[Question for Bo] about formal Ackermann laws

[MphLee]

Maybe it was wrong. Let's visualize the summands:
\[\begin{pmatrix}
        a_{00} & a_{01}Y & a_{02}Y^2 & a_{03}Y^3 & \cdots \\
        a_{10}X & a_{11}XY & a_{12}XY^2 & a_{13}XY^3 &\cdots \\
        a_{20}X^2 & a_{21}X^2Y & a_{22}X^2Y^2 & a_{23}X^2Y^3 &\cdots \\
        a_{30}X^3 & a_{31}X^3Y & a_{32}X^3Y^2 & a_{33}X^3Y^3 &\cdots \\
        \vdots & \vdots  & \vdots  & \vdots  & \ddots \\
        \end{pmatrix}\]
The condition \(A(0,Y)=1\) is
\[\begin{pmatrix}
        a_{00} & a_{01}Y & a_{02}Y^2 & a_{03}Y^3 & \cdots \\
        0 & 0 & 0 & 0 &\cdots \\
        0&0 &0 & 0 &\cdots \\
       0& 0 & 0 & 0 &\cdots \\
        \vdots & \vdots  & \vdots  & \vdots  & \ddots \\
        \end{pmatrix}=1+Y
\]
I'd deduce from this that as formal powerseries \(a_{00}=1\) and \(a_{01}=1\) with all the higher terms equal zero.

Let me try to explore the tho other possible boundary values and how they change the infinite matrix. We look for three infinite matrices \(S,A, G\in \mathbb R[[X,Y]]\). Assume that they have trivial zeration.

• (Simple Bounday condition) \(S(X+1,0)=1\) implies that \(1+\sum_{0\lt n}s_{n,0}(X+1)^n=1\) i.e. \[\sum_{0\lt n}s_{n,0}(X+1)^n=0\].
  
• (Ackermann Bounday condition) \(A(X+1,0)=A(X,1)\) implies that \(1+\sum_{0\lt n}a_{n,0}(X+1)^n=2+\sum_{0\lt n,m}a_{n,m}X^n\) i. e \[\sum_{0\lt n}a_{n,0}(X+1)^n=1+\sum_{0\lt n,m}a_{n,m}X^n\]
  \[\begin{pmatrix}
          1 & 0 & 0 & 0 & \cdots \\
          a_{10}(X+1) & 0 & 0 & 0 &\cdots \\
          a_{20}(X+1)^2 & 0& 0 & 0 &\cdots \\
          a_{30}(X+1)^3 & 0 & 0 & 0 &\cdots \\
          \vdots & \vdots  & \vdots  & \vdots  & \ddots \\
          \end{pmatrix}-\begin{pmatrix}
          1 & 1 & 0 & 0 & \cdots \\
          a_{10}X & a_{11}X & a_{12}X & a_{13}X &\cdots \\
          a_{20}X^2 & a_{21}X^2 & a_{22}X^2 & a_{23}X^2 &\cdots \\
          a_{30}X^3 & a_{31}X^3 & a_{32}X^3 & a_{33}X^3 &\cdots \\       
        \vdots & \vdots  & \vdots  & \vdots  & \ddots \\         \end{pmatrix}=0\]
  \[\sum_{n=1}^\infty a_{n,0}\left (\sum_{k=0}^n\binom{n}{k} X^k\right )-\sum_{n=1}^\infty \left (\sum_{m=0}^\infty a_{nm}\right )X^n=1  \]
  
  
• (Goodstein Bounday conditions) \(G(1,0)=b,\,G(2,0)=0,\,G(X+2,0)=1,\,\) implies, considering \(2\leq j\) as a constant powerseries, that
  \[\begin{pmatrix}
          1-b &  0 & 0 & 0 & \cdots \\
          g_{10}&  0 & 0 & 0 &\cdots \\
          g_{20} &  0 & 0 & 0 &\cdots \\
          g_{30} &  0 & 0 & 0 &\cdots \\
          \vdots & \vdots  & \vdots  & \vdots  & \ddots \\
          \end{pmatrix}=0\,\quad
  \begin{pmatrix}
          1 &  0 & 0 & 0 & \cdots \\
          g_{10}2 &  0 & 0 & 0 &\cdots \\
          g_{20}4 &  0 & 0 & 0 &\cdots \\
          g_{30}8 &  0 & 0 & 0 &\cdots \\
          \vdots & \vdots  & \vdots  & \vdots  & \ddots \\
          \end{pmatrix}=0\,\quad
  \begin{pmatrix}
          0 &  0 & 0 & 0 & \cdots \\
          g_{10}j &  0 & 0 & 0 &\cdots \\
          g_{20}j^2 &  0 & 0 & 0 &\cdots \\
          g_{30}j^3 &  0 & 0 & 0 &\cdots \\
          \vdots & \vdots  & \vdots  & \vdots  & \ddots \\
          \end{pmatrix}=0\]
  \[1-b+\sum_{n=1}g_{n0}=0;\quad 1+\sum_{n=1}g_{n0}2^n=0;\quad \sum_{n=1}g_{n0}j^n=0\]
  

Not sure how to interpret the last expressions. It's is not evaluation of the formal powerseries, because it may not exists... but composition with in the first formal variable with a constant powerseries.>

Can someone extract some insight from the Ackermann version?