It's true (as the answer below and some of the commenters note) that it's easy to interpret this question in a way that makes it seem trivial and uninteresting. I'm quite sure, however, that pursuing typographical similarity between $e^{x^2}$ and $\zeta^{m^2}$ leads to interesting mathematics, and so here's a more serious attempt at propoganda for some of Ivan Cherednik's work.

Pages 6,7,8 and 9 of his Cherednik's paper "Double affine Hecke algebras and difference Fourier transforms" explain how to ``interpolate'' between integral formulas relating the Gaussian to the Gamma function and (a certain generalization of) Gauss sums.

More explicitly, he shows that the formula (for many people, it's really just the definition of the Gamma function)

$$\int_{-\infty}^{\infty} e^{-x^2} x^{2k} dx=\Gamma \left( k+\frac{1}{2} \right)$$

(for $k \in \mathbb{C}$ with real part $>-1/2$) and the Gauss-Selberg sum

$$\sum_{j=0}^{N-2k} \zeta^{(k-j)^2/4} \frac{1-\zeta^{j+k}}{1-\zeta^k} \prod_{l=1}^j \frac{1-\zeta^{l+2k-1}}{1-\zeta^l}=\prod_{j=1}^k (1-\zeta^j)^{-1} \sum_{m=0}^{2N-1} \zeta^{m^2/4}$$

(where $N$ is a positive integer, $\zeta=e^{2\pi i/N}$ is a prim. $N$th root of $1$, and $k$ is a positive integer at most $N/2$) can both be obtained as limiting cases of the same $q$-series identity. The common generalization of the Gaussian and the function $k \mapsto \zeta^{k^2}$ is the function $x \mapsto q^{x^2}$, and the measures weighting the integral and sum get replaced by Macdonald's measure---essentially the same one that shows up in the constant term conjecture for $A_1$, and that produces the Macdonald polynomials and kick-started the DAHA. The Fourier transform is deformed along with everything else to produce the "Cherednik-Fourier" transform.

I don't know how much of the roots of unity story generalizes to higher rank root systems.

Note: In the Gauss-Selberg sum, replacing $k$ by the integer part of $N/2$ and manipulating a little (as in the nice exposition by David Speyer linked to in the question above) gives the usual formula for the Gauss sum.

Ivan Cherednik has thought about

It's true (as the analogy answer below some of the commenters note) that it's easy to interpret this question in a way that makes it seem trivial and uninteresting. I'm quite sure, however, that pursuing typographical similarity between $e^{x^2}$ and pages $\zeta^{m^2}$ leads to interesting mathematics, and so here's a more serious attempt at propoganda for some of Cherednik's work.

Pages 6,7,8 and 9 of his paper "Double affine Hecke algebras and difference Fourier transforms" explain how to ``interpolate'' using Jackson sums between integral formulas for relating the Gaussian to the Gamma function and (a certain generalization of) Gauss sums. It's good motivation for learning about the simplest double affine Hecke algebra!

Edit:

More explicitly, he shows that the formula (for many people, it's really just the definition of the Gamma function)

(where $N$ is a positive integer, $\zeta=e^{2\pi i/N}$ is a prim. $N$th root of $1$, and $k$ is a positive integer at most $N/2$) can both be obtained as limiting cases of the same $q$-series identity(I'm not going to transcribe it) which itself comes from . The common generalization of the 1-dimensional Gaussian and the function $k \mapsto \zeta^{k^2}$ is the function $x \mapsto q^{x^2}$, and the measures weighting the integral and sum get replaced by Macdonald's measure---essentially the same one that shows up in the constant term conjecture for $A_1$, and that produces the Macdonald polynomials and kick-started the DAHAtheory. The Fourier transform is deformed along with everything else to produce the "Cherednik-Fourier" transform.

I don't know how much of the roots of unity story generalizes to higher rank root systems.

Ivan Cherednik has thought about the analogy some, and pages 6,7,8 and 9 of his paper "Double affine Hecke algebras and difference Fourier transforms" explain how to ``interpolate'' using Jackson sums between integral formulas for the Gaussian and (a certain generalization of) Gauss sums. It's good motivation for learning about the simplest double affine Hecke algebra!

Edit:

More explicitly, he shows that the formula

$$\int_{-\infty}^{\infty} e^{-x^2} x^{2k} dx=\Gamma \left( k+\frac{1}{2} \right)$$

(for $k \in \mathbb{C}$ with real part $>-1/2$) and the Gauss-Selberg sum

$$\sum_{j=0}^{N-2k} \zeta^{(k-j)^2/4} \frac{1-\zeta^{j+k}}{1-\zeta^k} \prod_{l=1}^j \frac{1-\zeta^{l+2k-1}}{1-\zeta^l}=\prod_{j=1}^k (1-\zeta^j)^{-1} \sum_{m=0}^{2N-1} \zeta^{m^2/4}$$

(where $N$ is a positive integer, $\zeta=e^{2\pi i/N}$ is a prim. $N$th root of $1$, and $k$ is a positive integer at most $N/2$) can both be obtained as limiting cases of the same $q$-series identity (I'm not going to transcribe it) which itself comes from the 1-dimensional DAHA theory.

