Discrete Invariant Curve Flows, Orthogonal Polynomials, and Moving Frame

Abstract

In this paper, an orthogonal polynomials-based (OPs-based) approach to generate discrete moving frames and invariants is developed. It is shown that OPs can provide explicit expressions for the discrete moving frame as well as the associated difference invariants, and this approach enables one to obtain the corresponding discrete invariant curve flows simultaneously. Several examples in the cases of centro-affine plane, pseudo-Euclidean plane, and high-dimensional centro-affine space are presented.

Hungry Lotka–Volterra lattice under nonzero boundaries, block‐Hankel determinant solution, and biorthogonal polynomials

“…When we consider 𝑀 = 2 of the hLV lattice (1), our conclusions give generalizations of the lattice and its Lax pair discussed in Ref. [14] to a case of nonzero boundary functions 𝑎 −𝑠 , 𝑠 = 0, 1.…”

“…The biorthogonal polynomials we shall discuss in this paper follow the definition in (14). It is a special case of the symmetric (𝑝, 𝑞)-biorthogonal polynomials mentioned by Maeda et al in Ref.…”

“…In particular, for a fixed positive integer 𝑀, let us consider two sets of symmetric (M,1)-biorthogonal polynomials {𝑃 𝑛 (𝑧)} ∞ 𝑛=0 and {𝑄 𝑛 (𝑧)} ∞ 𝑛=0 satisfying the biorthogonal condition (see also in Refs. [14,22,31])…”

“…Semidiscrete integrable systems are closely related to many branches of mathematics, such as orthogonal polynomials, [1][2][3][4][5][6] differential geometry, [7][8][9][10] dynamical system, [11][12][13] algebraic curve, 14,15 combinatorial numbers, 16,17 and so on. More examples can be found in Refs.…”

“…For simplicity, we shall call Equation (1) as the hLV lattice throughout this paper. It is a semidiscrete completely integrable system 14 and can be seen as an extension of the well-known Lotka-Volterra lattice 31 𝑑𝑎 𝑛 𝑑𝑡 = 𝑎 𝑛 (𝑎 𝑛+1 − 𝑎 𝑛−1 ), 𝑛 = 1, 2, 3, … ,…”