Right now I'm writing out some assumptions we have to put away.
For example:
\( 2 \,\,\bigtriangleup_s\,\,2 = 4 \,\,\Rightarrow\,\,2\,\,\bigtriangleup_{s+1}\,\,1 = 2 \)
This implies that \( 2\,\,\bigtriangleup_s\,\,1 \) is not analytic because it is 2 for all s with real part greater than or equal to 1 and 3 when s is 0.
Another one is that any continuous segment of operators is commutative or associative all the operators have to be. As well; operators in a continuous segment cannot have the same identity.
The functional requirement is the following:
\( \vartheta_n(s+1) = \sum_{k=0}^{\infty} \vartheta_n(\mu_k) \vartheta_k(s) \)
where we have:
\( \Pi(s) = x\,\,\bigtriangleup_s\,\,y \)
\( \Pi(\mu_k) = x\,\,\bigtriangleup_k\,\,\ell \)
\( \ell = x\,\,\bigtriangleup_{s+1}\,\,(y-1)\,\,\in\,\mathbb{N} \)
And \( \vartheta \) is as before.
I can obtain \( \Pi^{-1}(s) \) as a taylor series using lagrange inversion. So all of these functions are theoretically computable besides \( \psi_n \) which is in \( \vartheta_n \). So the requirement is restricted to it. I'm writing this all out trying to solve for the taylor series coefficients of \( \psi \). \( \vartheta_n \) is entire if \( \psi_n \) is entire; so I hope it is.
Thanks for the encouragement Gottfried. Like all math; it's slow progress. Little breakthroughs from time to time.
For example:
\( 2 \,\,\bigtriangleup_s\,\,2 = 4 \,\,\Rightarrow\,\,2\,\,\bigtriangleup_{s+1}\,\,1 = 2 \)
This implies that \( 2\,\,\bigtriangleup_s\,\,1 \) is not analytic because it is 2 for all s with real part greater than or equal to 1 and 3 when s is 0.
Another one is that any continuous segment of operators is commutative or associative all the operators have to be. As well; operators in a continuous segment cannot have the same identity.
The functional requirement is the following:
\( \vartheta_n(s+1) = \sum_{k=0}^{\infty} \vartheta_n(\mu_k) \vartheta_k(s) \)
where we have:
\( \Pi(s) = x\,\,\bigtriangleup_s\,\,y \)
\( \Pi(\mu_k) = x\,\,\bigtriangleup_k\,\,\ell \)
\( \ell = x\,\,\bigtriangleup_{s+1}\,\,(y-1)\,\,\in\,\mathbb{N} \)
And \( \vartheta \) is as before.
I can obtain \( \Pi^{-1}(s) \) as a taylor series using lagrange inversion. So all of these functions are theoretically computable besides \( \psi_n \) which is in \( \vartheta_n \). So the requirement is restricted to it. I'm writing this all out trying to solve for the taylor series coefficients of \( \psi \). \( \vartheta_n \) is entire if \( \psi_n \) is entire; so I hope it is.
Thanks for the encouragement Gottfried. Like all math; it's slow progress. Little breakthroughs from time to time.
