For anyone who know the hyperfactorial, is it a way to prove that 0^(0)=1? We can show that H(1)=1^(1)H(0) H(0)=1

Question

For anyone who know the hyperfactorial, is it a way to prove that
`0^(0)=1`? We can show that


`H(1)=1^(1)H(0)`


`H(0)=1`

Answer 1

No, the Hyperfactorial is not a way to prove that.

0 to the 0 power is just undefined, but it cannot be 1.

Explanation:

H(1)=1^{1}H(0)

The Hyperfactorial is defined as the result of multiplying a given number of consecutive integers from 1 to the given number, each raised to its own power.

No, since 1^1=1, H(1)—Consecutive Integer is 0.

0 is what you get in the end, no matter what.

Answer 2

I don't think that there's a way to "prove" that
`0^0` equals anything, since that quantity is undefined.

Moreover, the hyperfactorials are defined as

`H(n) = 1^1\cdot2^2\cdot3^3\cdot\ldots\cdot n^n`

So, claiming that
`H(1)=1^1\cdot H(0)` wouldn't be true, because
`H(1)` is already the case that solves the recursion.