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  • What are some of interesting results that arise from using difference equations in number theory , Combinatorics or any other field ?
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You may want to clarify what qualifies as outside the theory of difference equations. Would you consider discrete integrable systems, q-Painleve equations, discrete complex analysis etc. as examples? –  Gjergji Zaimi Dec 26 '11 at 10:54
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Since you're after a range of answers, and not a single "correct" one, I suggest that you make this question "community wiki" - you should be able to edit the question and click an appropriate box to do so –  Yemon Choi Dec 26 '11 at 10:55
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In nonstandard analysis, what is classically called a differential equation becomes a difference equation. –  Robert Kucharczyk Dec 26 '11 at 15:48
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Hrushovski used the model theory of difference fields to give another proof of the Manin-Mumford conjecture.

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Along the same lines the model theory of difference fields has been used to study arithmetic properties of algebraic dynamical systems, algebraic relations amongst special functions, difference Galois theory, the Tate-Voloch conjecture and other diophantine problems of Manin-Mumford/André-Oort-type. (See some of the notes at: math.u-psud.fr/~bouscare/workshop_diff/index.html ) –  Thomas Scanlon Dec 27 '11 at 7:52
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In signal processing, digital filtering, ARMA modelesation, use difference equations. In Linear prediction Coding, we seek a difference operator such that:

$x(n)=\sum_p{a(n)x(n-p)}+e(n)$

where $x(n)$ is for example a time serie (the uknown function in the difference equation). The goal of this coding is to find $a(n)$ by minimizing the error $e(n)$ so this is an example of an inverse problem for the difference operator given by the above equation. Like diffrential operators which are diagonalized by the Fourier transform, difference operators are diagonalized by the Z-Transform defined by:

$X(Z)= \sum_n{x(n)Z^{-n}}$

where $Z$ is a complex variable lying on the unit circle of the complex plane ($Z=\exp(-if)$). So a difference operator is transformed via the Z-transform to a multiplication operator by a polynom $P(Z)=\sum_p{a(p)Z^{-p}}$ and one method for the solution of the problem is the use of orthogonal polynoms over the unit circle.

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In analysis over fields of positive characteristic, the role of differential operators is played by special difference operators (the Carlitz derivative and its generalizations). In particular, the main special functions of that theory satisfy some difference equations. For the details see my book "Analysis in Positive Characteristic" (Cambridge University Press, 2009).

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The three-term recurrence relation satisfied by a family of orthogonal polynomials is a crucial fact which brings together classical analysis, spectral theory and other branches of mathematics. This recurrence relation is obviously an example of a difference equation.

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To expand on just one of the many directions implicit in Andrei's answer, Cherednik's proof of the Macdonald constant term conjecture relies fundamentally on double affine Hecke algebras (DAHAs). To quote Opdam's beautiful Math Review of Cherednik's 1995 Annals paper, "It is the great contribution of Cherednik that he perceived what the constant term conjecture has been trying to convey to us all the time, which is the existence of the double affine Hecke algebra." DAHAs are, <i>very</i> roughly, algebras of (twisted) difference operators (with an additional Weyl group built in). –  Thomas Nevins Dec 28 '11 at 15:01
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