$f^{\alpha }\left( \overrightarrow {0}\right) +f^{\beta }\left( \overrightarrow {0}\right) =f^{\alpha \beta +\alpha +\beta }\left( \overrightarrow {0}\right)$

Is there a function $f$ from $R^{\infty}$ to $R^{\infty}$ that satisfies this equation for all natural ${\alpha}$ and ${\beta}$ ?

I already know that any function that has $f\left( \overrightarrow {0}\right)= \overrightarrow {0}$ satisfies the equation, so are there any other functions that satisfy the equation?

Thank you in advance!

*My phrasing for this being a "funtional equation" was flawed. All I really wanted to know was the existance of a function that satisfies the equation above.

*The superscripts indicate iterations of $f$.

$f^{\alpha }\left( \overrightarrow {0}\right) +f^{\beta }\left( \overrightarrow {0}\right) =f^{\alpha \beta +\alpha +\beta }\left( \overrightarrow {0}\right)$

Is there a function $f$ from $R^{\infty}$ to $R^{\infty}$ that satisfies this equation for all natural ${\alpha}$ and ${\beta}$ ?

I already know that any function that has $f\left( \overrightarrow {0}\right)= \overrightarrow {0}$ satisfies the equation, so are there any other functions that satisfy the equation?

Thank you in advance!

*My phrasing for this being a "functional equation" was flawed. All I really wanted to know was the existance of a function that satisfies the equation above.

*The superscripts indicate iterates of $f$.

$f^{\alpha }\left( \overrightarrow {0}\right) +f^{\beta }\left( \overrightarrow {0}\right) =f^{\alpha \beta +\alpha +\beta }\left( \overrightarrow {0}\right)$

Is there a function $f$ from $R^{\infty}$ to $R^{\infty}$ that satisfies this equation for all natural ${\alpha}$ and ${\beta}$ ?

I already know that any function that has $f\left( \overrightarrow {0}\right)= \overrightarrow {0}$ satisfies the equation, so are there any other functions that satisfy the equation?

Thank you in advance!

*My phrasing for this being a "funtional equation" was flawed. All I really wanted to know was the existance of a function that satisfies the equation above.

*The superscripts indicate iterates of $f$.

$f^{\alpha }\left( \overrightarrow {0}\right) +f^{\beta }\left( \overrightarrow {0}\right) =f^{\alpha \beta +\alpha +\beta }\left( \overrightarrow {0}\right)$

Is there a function $f$ from $R^{\infty}$ to $R^{\infty}$ that satisfies this equation for all natural ${\alpha}$ and ${\beta}$ ?

I already know that any function that has $f\left( \overrightarrow {0}\right)= \overrightarrow {0}$ satisfies the equation, so are there any other functions that satisfy the equation?

Thank you in advance!

*My phrasing for this being a "funtional equation" was flawed. All I really wanted to know was the existance of a function that satisfies the equation above.

*The superscripts indicate iterations of $f$.

$f^{\alpha }\left( \overrightarrow {0}\right) +f^{\beta }\left( \overrightarrow {0}\right) =f^{\alpha \beta +\alpha +\beta }\left( \overrightarrow {0}\right)$

Is there a function $f$ from $R^{\infty}$ to $R^{\infty}$ that satisfies this equation for all natural ${\alpha}$ and ${\beta}$ ?

I already know that any function that has $f\left( \overrightarrow {0}\right)= \overrightarrow {0}$ satisfies the equation, so are there any other functions that satisfy the equation?

Thank you in advance!

*My phrasing for this being a "funtional equation" was flawed. All I really wanted to know was the existance of a function that satisfies the equation above.

$f^{\alpha }\left( \overrightarrow {0}\right) +f^{\beta }\left( \overrightarrow {0}\right) =f^{\alpha \beta +\alpha +\beta }\left( \overrightarrow {0}\right)$

Is there a function $f$ from $R^{\infty}$ to $R^{\infty}$ that satisfies this equation for all natural ${\alpha}$ and ${\beta}$ ?

I already know that any function that has $f\left( \overrightarrow {0}\right)= \overrightarrow {0}$ satisfies the equation, so are there any other functions that satisfy the equation?

Thank you in advance!

*My phrasing for this being a "funtional equation" was flawed. All I really wanted to know was the existance of a function that satisfies the equation above.

*The superscripts indicate iterates of $f$.