Floreanini, Roberto; Vinet, Luc Symmetries of the \(q\)-difference heat equation. (English) Zbl 0805.39008 Lett. Math. Phys. 32, No. 1, 37-44 (1994). 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. Reviewer: E.Trofimtchouk (Kiev) Cited in 3 ReviewsCited in 12 Documents 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 Keywords:\(q\)-difference heat equation; \(q\)-Hermite polynomials; symmetry operators; symmetry algebra; dilatation symmetry PDF BibTeX XML Cite \textit{R. Floreanini} and \textit{L. Vinet}, Lett. Math. Phys. 32, No. 1, 37--44 (1994; Zbl 0805.39008) Full Text: DOI OpenURL References: [1] Gasper, G. and Rahman, M.,Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990. · Zbl 0695.33001 [2] Floreanini, R. and Vinet, L., Quantum algebras andq-special functions,Ann. Phys. 221 (1993), 53-79. · Zbl 0773.33010 [3] Miller, W.,Symmetry and Separation of Variables, Addison-Wesley, Reading, Mass, 1977. · Zbl 0368.35002 [4] Blumen, G. and Cole, J., The general similarity solution of the heat equation,J. Math. Mech. 18, 1025-1042 (1969); Kalnins, E. and Miller, W., Lie theory and separation of variables, 5: The equationsiU t +U xx = 0 andiU t +U xx ?c/x 2 U = 0,J. Math. Phys. 15, 1728-1737 (1974). [5] Al-Salam, W. A. and Carlitz, L., Some orthogonalq-polynomials,Math. Nachr. 30, 47-61 (1965). · Zbl 0135.27802 [6] Floreanini, R. and Vinet, L., Quantum symmetries ofq-difference equations, CRM-preprint, 1994. · Zbl 0805.39008 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.