Note: In the Gauss-Selberg sum, replacing $k$ by the integer part of $N/2$ and manipulating a little (as in the nice exposition by David Speyer linked to in the question above) gives the usual formula for the Gauss sum.

Ivan Cherednik has thought about the analogy some, and pages 6,7,8 and 9 of his paper "Double affine Hecke algebras and difference Fourier transforms" explain how to ``interpolate'' using Jackson sums between integral formulas for the Gaussian and (a certain generalization of) Gauss sums. It's good motivation for learning about the simplest double affine Hecke algebra!

Edit:

More explicitly, he shows that the formula

$$\int_{-\infty}^{\infty} e^{-x^2} x^{2k} dx=\Gamma \left( k+\frac{1}{2} \right)$$

(for $k \in \mathbb{C}$ with real part $>-1/2$) and the Gauss-Selberg sum

$$\sum_{j=0}^{N-2k} \zeta^{(k-j)^2/4} \frac{1-\zeta^{j+k}}{1-\zeta^k} \prod_{l=1}^j \frac{1-\zeta^{l+2k-1}}{1-\zeta^l}=\prod_{j=1}^k (1-\zeta^j)^{-1} \sum_{m=0}^{2N-1} \zeta^{m^2/4}$$

(where $N$ is a positive integer, $\zeta$ \zeta=e^{2\pi i/N}$ is a prim. $N$th root of $1$, and $k$ is a positive integer at most $N/2$) can both be obtained as limiting cases of the same $q$-series identity (I'm not going to transcribe it) which itself comes from the 1-dimensional DAHA theory.

Note: In the Gauss-Selberg sum, replacing $k$ by the integer part of $N/2$ and manipulating a little gives the usual formula for the Gauss sum.

Ivan Cherednik has thought about the analogy some, and pages 6,7,8 and 9 of his paper "Double affine Hecke algebras and difference Fourier transforms" explain how to ``interpolate'' using Jackson sums between integral formulas for the Gaussian and (a certain generalization of) Gauss sums. It's good motivation for learning about the simplest double affine Hecke algebra!

Edit:

More explicitly, he shows that the formula

$$\int_{-\infty}^{\infty} e^{-x^2} x^{2k} dx=\Gamma(k+\frac{1}{2})$$dx=\Gamma \left( k+\frac{1}{2} \right)$$

(for $k \in \mathbb{C}$ with real part $>-1/2$) and the Gauss-Selberg sum

$$\sum_{j=0}^{N-2k} \zeta^{(k-j)^2/4} \frac{1-\zeta^{j+k}}{1-\zeta^k} \prod_{l=1}^j \frac{1-\zeta^{l+2k-1}}{1-\zeta^l}=\prod_{j=1}^k (1-\zeta^j)^{-1} \sum_{m=0}^{2N-1} \zeta^{m^2/4}$$

(where $N$ is a positive integer, $\zeta$ is a prim. $N$th root of $1$, and $k$ is a positive integer at most $N/2$) can both be obtained as limiting cases of the same $q$-series identity (I'm not going to transcribe it) which itself comes from the 1-dimensional DAHA theory.

Ivan Cherednik has thought about the analogy some, and pages 6,7,8 and 9 of his paper explain how to ``interpolate'' using Jackson sums between integral formulas for the Gaussian and (a certain generalization of) Gauss sums. It's good motivation for learning about the simplest double affine Hecke algebra!

Edit:

More explicitly, he shows that the formula

$$\int_{-\infty}^{\infty} e^{-x^2} x^{2k} dx=\Gamma(k+\frac{1}{2})$$

(for $k \in \mathbb{C}$ with real part $>-1/2$) and the Gauss-Selberg sum

$$\sum_{j=0}^{N-2k} \zeta^{(k-j)^2/4} \frac{1-\zeta^{j+k}}{1-\zeta^k} \prod_{l=1}^j \frac{1-\zeta^{l+2k-1}}{1-\zeta^l}=\prod_{j=1}^k (1-\zeta^j)^{-1} \sum_{m=0}^{2N-1} \zeta^{m^2/4}$$

(where $N$ is a positive integer, $\zeta$ is a prim. $N$th root of $1$, and $k$ is a positive integer at most $N/2$) can both be obtained as limiting cases of the same $q$-series identity (I'm not going to transcribe it) which itself comes from the 1-dimensional DAHA theory.

Ivan Cherednik has thought about the analogy some, and pages 6,7,8 and 9 of his paper explain how to ``interpolate'' using Jackson sums between integral formulas for the Gaussian and (a certain generalization of) Gauss sums. It's good motivation for learning about the simplest double affine Hecke algebra!