An explicit series for the tetration of a complex height

[Vladimir Reshetnikov]

Secondly, do you have any thoughts on how to show convergence. This seems to be the only thing blocking the path.

Unfortunately, I do not yet have a proof of convergence. But the intuition that lead me to this series is as follows. The difference between a tetration value of a finite integer height \( ^n a \) and its limiting value (fixpoint) \( L={^\infty a} \) decays exponentially, where the base of the exponent (I named it q, but it is commonly denoted by λ) is the logarithm of the limiting value. Now we want to find an expression for tetration \( {^z a} \) of a complex height z. Exponential decay is not approximated well by polynomials, so we need to switch from z to a new variable \( w=q^z \). After this, the values of the discrete tetration all lie almost on a straight line. It seemed plausible that they could be interpolated by Lagrange interpolating polynomials without causing Runge's phenomenon (erratic oscillations between sample points). Because with the new variable w sample points are not equally spaced, but rather form a geometric progression, the products in numerators and denominators of terms in the Lagrange interpolating polynomial turn into q-Pochhammer symbols and then can be combined into q-binomial coefficients (so that's the moment when q comes into play). The series I proposed is just the limit of the Lagrange interpolating polynomial when number of sample points tends to infinity, and its partial sums are the Lagrange interpolating polynomials built on a finite number of points. By construction, the series converges and exactly reproduces sample values at integer points. Convergence at non-integer points (that I can confirm numerically, but cannot prove) is basically equivalent to not having Runge's phenomenon during polynomial interpolation. Obeying the functional equation for tetration at all points, including non-integer, does not immediately follow from the construction, but was initially just a nice conjecture supported by numerical computations.