I was looking at the summation identity that is equal to e^x and from that I derived some other summation identities.

I used the summation identities that is equal to a^x to derive some summation identities that are equal to different size power towers. I found that the summation identities for power towers or tetration involve summations within summations and sometimes summations within summations within summations. I'm wondering if there would be a simpler way to express the summations within summations as if there is then I'm thinking it could help with tetration involving numbers other than the positive integers.

I used the summation identities that is equal to a^x to derive some summation identities that are equal to different size power towers. I found that the summation identities for power towers or tetration involve summations within summations and sometimes summations within summations within summations. I'm wondering if there would be a simpler way to express the summations within summations as if there is then I'm thinking it could help with tetration involving numbers other than the positive integers.