Another way to continuum sum!

[JmsNxn]

Does this work to find 

t(s+1) = exp(-s^2) + t(s) 

??

I ask this because of the connection to

https://math.eretrandre.org/tetrationfor...p?tid=1652

Many ways for continuum sum exist but I was looking for an integral type.

regards

tommy1729

So tommy, unfortunately it does not. BUT!

It works for \(f(s)\) if \(|f(s)| \le O(e^{\tau |\Im(s)|})\) in the strip \(a \le \Re(s) \le b\) and \(0 \le \tau < \pi/2\). So it works for \(f(s) = e^{\lambda s^2}\) so long as \(\Re(\lambda) \ge 0\). It fails for \(\Re\lambda < 0\). I imagine it would be possible to massage this though to get it to work for \(e^{-s^2}\). I remember that was a big frustrating moment for me. I could get it for \(e^{s^2}\) but not when you put the negative. But I did find work arounds, I just never published much about it, I believe because \(e^{-s^2}\) is fourier transformable there are clever ways to actually indefinite sum this using the Fourier transform. Though it's a little tricky. I'll have to dig through 10 year old notes ....

This post is a blast from the past, here's the paper that came from it, in case your interested in understanding how this works:

https://arxiv.org/abs/1503.06211