Doubly Almost Bipartite Leonard Pairs

In this paper, we introduce the concepts of compatibility and companion for Leonard pairs. These concepts are roughly described as follows. Let $\mathbb{F}$ denote a field, and let $V$ denote a vector space over $\mathbb{F}$ with finite positive dimension.A Leonard pair on $V$ is an ordered pair of diagonalizable $\mathbb{F}$-linear maps $A : V \to V$ and $A^* : V \to V$ that each act in an irreducible tridiagonal fashion on an eigenbasis for the other one. Leonard pairs $A,A^*$ and $B,B^*$ on $V$ are said to be compatible whenever $A^* = B^*$ and $[A,A^*] = [B,B^*]$, where $[r,s] = r s - s r$. For a Leonard pair $A,A^*$ on $V$, by a companion of $A,A^*$ we mean an $\mathbb{F}$-linear map $K: V \to V$ such that $K$ is a polynomial in $A^*$ and $A-K, A^*$ is a Leonard pair on $V$. The concepts of compatibility and companion are related as follows. For compatible Leonard pairs $A,A^*$ and $B,B^*$ on $V$, define $K = A-B$. Then $K$ is a companion of $A,A^*$. For a Leonard pair $A,A^*$ on $V$ and a companion $K$ of $A,A^*$,define $B = A-K$ and $B^* = A^*$. Then $B,B^*$ is a Leonard pair on $V$ that is compatible with $A,A^*$. Let $A,A^*$ denote a Leonard pair on $V$. We find all the Leonard pairs $B, B^*$ on $V$ that are compatible with $A,A^*$.For each solution $B, B^*$, we describe the corresponding companion $K = A-B$.

Two linear transformations each tridiagonal with respect to an eigenbasis of the other; the TD–D canonical form and the LB–UB canonical form

We first summarize the basic structure of the outer distribution module of a completely regular code. Then, employing a simple lemma concerning eigenvectors in association schemes, we propose to study the tightest case, where the indices of the eigenspace that appear in the outer distribution module are equally spaced. In addition to the arithmetic codes of the companion paper, this highly structured class includes other beautiful examples and we propose the classification of $Q$-polynomial completely regular codes in the Hamming graphs. A key result is Theorem 3.10 which finds that the $Q$-polynomial condition is equivalent to the presence of a certain Leonard pair. This connection has impact in two directions. First, the Leonard pairs are classified and we gain quite a bit of information about the algebraic structure of any code in our class. But also this gives a new setting for the study of Leonard pairs, one closely related to the classical one where a Leonard pair arises from each thin/dual-thin irreducible module of a Terwilliger algebra of some $P$- and $Q$-polynomial association scheme, yet not previously studied. It is particularly interesting that the Leonard pair associated to some code $C$ may belong to one family in the Askey scheme while the distance-regular graph in which the code is found may belong to another.

Let V denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations and that satisfy (i) and (ii) below. There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing is diagonal. There exists a basis for V with respect to which the matrix representing is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V. In the literature, there are some parameters that are used to describe Leonard pairs called the intersection numbers , , , and the dual eigenvalues . In this paper, we provide two characterizations of Leonard pairs. For the first characterization, the focus is on the and . For the second characterization, the focus is on the , , and .

The universal DAHA of type [formula omitted] and Leonard pairs of q-Racah type

A characterization of Leonard pairs using the parameters [formula omitted

Generalizing previous work of Hora (1998) on the asymptotic spectral analysis for the Hamming graph $H(n,q)$ which is the $n^{\mathrm{th}}$ Cartesian power $K_q^{\square n}$ of the complete graph $K_q$ on $q$ vertices, we describe the possible limits of the joint spectral distribution of the pair $(G^{\square n},\overline{G}\vphantom{G}^{\square n})$ of the $n^{\mathrm{th}}$ Cartesian powers of a strongly regular graph $G$ and its complement $\overline{G}$, where we let $n\rightarrow\infty$, and $G$ may vary with $n$. This result is an analogue of the bivariate central limit theorem, and we obtain in this way the bivariate Poisson distributions and the standard bivariate Gaussian distribution, together with the product measures of univariate Poisson and Gaussian distributions. We also report a family of bivariate hypergeometric orthogonal polynomials with respect to the last distributions, which we call the bivariate Charlier-Hermite polynomials, and prove basic formulas for them. This family of orthogonal polynomials seems previously unnoticed, possibly because of its peculiarity.

