11/20/2007, 08:43 PM
Just at a casual glance, I'm having trouble understanding the relationship. I mean, besides the obvious use of the natural logarithm.
For example, my series is related to the power series of the sum of the logarithms at the fixed points, using as the base for each logarithm the associated fixed point itself. I would have been less surprised by something of the form ln(1-exp(x)/x), as but an example. In that case, we'd have singularities everywhere that exp(x)=x, which is of course at the fixed points.
Changing things around, ln(1-exp(x)/x) is the same thing as ln(1+exp(-1/w)*w), where w=-1/x. But this still gets me no closer to the form used in the Sloane series.
For example, my series is related to the power series of the sum of the logarithms at the fixed points, using as the base for each logarithm the associated fixed point itself. I would have been less surprised by something of the form ln(1-exp(x)/x), as but an example. In that case, we'd have singularities everywhere that exp(x)=x, which is of course at the fixed points.
Changing things around, ln(1-exp(x)/x) is the same thing as ln(1+exp(-1/w)*w), where w=-1/x. But this still gets me no closer to the form used in the Sloane series.
~ Jay Daniel Fox
