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 Rational sums of inverse powers of fixed points of e bo198214 Administrator Posts: 1,386 Threads: 90 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

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