We report the discovery in the Galactic Centre region of two hard X-ray sources, designated as XMM J174457−2850.3 and XMM J174544-2913.0, which exhibit flux variations in the 2–10 keV band in excess of a factor of 100 in observations spanning roughly a year. In both cases the observed hydrogen column density is consistent with a location near to the Galactic Centre, implying peak X-ray luminosities of ∼5 × 1034 erg s−1. These objects may represent a new population of transient source, with properties very different from those of the much more luminous Galactic Centre transients associated with neutron star and black hole binary systems. Spectral analysis shows that XMM J174457−2850.3 has relatively weak iron-line emission set against a very hard continuum. XMM J174544-2913.0, on the other hand, has an extremely strong K-line from helium-like iron with an equivalent width of ∼2.4 keV. The nature of the latter source is of particular interest. Does it represent an entirely new class of object or does it correspond to a known class of source in a very extreme configuration?

A quantum exactly solvable non-linear oscillator with quasi-harmonic behaviour

In recent years, one of the most interesting developments in quantum mechanics has been the construction of new exactly solvable potentials connected with the appearance of families of exceptional orthogonal polynomials (EOP) in mathematical physics. In contrast with families of (Jacobi, Laguerre and Hermite) classical orthogonal polynomials, which start with a constant, the EOP families begin with some polynomial of degree greater than or equal to one, but still form complete, orthogonal sets with respect to some positive-definite measure. We show how they may appear in the bound-state wavefunctions of some rational extensions of well-known exactly solvable quantum potentials. Such rational extensions are most easily constructed in the framework of supersymmetric quantum mechanics (SUSYQM), where they give rise to a new class of translationally shape invariant potentials. We review the most recent results in this field, which use higher-order SUSYQM. We also comment on some recent re-examinations of the shape invariance condition, which are independent of the EOP construction problem.

For some families of classical orthogonal polynomials defined on appropriate intervals, it is shown that the corresponding Jacobi matrices are totally positive and their bidiagonal factorizations can be accurately computed. By exploiting these facts, an algorithm to compute with high relative accuracy the eigenvalues of those Jacobi matrices, and consequently the nodes of Gaussian quadrature formulae for those families of orthogonal polynomials, is presented. An algorithm is also presented for the computation of the eigenvectors of these Jacobi matrices, and hence the weights of Gaussian quadrature formulae. Although in this case high relative accuracy is not theoretically guaranteed, the numerical experiments with our algorithm provide very accurate results.

Eigenvalues of Toeplitz matrices associated with orthogonal polynomials

Two families of orthogonal polynomials in two discrete variables are constructed for a weight function of q-hypergeometric type. The polynomials are expressed in terms of q-Hahn polynomials. The connection coefficients between the two families involve Askey and Wilson’s ${}_4 \phi _3 $-polynomials, which are certain balanced, terminating basic hypergeometric series. The method is to consider functions on the lattice of subspaces of a finite vector space which are invariant under the subgroup of the corresponding general linear group which fixes a pair of nested subspaces. By limiting methods, corresponding results are obtained for Andrews and Askey’s little q-Jacobi polynomials, which are orthogonal on a countable compact set. The classical Hahn version of the theory, where the underlying group is the symmetric group, has been worked out by the author in a previous paper.

Wronskian type determinants of orthogonal polynomials, Selberg type formulas and constant term identities

We compute the limiting distributions of the lengths of the longest monotone subsequences of random (signed) involutions with or without conditions on the number of fixed points (and negated points) as the sizes of the involutions tend to infinity. The resulting distributions are, depending on the number of fixed points, (1) the Tracy-Widom distributions for the largest eigenvalues of random GOE, GUE, GSE matrices, (2) the normal distribution, or (3) new classes of distributions which interpolate between pairs of the Tracy-Widom distributions. We also consider the second rows of the corresponding Young diagrams. In each case the convergence of moments is also shown. The proof is based on the algebraic work of J. Baik and E. Rains in [7] which establishes a connection between the statistics of random involutions and a family of orthogonal polynomials, and an asymptotic analysis of the orthogonal polynomials which is obtained by extending the Riemann-Hilbert analysis for the orthogonal polynomials by P. Deift, K. Johansson, and Baik in [3].