Lie symmetries of finite‐difference equations

Discretizations of the Helmholtz, heat, and wave equations on uniform lattices are considered in various space–time dimensions. The symmetry properties of these finite‐difference equations are determined and it is found that they retain the same Lie symmetry algebras as their continuum limits. Solutions with definite transformation properties are obtained; identities and formulas for these functions are then derived using the symmetry algebra.

[1]  J. Miller.  Lie Theory and Generalizations of the Hypergeometric Functions , 1973 .

[2]  L. Vinet,et al.  Quantum Mechanics and Polynomials of a Discrete Variable , 1993 .

[3]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[4]  V. Kac,et al.  Quasifinite highest weight modules over the Lie algebra of differential operators on the circle , 1993, hep-th/9308153.

[5]  L. Vinet,et al.  Symmetries of theq-difference heat equation , 1994 .

[6]  N. Vilenkin.  Special Functions and the Theory of Group Representations , 1968 .

[7]  L. Vinet,et al.  Quantum symmetries of q‐difference equations , 1995 .

[8]  G. Quispel,et al.  Factorizable Lie symmetries and the linearization of difference equations , 1995 .

[9]  H. Feshbach,et al.  Finite Difference Equations , 1959 .

[10]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[11]  D. Levi,et al.  Continuous symmetries of discrete equations , 1991 .

[12]  W. Rheinboldt,et al.  Generalized hypergeometric functions , 1968 .

[13]  Michio Jimbo,et al.  Algebraic Analysis of Solvable Lattice Models. , 1994 .

[14]  L. Vinet,et al.  Symmetries of the heat equation on the lattice , 1996 .

[15]  Wilhelm Magnus,et al.  Lie Theory and Special Functions , 1969 .

[16]  V. B. Uvarov,et al.  Special Functions of Mathematical Physics: A Unified Introduction with Applications , 1988 .

[17]  F. H. Jackson.  q-Difference Equations , 1910 .

[18]  W. Miller,et al.  Canonical equations and symmetry techniques for q -series , 1987 .

[19]  W. Miller.  Lie theory and difference equations. II , 1969 .

[20]  W. Miller.  Symmetry Techniques and Orthogonality for q -Series , 1989 .