From: genseber@fwi.uva.nl (Menno Genseberger (W90))
Newsgroups: sci.math
Subject: Sum To Be Solved
Date: 3 Apr 92 12:30:34 GMT

Is there a solution for this sum

  oO 
 ___
 \    -n(n+a)/2
 /__ e
 n=0
 
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Newsgroups: sci.math
From: rusin@mp.cs.niu.edu (David Rusin)
Subject: Re: Sum To Be Solved
Date: Fri, 3 Apr 1992 19:44:01 GMT

In article <1992Apr3.123034.231@fwi.uva.nl> genseber@fwi.uva.nl (Menno Genseberger (W90)) writes:
>
>Is there a solution for this sum
>
>  oO 
> ___
> \    -n(n+a)/2
> /__ e
> n=0
> 
Well, perhaps I can help rewrite it. Can you dispense with the "n>= 0"?
In that case you can try Jacobi's identity.
I quote from the fascinating book "Fourier Series and Integrals", 
Dym and McKean, pp 52-4:
 
 +oo                                       +oo
Sum  exp[ - (n-x)^2 / 2t ] = Sqrt(2 pi t) Sum   exp (-2 pi^2 n^2 t)e_n
 -oo                                       -oo

where e_n = exp(2 pi i n x). This is an identity "known to Jacobi". The two 
sums converge with amazingly different rates of convergence for  t  far
from 1. Your sum, unfortunately, has t=1, but the right hand side is
different and, at least for some values of your  a,  will behave a
little differently from the original sum.

I can't imagine a closed form for the sum which does not mention the
theta function.

Dave Rusin@math.niu.edu

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