Multi-indexed Orthogonal Polynomials Research Articles

‘Diophantine’ and factorization properties of finite orthogonal polynomials in the Askey scheme

A new interpretation and applications of the ‘Diophantine’ and factorization properties of finite orthogonal polynomials in the Askey scheme are explored. The corresponding twelve polynomials are the (q-)Racah, (dual, q-)Hahn, Krawtchouk and five types of q-Krawtchouk. These (q-)hypergeometric polynomials, defined only for the degrees of 0 , 1 , … , N , constitute the main part of the eigenvectors of N + 1-dimensional tri-diagonal real symmetric matrices, which correspond to the difference equations governing the polynomials. The monic versions of these polynomials all exhibit the ‘Diophantine’ and factorization properties at higher degrees than N. This simply means that these higher degree polynomials are zero-norm ‘eigenvectors’ of the N + 1-dimensional tri-diagonal real symmetric matrices. A new type of multi-indexed orthogonal polynomials belonging to these twelve polynomials could be introduced by using the higher degree polynomials as the seed solutions of the multiple Darboux transformations for the corresponding matrix eigenvalue problems. The shape-invariance properties of the simplest type of the multi-indexed polynomials are demonstrated. The explicit transformation formulas are presented.

New finite-type multi-indexed orthogonal polynomials obtained from state-adding Darboux transformations

Abstract The Hamiltonians of finite-type discrete quantum mechanics with real shifts are real symmetric matrices of order N + 1. We discuss the Darboux transformations with higher-degree (>N) polynomial solutions as seed solutions. They are state-adding and the resulting Hamiltonians after M steps are of order N + M + 1. Based on 12 orthogonal polynomials ((q-)Racah, (dual, q-)Hahn, Krawtchouk, and five types of q-Krawtchouk), new finite-type multi-indexed orthogonal polynomials are obtained, which satisfy second-order difference equations, and all the eigenvectors of the deformed Hamiltonian are described by them. We also present explicit forms of the Krein–Adler-type multi-indexed orthogonal polynomials and their difference equations, which are obtained from the state-deleting Darboux transformations with lower-degree (≤N) polynomial solutions as seed solutions.

Discrete orthogonality relations for the multi-indexed orthogonal polynomials in discrete quantum mechanics with pure imaginary shifts

The discrete orthogonality relations for the multi-indexed orthogonal polynomials in discrete quantum mechanics with pure imaginary shifts are investigated. We show that the discrete orthogonality relations hold for the case-(1) multi-indexed orthogonal polynomials of continuous Hahn, Wilson, and Askey–Wilson types, and we conjecture their normalization constants.

Discrete orthogonality relations for multi-indexed Laguerre and Jacobi polynomials

The discrete orthogonality relations hold for all the orthogonal polynomials obeying three term recurrence relations. We show that they also hold for multi-indexed Laguerre and Jacobi polynomials, which are new orthogonal polynomials obtained by deforming these classical orthogonal polynomials. The discrete orthogonality relations could be considered as a more encompassing characterization of orthogonal polynomials than the three term recurrence relations. As the multi-indexed orthogonal polynomials start at a positive degree ℓD≥1, the three term recurrence relations are broken. The extra ℓD “lower degree polynomials,” which are necessary for the discrete orthogonality relations, are identified. The corresponding Christoffel numbers are determined. The main results are obtained by the blow-up analysis of the second order differential operators governing the multi-indexed orthogonal polynomials around the zeros of these polynomials at a degree ℓD+N. The discrete orthogonality relations are shown to hold for another group of “new” orthogonal polynomials called Krein–Adler polynomials based on the Hermite, Laguerre, and Jacobi polynomials.

Recurrence relations of the multi-indexed orthogonal polynomials. VI. Meixner–Pollaczek and continuous Hahn types

In previous papers, we discussed the recurrence relations of the multi-indexed orthogonal polynomials of the Laguerre, Jacobi, Wilson, Askey–Wilson, Racah, and q-Racah types. In this paper, we explore those of the Meixner–Pollaczek and continuous Hahn types. For the M-indexed Meixner–Pollaczek and continuous Hahn polynomials, we present 3 + 2M term recurrence relations with variable dependent coefficients and 1 + 2L term (L ≥ M + 1) recurrence relations with constant coefficients. Based on the latter, the generalized closure relations and the creation/annihilation operators of the quantum mechanical systems described by the multi-indexed Meixner–Pollaczek and continuous Hahn polynomials are obtained.

