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Symmetries of the \(q\)-difference heat equation. (English) Zbl 0805.39008
The authors consider three \(q\)-difference analogs of the heat equation in one space dimension. The symmetry operators of these \(q\)-difference equations as well as relations defining a symmetry algebra are determined. (Note that all considered \(q\)-deformations of the heat equation have the same symmetry algebra). For the \(q\)-difference heat equation, which has symmetry operators of the simplest form, an interesting representation of solutions involving \(q\)-Hermite polynomials are obtained. For this purpose the authors perform the separation of variables associated to the dilatation symmetry.

MSC:
39A10 Additive difference equations
81R99 Groups and algebras in quantum theory
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
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