Hello,

I happened upon your question and noticed that your integrals look a lot like fractional derivatives. I think they provide the generalization your looking for.

http://en.wikipedia.org/wiki/Fractional_derivative

I know this doesn't fully answer your question, but should be a good place to start looking. Good luck!

Hello,

I happened upon your question and noticed that your integrals look a lot like fractional derivatives. I think they provide the generalization you're looking for.

http://en.wikipedia.org/wiki/Fractional_derivative

I know this doesn't fully answer your question, but should be a good place to start looking. Good luck!