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.


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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