A new notation for operations and some questions

Question

let $x^{/1/}=x+x$ (addition)

$x^{/2/}=x.x$ (multiplication)

$x^{/3/}=x^x$ (exponentiation)

$x^{/4/}={}^xx$ (tetration)

and so on.....

I have the following questions:

$(1)$ Can we define an operation between two known operations (Ex.addition and multiplication), where $n$ in $x^{/n/}$ is fractional (Ex. $x^{/1.5/}$)?

$(2)$ Is there a closed form for $x^{/x/}$?

$(3)$ Is infinitation ($x^{/\infty/}$) equal to infinite tetration $\left( x^{x^{x...}}\right)$?

I think the answer to the last question is true as infinite tetration is equal to infinite pentation (as both of them equal to $x^{x^{x^{.^{.^.}}}}$ when we convert them to their exponentiation form.) Following the same logic, any $n$-ation is equal to $x^{x^{x^{.^{.^.}}}}$, and therefore infiniation is also equal to infinite tetration.

Edit: Does this notation seem useful for defining operations?

Answer 1

The standard notation for your $x^{/n/}$ is $H_n(x, x)$.

(3) Is infinitation ($x^{/\infty/}$) equal to infinite tetration $\left( x^{x^{x...}}\right)$?

You're asking whether $H_\infty(x, x) = H_4(x, \infty)$. The answer is no. Notice that

\begin{align*} H_n(a, -1) &= 0 & n \geq 4 \\ H_n(a, 0) &= 1 & n \geq 3 \\ H_n(a, 1) &= a & n \geq 2 \\ H_n(2, 2) &= 4 & n \geq 1 \end{align*}

Therefore \begin{align*} H_\infty(-1, -1) &= 0 \\ H_\infty(0, 0) &= 1 \\ H_\infty(1, 1) &= 1 \\ H_\infty(2, 2) &= 4 \end{align*}

But \begin{align*} H_4(-1, \infty) &= -1 \\ H_4(0, \infty) &= \frac{1}{2} & \lim_{\varepsilon \rightarrow 0^+} \lim_{b \rightarrow \infty} H_4(\varepsilon, b) \\ H_4(1, \infty) &= 1 \\ H_4(2, \infty) & \quad \text{diverges} \end{align*}

so they disagree at 3 of these 4 points.

$(1)$ Can we define an operation between two known operations (Ex.addition and multiplication), where $n$ in $x^{/n/}$ is fractional (Ex. $x^{/1.5/}$)?

The names typically given to $H_{0.5}$ and $H_{1.5}$ are "halfation" and "sesquation", though there's little information about them.