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Satoshi Tsujimoto

Publications and source records attributed to Satoshi Tsujimoto.

At least 19 recordsLinked to original sources
arXiv2026

Rational Heun operators on $q$-linear grids

Rational Heun operators on the $q-$linear grid are presented. They are second-order $q-$difference operators $W_q$ constructively defined from the requirement that they have a raising action on rational functions of type $[n/n]$, namely $W_q: [n/n] \rightarrow [n+1/n+1]$, with poles on $q-$linear grids. It will be observed that these operators are related to one family of the Ruijsenaars-van Diejen-Takemura Hamiltonians. A distinguished subclass of $W_q$ called classical which shifts the pole structure while preserving the rational function type and a prescribed basis is also characterized.

math-ph↗
arXiv2026

The dynamical algebra of the generic superintegrable model on the two-sphere

The rank two Jacobi algebra $\mathfrak{J}_2$ is identified as the dynamical algebra of the generic quadratic superintegrable model on the two-sphere. The physical representation of this algebra is obtained from its embedding in $\mathfrak{su}(1,1)^{\otimes 3}$. The exact solution of the model is derived algebraically from this representation. The wavefunctions are found to be expressed in terms of two-variable Jacobi polynomials whose characterization is a by-product of the algebraic treatment of the model.

math-ph↗
arXiv2026

Meta Algebras and Special Functions: the Racah Case

Finite families of biorthogonal rational functions and orthogonal polynomials of Racah-type are studied within a unified algebraic framework based on the meta Racah algebra and its finite-dimensional representations. These functions are identified as overlap coefficients between eigensolutions of generalized and standard eigenvalue problems posited on the representation space. The approach naturally yields their orthogonality relations and bispectral properties.

math.CA↗
arXiv2025

Algebras behind the bispectrality of the Wilson rational functions and their ${}_4ϕ_3$ limits

The properties of the Wilson rational functions ${}_{10}ϕ_9$ with three different normalizations are described. For one normalization, it satisfies an $R_{II}$ recurrence relation, whereas for the two other ones, they satisfy a generalized eigenvalue problem. The so-called Wilson rational algebra is introduced, which encodes algebraically the spectral properties of these special functions. Finally, different limits are considered, leading up to functions proportional to ${}_{4}ϕ_3$. For one of these, the spectral algebra simplifies to yield the meta $q$-Racah algebra.

math-ph↗
arXiv2025

Algebraic interpretation of the two-variable Jacobi polynomials on the triangle: the pentagonal way

The rank two Jacobi algebra $\mathcal{J}_2$ is used to provide an interpretation of the two-variable Jacobi polynomials $J_{n,k}^{(a,b,c)}(x,y)$ on the triangle, as overlaps between two representation bases. The subalgebra structure of $\mathcal{J}_2$ depicted via a pentagonal graph is exploited to find the explicit expression of the two-variable functions in terms of univariate Jacobi polynomials. It is also seen to provide an explanation for the fact that the expansion on the basis $J_{n,k}^{(a,b,c)}(x,y)$ of the polynomials obtained from the latter by permuting the variables $x,y, z=1-x-y$ and the parameters $(a,b,c)$ is given in terms of Racah polynomials. The underlying order-three symmetry is discussed.

math.RT↗
arXiv2025

Linearization of the box-ball system with box capacity L

We construct a bijection between the state of the box-ball system with box capacity L and a pair of two sequences. In time evolution, one of the sequences moves at speed 1, and the other follows the rules of the box-ball system with box capacity one, which can be linearized by the Kerov-Kirillov-Reshetikhin(KKR) bijection. Our method can be applied to a state including a negative value or a value greater than the box capacity.

nlin.SI↗
arXiv2025

The rank two Jacobi algebra

The quadratic rank two Jacobi algebra is identified from the relations obeyed by the bispectral operators of the two variable Jacobi polynomials orthogonal on the triangle. It is seen to admit as subalgebras Racah and Jacobi algebras of rank one. The dual realizations in terms of differential operators in the variable representation and in terms of difference operators in the degree representation are provided. Structure relations for the two variable Jacobi polynomials are obtained as a by product.

math-ph↗
arXiv2025

Cauchy-Jacobi orthogonal polynomials and the discrete CKP equation

This paper intends to construct discrete spectral transformations for Cauchy-Jacobi orthogonal polynomials, and find its corresponding discrete integrable systems. It turns out that the normalization factor of Cauchy-Jacobi orthogonal polynomials acts as the $τ$-function of the discrete CKP equation, which has applications in Yang-Baxter equation, integrable geometry, cluster algebra, and so on.

math-ph↗
arXiv2025

Change of basis for the tridiagonal pairs of type II

We study tridiagonal pairs of type II. These involve two linear transformations $A$ and $A^\star$. We define two bases. In the first one, $A$ acts as a diagonal matrix while $A^\star$ acts as a block tridiagonal matrix, and in the second one, $A$ acts as a block tridiagonal matrix while $A^\star$ acts as a diagonal matrix. We obtain the change of basis coefficients between these two bases. The coefficients are special functions that are written as a nested product of polynomials that resemble Racah polynomials but involve shift operators in their expression.