Exactly solvable discrete quantum mechanical systems and multi-indexed orthogonal polynomials of the continuous Hahn and Meixner–Pollaczek types

Abstract We present new exactly solvable systems of the discrete quantum mechanics with pure imaginary shifts, whose physical range of coordinates is a whole real line. These systems are shape invariant and their eigenfunctions are described by the multi-indexed continuous Hahn and Meixner–Pollaczek orthogonal polynomials. The set of degrees of these multi-indexed polynomials is $\{\ell_{\mathcal{D}},\ell_{\mathcal{D}}+1,\ell_{\mathcal{D}}+2,\ldots\}$, where $\ell_{\mathcal{D}}$ is an even positive integer ($\mathcal{D}$: a multi-index set), but they form a complete set of orthogonal basis in the weighted Hilbert space.

多添字(q-)Racah直交多項式の双対多項式と解ける離散量子力学模型

多添字(q-)Racah直交多項式の双対多項式と解ける離散量子力学模型

Recurrence relations of the multi-indexed orthogonal polynomials V: Racah and q -Racah types

In previous papers, we discussed the recurrence relations of the multi-indexed orthogonal polynomials of the Laguerre, Jacobi, Wilson, and Askey-Wilson types. In this paper, we explore those of the Racah and q-Racah types. For the M-indexed (q-)Racah polynomials, we derive 3 + 2M term recurrence relations with variable dependent coefficients and 1 + 2L term (L ≥ M + 1) recurrence relations with constant coefficients. Based on the latter, the generalized closure relations and the creation and annihilation operators of the quantum mechanical systems described by the multi-indexed (q-)Racah polynomials are obtained. In Appendix B and Appendix C, we present a proof and some data of the recurrence relations with constant coefficients for the multi-indexed Wilson and Askey-Wilson polynomials.

Dual polynomials of the multi-indexed (q-)Racah orthogonal polynomials

We consider dual polynomials of the multi-indexed ($q$-)Racah orthogonal polynomials. The $M$-indexed ($q$-)Racah polynomials satisfy the second order difference equations and various $1+2L$ ($L\geq M+1$) term recurrence relations with constant coefficients. Therefore their dual polynomials satisfy the three term recurrence relations and various $2L$-th order difference equations. This means that the dual multi-indexed ($q$-)Racah polynomials are ordinary orthogonal polynomials and the Krall-type. We obtain new exactly solvable discrete quantum mechanics with real shifts, whose eigenvectors are described by the dual multi-indexed ($q$-)Racah polynomials. These quantum systems satisfy the closure relations, from which the creation/annihilation operators are obtained, but they are not shape invariant.

New determinant expressions of multi-indexed orthogonal polynomials in discrete quantum mechanics

The multi-indexed orthogonal polynomials (the Meixner, little $q$-Jacobi (Laguerre), ($q$-)Racah, Wilson, Askey-Wilson types) satisfying second order difference equations were constructed in discrete quantum mechanics. They are polynomials in the sinusoidal coordinates $\eta(x)$ ($x$ is the coordinate of quantum system) and expressed in terms of the Casorati determinants whose matrix elements are functions of $x$ at various points. By using shape invariance properties, we derive various equivalent determinant expressions, especially those whose matrix elements are functions of the same point $x$. Except for the ($q$-)Racah case, they can be expressed in terms of $\eta$ only, without explicit $x$-dependence.

Multi-indexed Meixner and little q-Jacobi (Laguerre) polynomials

As the fourth stage of the project multi-indexed orthogonal polynomials, we present the multi-indexed Meixner and little q-Jacobi (Laguerre) polynomials in the framework of ‘discrete quantum mechanics’ with real shifts defined on the semi-infinite lattice in one dimension. They are obtained, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier, from the quantum mechanical systems corresponding to the original orthogonal polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum–Krein–Adler deletion of virtual state vectors. The virtual state vectors are the solutions of the matrix Schrödinger equation on all the lattice points having negative energies and infinite norm. This is in good contrast to the (q-)Racah systems defined on a finite lattice, in which the ‘virtual state’ vectors satisfy the matrix Schrödinger equation except for one of the two boundary points.

Recurrence relations of the multi-indexed orthogonal polynomials. IV. Closure relations and creation/annihilation operators

We consider the exactly solvable quantum mechanical systems whose eigenfunctions are described by the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson, and Askey-Wilson types. Corresponding to the recurrence relations with constant coefficients for the M-indexed orthogonal polynomials, it is expected that the systems satisfy the generalized closure relations. In fact we can verify this statement for small M examples. The generalized closure relation gives the exact Heisenberg operator solution of a certain operator, from which the creation and annihilation operators of the system are obtained.

