# Can Mathematica do this summation?

I was very pleased to discover that Mathematica could do this summation and produce a symbolic result.

s1 = 1/nn Sum[Cos[2 π a (n - 1)] E^(-I (n - 1) (s - 1)/nn), {n, nn}] // Simplify


The answer is long but very useful for me. I also discovered that Mathematica could do some variants of this sum. However, Mathematica cannot do this summation.

s2 = 1/nn Sum[
Cos[2 π ((a - ϵ) (n - 1) + ϵ/(nn - 1) (n - 1)^2)] E^(-I (n - 1) (s - 1)/nn),
{n, nn}]


Note that nn >> 1. Is my sum s2 in some way pathological so that it cannot be done? Can my sum be coerced into a solution?

These sums come from Fourier analysis, where it is useful to have a symbolic expression for results that are often calculated numerically using the discrete Fourier transform (Fourier in Mathematica). In order to investigate my sum, I looked at the continuous Fourier transform in which time is equal to n -1 and frequency is (s - 1)/nn. The continuous Fourier transform is given by the integral

1/t2 Integrate[Cos[2 π ( (a - ϵ) t + ϵ/t2 t^2)] E^(-I 2 π f t), {t, 0, t2}]


Mathematica can do this integral which comes out in terms of error functions. However, I can't see how to use this to help get the symbolic solution to the sum s2.

Is there any hope for getting a symbolic solution for my second sum? In a previous question what-summations-can-mathematica-do? I had the naive idea that if the integral is possible then a similar sum should also be possible but although comments on this would be interesting it is probably too broad for this forum. Thanks.

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