03/28/2010, 03:25 AM
Hi.
I'm curious: Is there any ideas about how a closed form or even a series formula, recurrence, etc. could be found for the Taylor coefficients of the tetrational function at 0 developed from, e.g. the Abel iteration, or the Cauchy integral (which seem to give the same thing though it's not proven), to base e?
I.e.
\( \mathrm{tet}(z) = \sum_{n=0}^{\infty} a_n z^n \)
and what is \( a_n \)? We have \( a_0 = 1 \) and then \( a_1 = 1.0917... \), etc. but is there any way that coefficient \( a_1 \) for example could be expressed in terms of elementary or "well-known" (used by the general maths. community, that is -- e.g. gamma, polylog, zeta, erf, etc.) non-elementary special functions and known mathematical constants (like e, pi, eulergamma, etc.)? Even if it can only be done with an infinite expansion like an infinite sum or something involving those? Or is this function simply so darned exotic that it will utterly refuse and defy any and all attempts to try to relate it to known mathematical functions?
I'm curious: Is there any ideas about how a closed form or even a series formula, recurrence, etc. could be found for the Taylor coefficients of the tetrational function at 0 developed from, e.g. the Abel iteration, or the Cauchy integral (which seem to give the same thing though it's not proven), to base e?
I.e.
\( \mathrm{tet}(z) = \sum_{n=0}^{\infty} a_n z^n \)
and what is \( a_n \)? We have \( a_0 = 1 \) and then \( a_1 = 1.0917... \), etc. but is there any way that coefficient \( a_1 \) for example could be expressed in terms of elementary or "well-known" (used by the general maths. community, that is -- e.g. gamma, polylog, zeta, erf, etc.) non-elementary special functions and known mathematical constants (like e, pi, eulergamma, etc.)? Even if it can only be done with an infinite expansion like an infinite sum or something involving those? Or is this function simply so darned exotic that it will utterly refuse and defy any and all attempts to try to relate it to known mathematical functions?