Recurrence relations of the multi-indexed orthogonal polynomials. III

In Paper II, we presented conjectures of the recurrence relations with constant coefficients for the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson, and Askey-Wilson types. In this paper we present a proof for the Laguerre and Jacobi cases. Their bispectral properties are also discussed, which gives a method to obtain the coefficients of the recurrence relations explicitly. This paper extends to the Laguerre and Jacobi cases the bispectral techniques recently introduced by Gómez-Ullate et al. [J. Approx. Theory 204, 1 (2016); e-print arXiv:1506.03651 [math.CA]] to derive explicit expressions for the coefficients of the recurrence relations satisfied by exceptional polynomials of Hermite type.

Recurrence relations of the multi-indexed orthogonal polynomials. II

In a previous paper, we presented 3 + 2M term recurrence relations with variable dependent coefficients for M-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson, and Askey-Wilson types. In this paper, we present (conjectures of) the recurrence relations with constant coefficients for these multi-indexed orthogonal polynomials. The simplest recurrence relations have 3 + 2ℓ terms, where ℓ (≥M) is the degree of the lowest member of the multi-indexed orthogonal polynomials.

Equivalences of the multi-indexed orthogonal polynomials

Multi-indexed orthogonal polynomials describe eigenfunctions of exactly solvable shape-invariant quantum mechanical systems in one dimension obtained by the method of virtual states deletion. Multi-indexed orthogonal polynomials are labeled by a set of degrees of polynomial parts of virtual state wavefunctions. For multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson, and Askey-Wilson types, two different index sets may give equivalent multi-indexed orthogonal polynomials. We clarify these equivalences. Multi-indexed orthogonal polynomials with both type I and II indices are proportional to those of type I indices only (or type II indices only) with shifted parameters.

Recurrence relations of the multi-indexed orthogonal polynomials

Ordinary orthogonal polynomials are uniquely characterized by the three term recurrence relations up to an overall multiplicative constant. We show that the newly discovered M-indexed orthogonal polynomials satisfy 3+2M term recurrence relations with non-trivial initial data of the lowest M+1 members. These include the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson, and Askey-Wilson types. The M = 0 case is the corresponding classical orthogonal polynomials.

Confluence of apparent singularities in multi-indexed orthogonal polynomials: the Jacobi case

The multi-indexed Jacobi polynomials are the main part of the eigenfunctions of exactly solvable quantum mechanical systems obtained by certain deformations of the Pöschl–Teller potential (Odake–Sasaki). By fine-tuning the parameter(s) of the Pöschl–Teller potential, we obtain several families of explicit and global solutions of certain second-order Fuchsian differential equations with an apparent singularity of characteristic exponents −2 and −1. They form orthogonal polynomials over x ∈ ( − 1, 1) with weight functions of the form (1 − x)α(1 + x)β/{(ax + b)4q(x)2}, in which q(x) is a polynomial in x.

Multi-indexed Wilson and Askey–Wilson polynomials

As the third stage of the project multi-indexed orthogonal polynomials, we present, in the framework of ‘discrete quantum mechanics’ with pure imaginary shifts in one dimension, the multi-indexed Wilson and Askey–Wilson polynomials. They are obtained from the original Wilson and Askey–Wilson polynomials by multiple applications of the discrete analogue of the Darboux transformations or the Crum–Krein–Adler deletion of ‘virtual state solutions’ of types I and II, in a similar way to the multi-indexed Laguerre, Jacobi and (q-)Racah polynomials reported earlier.

Multi-indexed (q-)Racah polynomials

As the second stage of the project multi-indexed orthogonal polynomials, we present, in the framework of ‘discrete quantum mechanics’ with real shifts in one dimension, the multi-indexed (q-)Racah polynomials. They are obtained from the (q-)Racah polynomials by the multiple application of the discrete analogue of the Darboux transformations or the Crum–Krein–Adler deletion of ‘virtual state’ vectors, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier. The virtual state vectors are the ‘solutions’ of the matrix Schrödinger equation with negative ‘eigenvalues’, except for one of the two boundary points.

Exactly solvable quantum mechanics and infinite families of multi-indexed orthogonal polynomials

Infinite families of multi-indexed orthogonal polynomials are discovered as the solutions of exactly solvable one-dimensional quantum mechanical systems. The simplest examples, the one-indexed orthogonal polynomials, are the infinite families of the exceptional Laguerre and Jacobi polynomials of types I and II constructed by the present authors. The totality of the integer indices of the new polynomials are finite and they correspond to the degrees of the ‘virtual state wavefunctions’ which are ‘deleted’ by the generalisation of Crum–Adler theorem. Each polynomial has another integer n which counts the nodes.