03/12/2014, 04:37 AM
I'll try to work with the equation you provided, but it will take me a while to figure it out. Thanks for the suggestion!
Regarding the relation provided:
\( a [x](\alpha + \beta) = (a[x] \alpha)[x-1](a[x] \beta) \)
It is only true when \( \alpha=\beta \), unfortunately, because
\( (a[x] \alpha)[x-1](a[x] \beta)\neq(a[x] \beta)[x-1](a[x] \alpha) \) in general.
For example (x=4):
\( (a[4]2)[3](a[4]4)\neq(a[4]4)[3](a[4]2)\leftrightarrow (a^a)^{\left ((a^a)^{(a^a)}\right )}\neq\left ((a^a)^{(a^a)}\right )^{(a^a) \)
Regarding the relation provided:
\( a [x](\alpha + \beta) = (a[x] \alpha)[x-1](a[x] \beta) \)
It is only true when \( \alpha=\beta \), unfortunately, because
\( (a[x] \alpha)[x-1](a[x] \beta)\neq(a[x] \beta)[x-1](a[x] \alpha) \) in general.
For example (x=4):
\( (a[4]2)[3](a[4]4)\neq(a[4]4)[3](a[4]2)\leftrightarrow (a^a)^{\left ((a^a)^{(a^a)}\right )}\neq\left ((a^a)^{(a^a)}\right )^{(a^a) \)