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Alexei Zhedanov

Publications and source records attributed to Alexei Zhedanov.

At least 19 recordsLinked to original sources
arXiv2026

Morse momentum wavefunctions and rational functions

We revisit the bound states of the Morse potential in the momentum representation. After the ground-state factor is extracted, the remaining factors are finite rational functions of the momentum variable. These functions are eigenfunctions of a second-order difference operator and are identified with the symmetric specialization of a finite family of biorthogonal rational functions introduced by Koepf and Masjed-Jamei. They also satisfy a generalized eigenvalue problem in the degree variable, thereby placing the Morse momentum wavefunctions within the framework of rational bispectrality and $R_{II}$-type systems. Finally, after extraction of their poles, the same wavefunctions are expressed in terms of Meixner--Pollaczek polynomials with degree-dependent parameters. This gives a simple description of their zeros. The Morse potential thus provides a concrete quantum-mechanical realization of finite biorthogonal rational functions.

math-ph↗
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 2D Smorodinsky--Winternitz II system and the Laguerre--Heun algebra

We identify the quadratic symmetry algebra of the two-dimensional Smorodinsky--Winternitz II system with a Laguerre-type confluent Heun algebra. The system is separable in Cartesian and parabolic coordinates. The complementary Cartesian separation operator \[ Y=\partial_y^2-\omega^2y^2+\frac{1/4-c^2}{y^2} \] is of Laguerre type, while the parabolic integral \(W=L_2\) is its algebraic Heun partner. With \(Z=[Y,W]\), the defining relations are \[ [Y,Z]=16\omega^2W-2bY,\qquad [W,Z]=6Y^2-4HY+2bW+8\omega^2(1-c^2), \] where \(H\) is central. This gives a direct superintegrable realization of the Laguerre--Heun algebra.

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

Mirror symmetric polynomials orthogonal on the unit circle

We introduce and study a special family of polynomials orthogonal on the unit circle (OPUC). These OPUC satisfy a mirror symmetry property of their Verblunsky coefficients. Several equivalent conditions for the OPUC to be mirror symmetric are presented. Corresponding unitary CMV matrices satisfy simple algebraic relations similar to relations for persymmetric tridiagonal matrices. We present three explicit examples of mirror symmetric OPUC.

math.CA↗
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

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

Eigenvalue equations for sieved polynomials or proving Askey right again

The sieved Jacobi polynomials have been introduced by Askey. These can be obtained from conveniently taking $q$ to be a root of unity in the Askey-Wilson polynomials. The question of determining if they are eigenfunctions of some operator has been lingering for a long time. Askey impressed on us his conviction that it had an affirmative answer. It is shown that he was right and that this operator is of Dunkl type with cyclic reflections corresponding to the powers of $q$.

math.CA↗
arXiv2025

Bispectrality of the sieved Jacobi polynomials

It is shown that the CMV Laurent polynomials associated to the sieved Jacobi polynomials on the unit circle satisfy an eigenvalue equation with respect to a first order differential operator of Dunkl type. Using this result, the sieved Jacobi polynomials on the real line are found to be eigenfunctions of a Dunkl differential operator of second order. Eigenvalue equations for the sieved ultraspherical polynomials of the first and second kind are obtained as special cases. These results mean that the sieved Jacobi polynomials (either on the unit circle or on the real line) are bispectral.

math.CA↗
arXiv2024

The CMV bispectrality of the Jacobi polynomials on the unit circle

We show that the Jacobi polynomials that are orthogonal on the unit circle (the Jacobi OPUC) are CMV bispectral. This means that the corresponding Laurent polynomials in the CMV basis satisfy two dual ordinary eigenvalue problems: a recurrence relation and a differential equation of Dunkl type. This is presumably the first nontrivial explicit example of CMV bispectral OPUC. We introduce the circle Jacobi algebra which plays the role of hidden symmetry algebra for the Jacobi OPUC. All fundamental properties of the Jacobi OPUC can be derived from representations of this algebra.

math.CA↗
arXiv2024

Spectral surgery and high-fidelity quantum state transfer in $XX$ chains

We consider an inhomogeneous $XX$ spin chain which interpolates between the Krawtchouk one with perfect state transfer and the homogeneous $XX$ chain. This model can be used to perform qubit state transfer with sufficiently high fidelity. The advantage of this model with respect to the Krawtchouk chain is that while maintaining high transfer fidelity, the coupling strengths are capped and do not become excessively large as the number of sites grows. The construction is fully analytic and is based on spectral surgery transformations of the homogeneous chain.

quant-ph↗
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

The $q-$Onsager algebra and multivariable $q-$special functions

Two sets of mutually commuting $q-$difference operators $x_i$ and $y_j$, $i,j=1, ...,N$ such that $x_i$ and $y_i$ generate a homomorphic image of the $q-$Onsager algebra for each $i$ are introduced. The common polynomial eigenfunctions of each set are found to be entangled product of elementary Pochhammer functions in $N$ variables and $N+3$ parameters. Under certain conditions on the parameters, they form two `dual' bases of polynomials in $N$ variables. The action of each operator with respect to its dual basis is block tridiagonal. The overlap coefficients between the two dual bases are expressed as entangled products of $q-$Racah polynomials and satisfy an orthogonality relation. The overlap coefficients between either one of these bases and the multivariable monomial basis are also considered. One obtains in this case entangled products of dual $q-$Krawtchouk polynomials. Finally, the `split' basis in which the two families of operators act as block bidiagonal matrices is also provided.

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arXiv2023

Para-Bannai-Ito Polynomials

New bispectral polynomials orthogonal on a Bannai-Ito bi-lattice (uniform quadri-lattice) are obtained from an unconventional truncation of the untruncated Bannai-Ito and complementary Bannai-Ito polynomials. A complete characterization of the resulting para-Bannai-Ito polynomials is provided, including a three term recurrence relation, a Dunkl-difference equation, an explicit expression in terms of hypergeometric series and an orthogonality relation. They are also derived as a $q\to -1$ limit of the $q$-para-Racah polynomials. A connection to the dual $-1$ Hahn polynomials is also established.

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