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This question is inspired by Why are Bessel function and Kloosterman sum similar? - it developed in me desire to understand Kloosterman sums better.

There seems to be common knowledge that Gauss, Jacobi and Kloosterman sums are analogs of the, respectively, Gamma, Beta and Bessel functions, and I would like to understand this better.

(Subquestion one: are there other similar analogies known?)

For example, are there any "finite sum analogs" of hypergeometric functions in this context?

Now there is a whole branch of the theory of special functions - their common treatment as matrix elements of Lie group representations; most comprehensive treatment that I know is in several volumes by Klimyk and Vilenkin

(subquestion two: are there other texts you would recommend?)

For example, Gamma appears in representations $R_\lambda$ of the group of affine transformations of the real line on functions via $R_\lambda[aX+b]:\varphi(x)\mapsto e^{\lambda bx}\varphi(ax)$, and there is (in the end of the second volume) a parallel definition of Gamma and Beta functions for any locally compact totally disconnected non-discrete field. (Seems like they in particular obtain $p$-adic $\Gamma$, although I am not sure.) Bessel functions are obtained from representations of $SO(n)$ on functions on the unit sphere, although I could not get a clear picture of it for me in the book.

So I thought - maybe there is some still more general unified treatment which would give all of the above but in addition would encompass the case of the field with one element; or, in more conventional terms, would give Gamma, Beta, Bessel, etc. functions for simple Lie groups and in parallel would give, respectively, Gauss, Jacobi, Kloosterman, etc. sums for their respective Weyl groups?

Or maybe $\mathbb F_1$ is not related in any way and one has simply to ask for a unified treatment including representations of linear groups over finite fields into above picture. Still in that case a question remains what would(could) one get in parallel from representations of Weyl groups.

And, since I am asking anyway - what about $q$-analogs? It seems that applying the above to $q$-deformations of representations of universal enveloping algebras one may get things like $q$-Gamma and $q$-hypergeometric functions; are there any finite field or $\mathbb F_1$-versions of these $q$-analogs known?

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The analogy is obtained by placing the special function in a larger context of not-so-special functions. For instance I could compare the analytic function

$$\int_0^\infty e^{t^3x - t^2 x^2 + t x^4- x^8} x^{1/3} dx$$

which presumably has no special significance, to the exponential sum

$$ \sum_{x\in \mathbb F_q} \psi(t^3x-t^2x^2+ t x^4-x^8) \chi(x)$$

for $\psi$ an additive character and $\chi$ a multiplicative character of order $3$.

This transformation is a fairly straightforward process. The only subtlety is that we may have multiple choices for the contour of integration. Really one should compare the finite field function to a whole family of analytic functions.

Katz studied many examples in his book Exponential Sums and Differential Equations, including the hypergeometric ones.

I'm not familiar with the representation-theoretic interpretation of special functions, but it should generalize as long as the representation theory perspective gives us integrals that we can transform to exponential sums in this way.

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    $\begingroup$ I now realize I failed to mention one essential feature of that representation-theoretic approach: various functional equations satisfied by special functions become explained by them being matrix elements of group representations, and so obliged to play their part in turning group operation into matrix multiplication. I say this since your example, I think, does not address this. $\endgroup$ – მამუკა ჯიბლაძე Mar 19 '17 at 18:54
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    $\begingroup$ @მამუკაჯიბლაძე Generally proofs of identities by algebraic manipulations (to check a given formula describes matrix coefficients, and to check that the matrices described are a representation) carry over from the integral world to the finite field sum world. So that should be fine as well. Have you tried a straightforward translation of the formulas? $\endgroup$ – Will Sawin Mar 19 '17 at 23:15
  • $\begingroup$ Well in most cases I am not sure which representations correspond to which - like for Kloosterman sums, what is the analog of functions on the unit sphere. But I agree, in each case it must not be difficult to pick the correct one. $\endgroup$ – მამუკა ჯიბლაძე Mar 20 '17 at 4:47

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