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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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##### References:
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