Modified Extensible Lattice Sequence

• The paper introduces a two-dimensional integrable system via a bilinear recurrence that generalizes the Heideman–Hogan recurrence.
• It employs Dodgson condensation to derive finite 6-term linear relations in both lattice directions, ensuring linearizability despite nonlinear dynamics.
• The sequence maintains the Laurent property, irreducibility, and coprimeness, with extensions linking it to Dana–Scott frieze patterns and one-dimensional reductions.

A modified extensible lattice sequence refers to a two-dimensional discrete integrable system that generalizes the linearizable Somos-like Heideman–Hogan recurrence, introduced and studied in depth by Kamiya, Kanki, Mase, and Tokihiro (Kamiya et al., 2017). Its defining equation, bilinear in character, presents rich algebraic and integrability properties, notably linearizability in both lattice directions, the Laurent property, and coprimeness of iterates. The model is deeply connected to the combinatorial structure of frieze patterns and links to higher-order integrable lattice equations such as those in the Dana–Scott family.

1. Definition and Fundamental Equation

The modified extensible lattice sequence is defined on $\mathbb{Z}^2$ via the bilinear relation

$$I_{n+2,t+1}\,I_{n,t} = I_{n,t+1}\,I_{n+2,t} + I_{n+1,t+1} + I_{n+1,t}, \qquad (n,t) \in \mathbb{Z}^2.$$

Alternatively, this can be written as the nonlinear recurrence

$$I_{n+2,t+1} = \frac{I_{n,t+1} I_{n+2,t} + I_{n+1,t+1} + I_{n+1,t}}{I_{n,t}},$$

so iteration at each site is determined by nearest-neighbor data and values along three “half-lines” serving as initial boundaries: $t=0$, $n=0$, and $n=1$.

This lattice equation is a two-dimensional generalization of the Heideman–Hogan recurrence, which in one dimension is a linearizable mapping connected to Somos recurrences, thereby relating to the Laurent phenomenon and frieze patterns (Kamiya et al., 2017).

2. Linearizability via Dodgson Condensation

A salient property of the modified extensible lattice sequence is linearizability in both lattice directions. This feature is established using Dodgson condensation (Desnanot–Jacobi identity), which asserts that for the $4 \times 4$ stencil matrix built from $I_{n,t}$ and its translates,

$X_4(n,t) = \left( I_{n+2i,t+2j} \right)_{i,j=0}^3,$

the determinant $D_4(n,t) = \det X_4(n,t)$ vanishes. This rank deficiency yields nontrivial linear relations. By expanding the determinant, one obtains a 6-term linear relation in $n$ and, analogously, in $t$: $$I_{n+6,t} + a(n) I_{n+4,t} + B(n) I_{n+2,t} - I_{n,t} = 0,$$

$$I_{n,t+6} + \alpha(n) I_{n,t+4} + \beta(n) I_{n,t+2} - I_{n,t} = 0,$$

where coefficients depend only on one coordinate. The system is therefore linearizable along each direction, despite its original nonlinear, bilinear formulation (Kamiya et al., 2017). This mechanism generalizes to higher-order stencils in Dana–Scott-type extensions.

3. Laurent Property, Irreducibility, and Coprimeness

The sequence $(I_{n,t})$ exhibits the strong Laurent property. Every iterate $I_{n,t}$, evolved from initial values along the three half-lines, is a Laurent polynomial in those variables with integer coefficients. Furthermore, the sequence demonstrates irreducibility: each $I_{n,t}$ is irreducible as a Laurent polynomial, and any two distinct iterates $I_{n,t}$ and $I_{m,s}$ have no non-monomial common factors. The proof proceeds by double induction using the bilinear recurrence, showing closure under the Laurent ring structure and ensuring that factorization and coprimeness propagate globally from local structure (Kamiya et al., 2017).

4. Dimensional Reduction to Generalized Heideman–Hogan Recurrences

Imposing a periodicity constraint

$I_{n,t} = I_{n+M, t-K}, \qquad \gcd(M,K) = 1,$

enables reduction of the two-dimensional system to a one-dimensional recurrence. Defining $a_j := I_{n,t}$ along the line $j = nK + tM$ yields

$$a_{j+2K+M} a_j = a_{j+M} a_{j+2K} + a_{j+M+K} + a_{j+K},$$

which includes the original Heideman–Hogan recurrence as a special case. The associated linear relations reduce similarly, giving rise to periodic-coefficient linear recurrences in the one-dimensional setting (Kamiya et al., 2017).

5. Higher-Order and Dana–Scott Family Extensions

The framework of the modified extensible lattice sequence admits generalization to the Dana–Scott family of two-frieze equations. The classical case,

$$I_{n,t} I_{n+2,t+2} = I_{n+2,t} I_{n,t+2} + I_{n+1,t+1},$$

serves as the template. For order $k$, the determinant

$$F_k(n,t) = \det \bigl[I_{n+2(i-1),\,t+2(j-1)}\bigr]_{i,j=1}^k$$

is considered, and higher minors are set to prescribed shifts. This approach generates new equations with $2k+2$ or $2k+3$ term linear relations and admits reductions to nonlinear recurrences of order $4k+4$ or $4k+6$. While these higher-order extensions preserve linearizability, the Laurent property is partially lost, constrained to allow a finite set of fixed monomial denominators (Kamiya et al., 2017).

6. Illustrative Special Cases

Notable special situations include:

• $k=0$ in the Dana–Scott family yields the cross-ratio equation

$I_{n,t}\,I_{n+2,t+2} = I_{n+2,t}\,I_{n,t+2} + 1.$

• $k=1$ gives rise to a $3\times3$ determinant formula, directly producing a 6-term linear relation.
• Reduction parameters $M=K=1$ in the core recursion recover the standard Heideman–Hogan mapping.

These examples clarify the unifying algebraic structure underpinning the modified extensible lattice sequence and its relatives. In all circumstances, Dodgson condensation and the vanishing of higher minors force finite-length linear relations, ensuring linearizability; meanwhile, the nonlinear forms maintain the Laurent and coprimeness properties except in certain generalized examples (Kamiya et al., 2017).