• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Rational sums of inverse powers of fixed points of e bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 11/23/2007, 08:22 AM This is indeed an interesting connection. Now my quick 2 cents about it. First, we want the explicit function $f$ of which the sums of the powers of the inverted fixed points are the coefficients. We compare Jay's beginnin with index 1 in the first row with Sloane's beginning at index 1 in the second row: $\begin{pmatrix} -1 & -1 & 2 & 9 & -6 & -155 & -232 & 3969 & 20870 & -118779 & -1655028\\ 1 & 1 & -1 & -2 & 9 & 6 & -155 & 232 & 3969 & -20870 & -118779 & 1655028\end{pmatrix}$ Obviously we have to move the lower row to the left which is the same as dividing Sloane's function $S(z)=\ln\left(1+ze^z\right)$ by $z$. We get then $\begin{pmatrix} -1 & -1 & 2 & 9 & -6 & -155 & -232 & 3969 & 20870 & -118779 & -1655028\\ 1 & -1 & -2 & 9 & 6 & -155 & 232 & 3969 & -20870 & -118779 & 1655028\end{pmatrix}$ and see that the sign is swapped for each uneven power, which can be achieved by using $-z$ instead of $z$. So we get $f(z)=-\frac{1}{z} \ln\left(1-\frac{z}{e^z}\right)$ with $f_n = \sum_{k=0} \left(\frac{1}{\overline{c_k}^n}+\frac{1}{c_k^n}\right)$. $f(z)=\sum_{n=1}^\infty z^n\sum_{k=0}^\infty \left(\frac{1}{\overline{c_k}^n}+\frac{1}{c_k^n}\right) =\sum_{k=0}^\infty \sum_{n=1}^\infty \left(\frac{z}{\overline{c_k}}\right)^n+\left(\frac{z}{c_k}\right)^n =\sum_{k=0}^\infty \frac{\frac{z}{\overline{c_k}}}{1-\frac{z}{\overline{c_k}}}+ \frac{ \frac{z}{c_k} }{ 1-\frac{z}{c_k}}=\sum_{k=0}^\infty \frac{z}{\overline{c_k}-z}+\frac{z}{c_k-z}$ for $|z|<|c_k|$. If we transform this further via $e^{-zf(z)}=1-z/e^z$ we get $\prod_{k=0}^\infty e^{\frac{z^2}{z-\overline{c_k}}} e^{ \frac{z^2}{z-c_k}}=1-z/e^z$ to prove. Looks strange, perhaps I made an error somewhere. « Next Oldest | Next Newest »

 Messages In This Thread Rational sums of inverse powers of fixed points of e - by jaydfox - 11/20/2007, 07:55 PM RE: Rational sums of inverse powers of fixed points of e - by jaydfox - 11/20/2007, 08:11 PM RE: Rational sums of inverse powers of fixed points of e - by Gottfried - 11/20/2007, 08:14 PM RE: Rational sums of inverse powers of fixed points of e - by jaydfox - 11/20/2007, 08:31 PM RE: Rational sums of inverse powers of fixed points of e - by jaydfox - 11/20/2007, 08:43 PM RE: Rational sums of inverse powers of fixed points of e - by jaydfox - 11/20/2007, 09:31 PM RE: Rational sums of inverse powers of fixed points of e - by jaydfox - 11/21/2007, 01:15 AM RE: Rational sums of inverse powers of fixed points of e - by jaydfox - 11/21/2007, 08:14 AM RE: Rational sums of inverse powers of fixed points of e - by Gottfried - 11/21/2007, 08:50 AM RE: Rational sums of inverse powers of fixed points of e - by jaydfox - 11/21/2007, 09:35 AM RE: Rational sums of inverse powers of fixed points of e - by Gottfried - 11/21/2007, 12:52 PM RE: Rational sums of inverse powers of fixed points of e - by jaydfox - 11/21/2007, 06:20 PM RE: Rational sums of inverse powers of fixed points of e - by Gottfried - 11/22/2007, 06:25 PM RE: Rational sums of inverse powers of fixed points of e - by jaydfox - 11/22/2007, 08:32 PM

 Possibly Related Threads... Thread Author Replies Views Last Post tetration from alternative fixed point sheldonison 22 52,299 12/24/2019, 06:26 AM Last Post: Daniel Thoughts on hyper-operations of rational but non-integer orders? VSO 2 3,738 09/09/2019, 10:38 PM Last Post: tommy1729 Inverse Iteration Xorter 3 6,655 02/05/2019, 09:58 AM Last Post: MrFrety Inverse super-composition Xorter 11 24,535 05/26/2018, 12:00 AM Last Post: Xorter Are tetrations fixed points analytic? JmsNxn 2 6,264 12/14/2016, 08:50 PM Last Post: JmsNxn the inverse ackerman functions JmsNxn 3 9,669 09/18/2016, 11:02 AM Last Post: Xorter Rational operators (a {t} b); a,b > e solved JmsNxn 30 68,093 09/02/2016, 02:11 AM Last Post: tommy1729 Removing the branch points in the base: a uniqueness condition? fivexthethird 0 3,046 03/19/2016, 10:44 AM Last Post: fivexthethird Inverse power tower functions tommy1729 0 3,362 01/04/2016, 12:03 PM Last Post: tommy1729 Derivative of exp^[1/2] at the fixed point? sheldonison 10 20,167 01/01/2016, 03:58 PM Last Post: sheldonison

Users browsing this thread: 1 Guest(s)