$n$ & $m :=$ any value in $\{0,1,2\ldots\}$
$\Omega$ & $\beta :=$ any value in $\{1,2,3\ldots\}$
[1] If there is a function $F_\beta$ such that for some value $\Omega$ and some function $T_{n,m}$ that $\text{if } F_1(\sum_{i=0}^{\Omega}T_{i,\Omega-i}) = T_{1,0}+T_{0,1} \text{ and } \\F_2(\sum_{i=0}^{\Omega}T_{i,(\Omega-i)}) = T_{2,0}+2T_{1,1}+T_{0,2}\text{ and } \\F_3(\sum_{i=0}^{\Omega}T_{i,(\Omega-i)})=T_{3,0}+3T_{2,1}+3T_{1,2}+T_{0,3}\text{ and }\\ \ldots\\\text{then }F_\beta(\sum_{i=0}^{\Omega}T_{i,(\Omega-i)})=\sum_{i=0}^\beta\binom{\beta}{i}T_{i,(\beta-i)}$
[2] if $\phi_i$ represents a variation of some function for every value i. $F_\beta(\sum_{i=0}^{k}\phi_i)=\sum_{i=0}^{k}F_\beta(\phi_i)$ AND $F_1(T_{n,m})=T_{n+1,m}+T_{n,m+1}$ $$:.F_\beta(T_{n,m})=\sum_{i=0}^\beta\binom{\beta}{i}T_{(n+i),(m+\beta-i)}$$
example of [1]
If $T_{n,m}:=a^n*e^m$ and $F_\beta(T_{1,0}+T_{0,1})=F_\beta(\sum_{i=0}^{1}T_{i,(i-1)}):=(a+e)^{\beta}$
$F_1(\sum_{i=0}^{1}T_{i,(i-1)})\\=(a^{1}+e^{1})^{1}=a^{1}e^{0}+a^{0}e^{1}=T_{1,0}+T_{0,1}$
$F_2(\sum_{i=0}^{1}T_{i,(i-1)})\\=(a^{1}+e^{1})^{2}=a^{2}e^{0}+2a^{1}e^{1}+a^{0}e^{2}=T_{1,0}+2T_{1,1}+T_{0,2}$
$F_3(\sum_{i=0}^{1}T_{i,(i-1)})=(a^{1}+e^{1})^{3}\\=a^{3}e^{0}+3a^{2}e^{1}+3a^{1}e^{2}+a^{0}e^{3}=T_{3,0}+3T_{2,1}+3T_{1,2}+T_{0,3}\\\ldots$
$:.F_\beta(\sum_{i=0}^{1}T_{i,(i-1)})=(a+e)^{\beta} \\=\sum_{i=0}^\beta\binom{\beta}{i}a^{i}*e^{\beta-i}=\sum_{i=0}^\beta\binom{\beta}{i}T_{i,(\beta-i)}$
example of [2]
If $T_{n,m}=u^{(n)}v^{(m)}$ and $F_\beta(G):=\frac{d^\beta}{(dx)^\beta}(G)$
$F_1(T_{n,m})=\frac{d}{dx}(u^{(n)}v^{(m)}) \\=u^{(n+1)}v^{(m)}+u^{(n)}v^{(m+1)}=T_{(n+1),m}+T_{n,(m+1)}$
$:.F_\beta(T_{n,m})=\frac{d^\beta}{(dx)^\beta}(u^{(n)}v^{(m)}) \\=\sum_{i=0}^\beta\binom{\beta}{i}u^{(n+i)}v^{(m+\beta-i)}=\sum_{i=0}^\beta\binom{\beta}{i}T_{(n+i),(m+\beta-i)}$
What are these distinct properties called? for when $F_\beta(\sum_{i=0}^{\Omega}T_{i,(\Omega-i)})=\sum_{i=0}^\beta\binom{\beta}{i}T_{i,(\beta-i)}$ or $F_\beta(T_{n,m})=\sum_{i=0}^\beta\binom{\beta}{i}T_{(n+i),(m+\beta-i)}$.
Edit: I found that $F_\beta(T_{n,m})=\sum_{i=0}^\beta\binom{\beta}{i}T_{(n+i),(m+\beta-i)}$ is shown in for n=0, m=0.
Abstract binomial theorem. Assume that we have:
- A vector space (or even a module) $V$.
- A family of "general terms" $(T_{i,j}\in V)_{i,j\in\mathbb N}$.
- A linear operatior $F: V\to V$, such that $F(T_{i,j}) = T_{(i+1),j} + T_{i,(j+1)}$.
Then $F^n(T_{0,0}) = \sum_{i=0}^n \binom{n}{i} T_{i,(n-i)}$.
