• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 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

 Possibly Related Threads... Thread Author Replies Views Last Post 2015 Continuum sum conjecture tommy1729 3 3,728 05/26/2015, 12:24 PM Last Post: tommy1729 Another way to continuum sum! JmsNxn 6 7,044 06/06/2014, 05:09 PM Last Post: MphLee Continuum sum = Continuum product tommy1729 1 2,883 08/22/2013, 04:01 PM Last Post: JmsNxn applying continuum sum to interpolate any sequence. JmsNxn 1 3,082 08/18/2013, 08:55 PM Last Post: tommy1729 sexp by continuum product ? tommy1729 6 9,239 06/30/2011, 10:07 PM Last Post: tommy1729 continuum sum again tommy1729 0 2,410 02/10/2011, 01:25 PM Last Post: tommy1729 Continuum sums -- a big problem and some interesting observations mike3 17 21,270 10/12/2010, 10:41 AM Last Post: Ansus Continuum sum - a new hope kobi_78 10 13,970 06/13/2010, 11:23 PM Last Post: sheldonison New tetration method based on continuum sum and exp-series mike3 16 22,780 05/02/2010, 09:58 AM Last Post: andydude Borel summation and other continuation/summability methods for continuum sums mike3 2 5,505 12/30/2009, 09:51 PM Last Post: mike3

Users browsing this thread: 1 Guest(s)