Try it out with a very interesting example:
(Just press enter again when it looks like '....').
The flow matrix 'p' will give the coefficients of x and t in \( f^t(x) \).
The following will give the first diagonal of the coefficients of x and t in the continuous iteration of \( e^x - 1 \), which is known to be \( \frac{1}{2^k} \) as Jay Fox noticed here and we can ensure that it fits this pattern by dividing by it.
Now for the fun part! Since we defined the helper function 'c', we can do the same thing for the second diagonal, which up until now has been a mystery.
as you can see, the second diagonal of the power series of iterated \( e^x - 1 \) is related to harmonic numbers, see the implementation notes for more details. This means that the power series of iterated \( e^x - 1 \) is of the form:
\(
(e^x-1)^{\circ t}(x) = \text{dxp}^{\circ t}(x)
= \sum_{k=0}^{\infty} \frac{t^k x^{k+1}}{2^k}
+ \sum_{k=0}^{\infty} \frac{t^k x^{k+2}H^{(2)}_k}{-6\cdot2^k}
+ \cdots
\)
Note that this is not the normal generalized harmonic numbers, but the ones used by Conway & Guy, which are slightly different. These harmonic numbers are defined by:
\(
H^{(1)}_n = H_n = \sum_{i=1}^{n} \frac{1}{i}
\)
\(
H^{(k)}_n = \sum_{i=1}^{n} H^{(k-1)}_i
\)
The reason why I took the time to include such an advanced algorithm for computing harmonic numbers, is because I believe they hold the key to a new closed form, identity, or tautology for iterated-dxp.
Andrew Robbins
Code:
sage: from hyper.all import *
sage: t = var('t')
sage: p = parabolic_flow_matrix(exp(x) - 1, t, x, 0, 10)
sage: def c(k): return map(lambda x: x[-k], p[k:])
....:The flow matrix 'p' will give the coefficients of x and t in \( f^t(x) \).
The following will give the first diagonal of the coefficients of x and t in the continuous iteration of \( e^x - 1 \), which is known to be \( \frac{1}{2^k} \) as Jay Fox noticed here and we can ensure that it fits this pattern by dividing by it.
Code:
sage: [c(1)[k] for k in xrange(9)]
[1, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, 1/256]
sage: [c(1)[k]*2**k for k in xrange(9)]
[1, 1, 1, 1, 1, 1, 1, 1, 1]Now for the fun part! Since we defined the helper function 'c', we can do the same thing for the second diagonal, which up until now has been a mystery.
Code:
sage: [c(2)[k] for k in xrange(1,8)]
[-1/12, -5/48, -13/144, -77/1152, -29/640, -223/7680, -481/26880]
sage: [c(2)[k]*2**k*(-6)/harmonic_number(k, 2) for k in xrange(1,8)]
[1, 1, 1, 1, 1, 1, 1]\(
(e^x-1)^{\circ t}(x) = \text{dxp}^{\circ t}(x)
= \sum_{k=0}^{\infty} \frac{t^k x^{k+1}}{2^k}
+ \sum_{k=0}^{\infty} \frac{t^k x^{k+2}H^{(2)}_k}{-6\cdot2^k}
+ \cdots
\)
Note that this is not the normal generalized harmonic numbers, but the ones used by Conway & Guy, which are slightly different. These harmonic numbers are defined by:
\(
H^{(1)}_n = H_n = \sum_{i=1}^{n} \frac{1}{i}
\)
\(
H^{(k)}_n = \sum_{i=1}^{n} H^{(k-1)}_i
\)
The reason why I took the time to include such an advanced algorithm for computing harmonic numbers, is because I believe they hold the key to a new closed form, identity, or tautology for iterated-dxp.
Andrew Robbins