Date: Fri, 3 Feb 95 12:14:26 PST
From: [Permission pending]
To: rusin@math.niu.edu
Subject: Re: Expression of Sum a^k^2, k from 0 to n


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From rusin@math.niu.edu Wed Feb  1 07:38:02 1995
Date: Wed, 1 Feb 95 09:30:16 CST
From: rusin@math.niu.edu (Dave Rusin)
To: [Permission pending]
Subject: Re: Expression of Sum a^k^2, k from 0 to n
Newsgroups: sci.math
Organization: Northern Illinois University, Math
Cc: 
Content-Length: 753

In article <[identifier deleted]> you write:
>I wonder who knows the expression of follow series:
>
>	   n
>	   -      2
>	   \     k
>	   /    a
>	   -
>         k = 0
>
>here a<=1.

>> I don't believe there is a closed form for the solution. One suggestion
>> is to consider the Jacobi theta function
>>	theta(t) = Sum exp( - pi k^2 t)  (for t>0)
>> where the sum is over all integers  k  (positive and negative). Taking
>> t = (1/pi).ln(1/a) gives theta(t)=-1+2S  where  S  differs from your
>> sum by, say, a^(n^2)/(1-a) at most.
>>	For  a  close to  1  you're better off using the Jacobi identity:
>> theta(t) = t^(-1/2).theta(1/t). This is a version of the Poisson summation
>> formula. 
>>	A reference is Bellman, "A brief introduction to tehta functions" 1961

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Thank you very much for the enlightening. If you don't mind, I'd like to
ask you a further question.

On the above expression,
                     i w
                a = e    ,
it is a complex varible. According to my math handbook
	infinity
	   -      2
	   \     k
	   /    a      = theta (0)
	   -                  3
         k = 0
How could I determint theta3(0) by using the complex a.

Thank you very much.

[sig deleted -- djr]