Timeline for Involutions on $[0,1]$ given by power series (related to probability generating functions)
Current License: CC BY-SA 3.0
3 events
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| Aug 24, 2015 at 14:48 | comment | added | James Martin | btw, also interested in the rates of decay of the coefficients. In 1. above, $|a_k|$ decays exponentially, while in 2. and 3., the decay is like $k^{-(1+1/n)}$, I believe. Which types of decay are possible? | |
| Aug 24, 2015 at 14:45 | comment | added | James Martin | OK, so it seems there are more solutions. For example if $F(x,y)$ is a symmetric function which is increasing in $x$ and $y$ with $F(1,0)=F(0,1)=0$, then setting $F(x,y)=0$ and writing $y$ in terms of $x$ gives an involution. Some of these have power series expansions, and some of those seem to have all coefficients except the first negative, as desired. For example, $y^2+y+x^2+x-2=0$ gives $y=[\sqrt{9-4x-4x^2}-1]/2$ which seems to work. Is there some systematic way to see when you get a power series expansion, and when that expansion has all coefficients except the first negative? | |
| Aug 24, 2015 at 1:25 | history | asked | James Martin | CC BY-SA 3.0 |