Hey, Mphlee.
I'm not that well versed in the matrices you are talking about, but I am versed very well in what you are trying to construct.
To begin, let's write:
\[
g(X,Y) = \sum_{n,m=0}^\infty g_{nm} X^nY^m\\
\]
And let's refer to this solely as a Formal Series--by which there is no need to check for convergence. Now let's assume that:
\[
g(X,g(X+1,Y)) = g(X+1,Y+1)
\]
And let's pull a Gottfried and just collect coefficients, and create a matrix solution to all the values. (This is little different than solving an \(\infty \times \infty\) linear system (Heisenberg shit).
The trouble is, Mphlee, this shit will almost certainly not converge. Adding in boundary conditions makes this way way way fucking harder too.
BUT! Formally, yes. It all works fine, and you can definitely pull out coefficients here, and solve a formal series which "should" converge to semi operators. But sadly, it'll be divergent, and probably brutally divergent.
The biggest problem with this approach is largely that it's not computable. It's not something we can plug into a calculator. But as a formal system, and an algebraic construct, absolutely it works. And this is, in many senses, an Abstract Algebraic construction. But it is not an Analytic construction; unless you can prove that \(g_{nm} = O(1)\) or something like that.
I'm not that well versed in the matrices you are talking about, but I am versed very well in what you are trying to construct.
To begin, let's write:
\[
g(X,Y) = \sum_{n,m=0}^\infty g_{nm} X^nY^m\\
\]
And let's refer to this solely as a Formal Series--by which there is no need to check for convergence. Now let's assume that:
\[
g(X,g(X+1,Y)) = g(X+1,Y+1)
\]
And let's pull a Gottfried and just collect coefficients, and create a matrix solution to all the values. (This is little different than solving an \(\infty \times \infty\) linear system (Heisenberg shit).
The trouble is, Mphlee, this shit will almost certainly not converge. Adding in boundary conditions makes this way way way fucking harder too.
BUT! Formally, yes. It all works fine, and you can definitely pull out coefficients here, and solve a formal series which "should" converge to semi operators. But sadly, it'll be divergent, and probably brutally divergent.
The biggest problem with this approach is largely that it's not computable. It's not something we can plug into a calculator. But as a formal system, and an algebraic construct, absolutely it works. And this is, in many senses, an Abstract Algebraic construction. But it is not an Analytic construction; unless you can prove that \(g_{nm} = O(1)\) or something like that.
