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 Powerful way to perform continuum sum JmsNxn Long Time Fellow Posts: 291 Threads: 67 Joined: Dec 2010 08/11/2013, 04:05 PM (This post was last modified: 08/11/2013, 04:22 PM by JmsNxn.) Actually I wasn't thinking about Ramanujan. I was thinking about Euler and the integral representation for the Gamma function. I wanted to exploit integration by parts as beautifully as he did. Why yes the difference operator is how I found the inverse, but this gives us an integral transform: $\mathcal{Z}^{-1} f(s) = \int_{\sigma - i\infty}^{\sigma + i\infty }e^{-\pi i t}\Gamma(t) (f(t) - f(t-1))s^{-t}dt$ Is an expression for one of the inverses. We also have: $\mathcal{Z}^{-1} f(s) = \sum_{n=0}^{\infty} (f(-n) - f(-n-1))\frac{s^n}{n!} = \int_{-\infty}^{\infty} (f(-y) -f(-y-1))\frac{s^y}{y!}dy$ Each inverse operator works on different classes of functions. I'm having a little trouble finding the exact restrictions on the functions we can use. Being more general, we can change the Riemann-liouville differintegral to work on different functions by changing the limits of integration in the integral expression. We can solve the following continuum sum using different limits: $\frac{d^{-s}}{dt^{-s}_0} f(t) = \frac{1}{(s-1)!}\int_0^t f(u)(t-u)^{s-1}du$ $Rf(s) = \frac{d^{-s}}{dt^{-s}_0}f(t) |_{t=1}$ $R t^n = \frac{n!}{(s+n)!}$ Therefore: $\phi(s) = \int_0^\infty e^{-t} \frac{d^{-s}}{dt^{-s}_0}(t+1)^ndt$ $\phi(s) = \frac{n!}{(s+n)!} + \phi(s-1)$ I'm wondering how to apply this to tetration or hyper operators. This performs a fair amount of mathematical work and solves a nice iteration problem--maybe its related to hyper operators *fingers crossed* « Next Oldest | Next Newest »

 Messages In This Thread Powerful way to perform continuum sum - by JmsNxn - 08/10/2013, 09:06 PM RE: Powerful way to perform continuum sum - by tommy1729 - 08/11/2013, 02:00 AM RE: Powerful way to perform continuum sum - by JmsNxn - 08/11/2013, 04:05 PM RE: Powerful way to perform continuum sum - by tommy1729 - 08/11/2013, 07:31 PM RE: Powerful way to perform continuum sum - by JmsNxn - 08/11/2013, 08:18 PM RE: Powerful way to perform continuum sum - by tommy1729 - 08/11/2013, 10:29 PM RE: Powerful way to perform continuum sum - by tommy1729 - 08/11/2013, 11:05 PM RE: Powerful way to perform continuum sum - by JmsNxn - 08/12/2013, 07:17 PM

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