math.RA↗
arXiv2024

Solitons in 3-State Mealy Automata

Box--ball systems (BBS) are integrable systems with soliton solutions and other good properties. We will search for automata that belong to the same class as BBS automata by introducing some classes of automata through the features of BBS automaton. In particular, we would like to classify 3-state automata over a 2-letter alphabet.

nlin.SI↗
arXiv2024

Meta algebras and biorthogonal rational functions: the $q$-Hahn case

A unified algebraic interpretation of both finite families of orthogonal polynomials and biorthogonal rational functions of $q$-Hahn type is provided. The approach relies on the meta $q$-Hahn algebra and its finite-dimensional bidiagonal representations. The functions of $q$-Hahn type are identified as overlaps (up to global factors) between bases solving ordinary or generalized eigenvalue problems in the representation of the meta $q$-Hahn algebra. Moreover, (bi)orthogonality relations, recurrence relations, difference equations and some contiguity relations satisfied by these functions are recovered algebraically using the actions of the generators of the meta $q$-Hahn algebra on various bases.

math.RT↗
arXiv2024

Meta Algebras and Biorthogonal Rational Functions: The Hahn Case

The finite families of biorthogonal rational functions and orthogonal polynomials of Hahn type are interpreted algebraically in a unified way by considering the three-generated meta Hahn algebra and its finite-dimensional representations. The functions of interest arise as overlaps between eigensolutions of generalized and ordinary eigenvalue problems on the representation space. The orthogonality relations and bispectral properties naturally follow from the framework.

math-ph↗
arXiv2024

Recurrence relations of Exceptional Laurent biorthogonal polynomials

Exceptional extensions of a class of Laurent biorthogonal polynomials (the so-called Hendriksen-van Rossum polynomials) have been presented by the authors recently. This is achieved through Darboux transformations of generalized eigenvalue problems. In this paper, we discuss the recurrence relations satisfied by these exceptional Laurent biorthogonal polynomials and provide a type of recurrence relations with $3l_0+4$ terms explicitly, where the parameter $l_0$ corresponds to the degree of the polynomial part in the seed function used in the Darboux transformation. In the proof of these recurrence relations, the backward operator which maps an exceptional polynomial into a classical one plays a significant role.

math.CA↗
arXiv2023

Quantum walks on simplexes and multiple perfect state transfer

In this paper, we study quantum walks on the extension of association schemes. Various state transfers can be achieved on these graphs, such as multiple state transfer among extreme points of a simplex, fractional revival on subsimplexes. Since only few examples of multiple (perfect) state transfer are known, we aim to make some additions in this collection.

quant-ph↗
arXiv2023

An isospectral transformation between Hessenberg matrix and Hessenberg-bidiagonal matrix pencil without using subtraction

We introduce an eigenvalue-preserving transformation algorithm from the generalized eigenvalue problem by matrix pencil of the upper and the lower bidiagonal matrices into a standard eigenvalue problem while preserving sparsity, using the theory of orthogonal polynomials. The procedure is formulated without subtraction, which causes numerical instability. Furthermore, the algorithm is discussed for the extended case where the upper bidiagonal matrix is of Hessenberg type.

math.NA↗
arXiv2022

Exceptional Laurent biorthogonal polynomials through spectral transformations of generalized eigenvalue problems

A formulation is given for the spectral transformation of the generalized eigenvalue problem through the decomposition of the second-order differential operators. This allows us to construct some Laurent biorthogonal polynomial systems with gaps in the degree of the polynomial sequence. These correspond to an exceptional-type extension of the orthogonal polynomials, as an extension of the Laurent biorthogonal polynomials. Specifically, we construct the exceptional extension of the Hendriksen-van Rossum polynomials, which are biorthogonal analogs of the classical orthogonal polynomials. Similar to the cases of exceptional extensions of classical orthogonal polynomials, both of state-deletion and state-addition occur.

math.CA↗
arXiv2022

Bi-infinite solutions for KdV- and Toda-type discrete integrable systems based on path encodings

We define bi-infinite versions of four well-studied discrete integrable models, namely the ultra-discrete KdV equation, the discrete KdV equation, the ultra-discrete Toda equation, and the discrete Toda equation. For each equation, we show that there exists a unique solution to the initial value problem when the given data lies within a certain class, which includes the support of many shift ergodic measures. Our unified approach, which is also applicable to other integrable systems defined locally via lattice maps, involves the introduction of a path encoding (that is, a certain antiderivative) of the model configuration, for which we are able to describe the dynamics more generally than in previous work on finite size systems, periodic systems and semi-infinite systems. In particular, in each case we show that the behaviour of the system is characterized by a generalization of the classical 'Pitman's transformation' of reflection in the past maximum, which is well-known to probabilists. The picture presented here also provides a means to identify a natural 'carrier process' for configurations within the given class, and is convenient for checking that the systems we discuss are all-time reversible. Finally, we investigate links between the different systems, such as showing that bi-infinite all-time solutions for the ultra-discrete KdV (resp. Toda) equation may appear as ultra-discretizations of corresponding solutions for the discrete KdV (resp. Toda) equation.

nlin.SI↗