Oh foo, I've been had. This is a different Komatsu paper, http://www.researchgate.net/publication/225486369_Hurwitz_continued_fraction... and the double mover effect is a trivial consequence of the regular CF scaling rule, where you alternately multiply and divide. --rwg On Sun, Sep 30, 2012 at 11:22 PM, Bill Gosper <billgosper@gmail.com> wrote:
BG> By uncontrived I meant not explicitly constructed as a continued fraction.
NeilB>Just my two cents:http://ttk.pte.hu/mii/html/pannonica/index_elemei/mp17-1/mp17-1-091-110.pdf claims that (1;2,4,8,16...) is equal to
1+Sum[1/((-1)^n*2^(1 + n)^2*QPochhammer[4, 4, n]), {n, 0, Infinity}]/ Sum[1/((-1)^n*2^n^2*QPochhammer[4, 4, n]), {n, 0, Infinity}]
(The source, Takao Komatsu's "Hurwitz and Tasoev Continued Fractions with Long Period" cites another paper, "On Hurwitzian and Tasoev's Continued Fractions", by the same author, but which I can't seem to find) --Neil Bickford
On Thu, Sep 27, 2012 at 10:33 AM, Allan Wechsler <acwacw@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=acwacw%40gmail.com>> wrote:>> If x is Hurwitz then so is x/2, so, uh ...>> I think that by "contrived" Gosper means "with cf terms explicitly chosen> to be non-periodic". An example would be (1;2,4,8,16...). I'm assuming> that there is not a single such example that anybody knows a "closed form"> for; the closed form thereof would answer Gosper's query.>> On Thu, Sep 27, 2012 at 12:54 PM, meekerdb <meekerdb@verizon.net <http://gosper.org/webmail/src/compose.php?send_to=meekerdb%40verizon.net>> wrote:>> > Is the smallest contrived Hurwitz, thereby uncontrived? :-)> >> > Brent
> On 9/27/2012 5:38 AM, Bill Gosper wrote:> >> >> Can someone name a single (uncontrived) constant (e.g., π, e^3, 2^(1/3),> >> parity number,...)> >> that is provably nonHurwitz? And if Hurwitz, is not homographically> >> equivalent to a single-> >> mover, linear?> >>> >> Boy, are we ignorant. --rwg
YOW! This paper has the general "single-mover" formula, and also has double movers! --rwg
Out[584]= 1 Tanh[-------] Sqrt[2] ------------- Sqrt[2]
In[585]:= hursigs2[%584, 99, 1]
During evaluation of In[585]:= 0.006964
Out[585]= {1, 0 1 }
1 0 2 {3 + 4 k, 10 + 8 k}
In[586]:= hursigs2[%584, 99, 2]
During evaluation of In[586]:= 0.035164
Out[586]= 0 2
{3, 1 1 8 {1, 1, 8 k, 1, 1, 1 + 4 k, 20 + 32 k, 3 + 4 k}}
0 2
1 0 8 {1, 1, 3 + 4 k, 36 + 32 k, 5 + 4 k, 1, 1, 12 + 8 k}
0 1
2 0 2 {1 + 4 k, 6 + 8 k}
In[587]:= hursigs2[%585, 99, 3]
During evaluation of In[587]:= 0.082445
Out[587]= 0 3
1 2 14 {1, 2, 8 k, 2, 1, 4 k, 2, 1, 2 + 8 k, 1, 2, 2 + 4 k, 54 + 72 k, 3 + 4 k} {4, }
0 3
1 1 14 {1, 2, 3 + 4 k, 2, 1, 8 + 8 k, 45 + 36 k, 11 + 8 k, 2, 1, 5 + 4 k, 1, 2, 13 + 8 k}
0 3
1 0 10 {1, 3 + 4 k, 78 + 72 k, 5 + 4 k, 102 + 72 k, 6 + 4 k, 2, 1, 13 + 8 k, 2}
0 1
3 0 10 {1, 2, 4 k, 1, 2, 3 + 8 k, 21 + 36 k, 6 + 8 k, 33 + 36 k, 8 + 8 k}
In[588]:= hursigs2[%585, 99, 4]
During evaluation of In[588]:= 0.087023
Out[588]= 0 4
{7, 1 3 8 {1, 3, 4 k, 3, 1, 2 k, 40 + 64 k, 1 + 2 k} }
0 4
1 2 12 {1, 1, 2 k, 3, 1, 1 + 4 k, 1, 3, 1 + 2 k, 1, 1, 17 + 16 k}
0 4
1 1 12 {1, 1, 1 + 2 k, 3, 1, 3 + 4 k, 1, 3, 2 + 2 k, 1, 1, 25 + 16 k}
0 4
1 0 8 {1, 3, 2 + 4 k, 3, 1, 1 + 2 k, 72 + 64 k, 2 + 2 k}
0 2
2 1 8 {1, 1, 4 k, 12 + 32 k, 2 + 4 k, 1, 1, 6 + 8 k}
0 2
2 0 2 {3 + 4 k, 10 + 8 k}
0 1
4 0 8 {1, 1, 2 + 4 k, 28 + 32 k, 4 + 4 k, 1, 1, 10 + 8 k}