{(q; \; q)_n}\;({^k a}) \)
The coefficients also satisfy the recurrence
\( c_m=\frac{\log(a)}{m\left(1-q^{1-m}\right)}\sum_{k=1}^{m-1}kq^{-k}c_{k}c_{m-k},\;\;m>1 \)