Term Recurrence Relation Research Articles

Matrix valued Hermite polynomials, Burchnall formulas and non-abelian Toda lattice

A general family of matrix valued Hermite type orthogonal polynomials is introduced as the matrix orthogonal polynomials with respect to a weight. The matrix polynomials are eigenfunctions of a matrix differential equation. For the weight we derive Pearson equations, which allow us to derive many explicit properties of these matrix polynomials. In particular, the matrix polynomials are eigenfunctions to another matrix differential equation. We also obtain for these polynomials shift operators, a Rodrigues formula, explicit expressions for the squared norm, explicit three term recurrence relations, etc. The matrix entries of the matrix polynomials can be expressed in terms of scalar Hermite and dual Hahn polynomials. We also derive a connection formula for the matrix Hermite polynomials. Next we show that operational Burchnall formulas extend to matrix polynomials. We make this explicit for the matrix Hermite polynomials and for previously introduced matrix Gegenbauer type orthogonal polynomials. The Burchnall approach gives two descriptions of the matrix valued orthogonal polynomials for the Toda modification of the matrix Hermite weight. In particular, we obtain an explicit non-trivial solution to the non-abelian Toda lattice equations.

Filter-generating system of Zernike polynomials

This work proposes a new approach to find the generating function (GF) of the Zernike polynomials in two dimensional form. Combining the methods of GFs and discrete-time systems, we can develop two dimensional digital systems for systematic generation of entire orders of Zernike polynomials. We establish two different formulas for the GF of the radial Zernike polynomials based on both the degree and the azimuthal order of the radial polynomials. In this paper, we use four terms recurrence relation instead of the ordinary three terms recursion to calculate the radial Zernike polynomials and their GFs using unilateral 2D Z-transform. A spatio-temporal implementation scheme is developed for generation of the radial Zernike polynomials. The case study shows a reliable way to evaluate Zernike polynomials with arbitrary degrees and azimuthal orders.

Recurrence relations for polynomials obtained by arithmetic functions

Families of polynomials associated to arithmetic functions [Formula: see text] are studied. The case [Formula: see text], the divisor sum, dictates the non-vanishing of the Fourier coefficients of powers of the Dedekind eta function. The polynomials [Formula: see text] are defined by [Formula: see text]-term recurrence relations. For the case that [Formula: see text] is a polynomial of degree [Formula: see text], we prove that at most a [Formula: see text] term recurrence relation is needed. For the special case [Formula: see text], we obtain explicit formulas and results.

On Freud–Sobolev type orthogonal polynomials

In this contribution we deal with sequences of monic polynomials orthogonal with respect to the Freud Sobolev-type inner product \begin{equation*} \left\langle p,q\right\rangle _{s}=\int_{\mathbb{R}}p(x)q(x)e^{-x^{4}}dx+M_{0}p(0)q(0)+M_{1}p^{\prime }(0)q^{\prime }(0), \end{equation*}% where $p,q$ are polynomials, $M_{0}$ and $M_{1}$ are nonnegative real numbers. Connection formulas between these polynomials and Freud polynomials are deduced and, as a consequence, a five term recurrence relation for such polynomials is obtained. The location of their zeros as well as their asymptotic behavior is studied. Finally, an electrostatic interpretation of them in terms of a logarithmic interaction in the presence of an external field is given.

Strong linearizations of rational matrices with polynomial part expressed in an orthogonal basis

We construct a new family of strong linearizations of rational matrices considering the polynomial part of them expressed in a basis that satisfies a three term recurrence relation. For this purpose, we combine the theory developed by Amparan et al. (2018) [4], and the new linearizations of polynomial matrices introduced by Faßbender and Saltenberger (2017) [15]. In addition, we present a detailed study of how to recover eigenvectors of a rational matrix from those of its linearizations in this family. We complete the paper by discussing how to extend the results when the polynomial part is expressed in other bases, and by presenting strong linearizations that preserve the structure of symmetric or Hermitian rational matrices. A conclusion of this work is that the combination of the results in this paper with those in Amparan et al. (2018) [4], allows us to use essentially all the strong linearizations of polynomial matrices developed in the last fifteen years to construct strong linearizations of any rational matrix by expressing such a matrix in terms of its polynomial and strictly proper parts.

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.

On the behavior of roots of trinomial equations

The main purpose of this paper is to investigate the distribution, simplicity and monotonicity of the roots of a class of trinomial equations that appears in certain financial mathematics problems. Furthermore, we show that it is possible to decompose some classes of trinomial equations into polynomials generated by a three term recurrence relation.

Extended Relativistic Toda Lattice, L-Orthogonal Polynomials and Associated Lax Pair

When a measure $\varPsi(x)$ on the real line is subjected to the modification $d\varPsi^{(t)}(x) = e^{-tx} d \varPsi(x)$ , then the coefficients of the recurrence relation of the orthogonal polynomials in $x$ with respect to the measure $\varPsi^{(t)}(x)$ are known to satisfy the so-called Toda lattice formulas as functions of $t$ . In this paper we consider a modification of the form $e^{-t(\mathfrak{p}x+ \mathfrak{q}/x)}$ of measures or, more generally, of moment functionals, associated with orthogonal L-polynomials and show that the coefficients of the recurrence relation of these L-orthogonal polynomials satisfy what we call an extended relativistic Toda lattice. Most importantly, we also establish the so called Lax pair representation associated with this extended relativistic Toda lattice. These results also cover the (ordinary) relativistic Toda lattice formulations considered in the literature by assuming either $\mathfrak{p}=0$ or $\mathfrak{q}=0$ . However, as far as Lax pair representation is concern, no complete Lax pair representations were established before for the respective relativistic Toda lattice formulations. Some explicit examples of extended relativistic Toda lattice and Langmuir lattice are also presented. As further results, the lattice formulas that follow from the three term recurrence relations associated with kernel polynomials on the unit circle are also established.

On generalized Laguerre matrix polynomials

AbstractThe main object of the present paper is to introduce and study the generalized Laguerre matrix polynomials for a matrix that satisfies an appropriate spectral property. We prove that these matrix polynomials are characterized by the generalized hypergeometric matrix function. An explicit representation, integral expression and some recurrence relations in particular the three terms recurrence relation are obtained here. Moreover, these matrix polynomials appear as solution of a differential equation.

Two finite hypergeometric sequences of discrete orthogonal polynomials

ABSTRACTTwo finite hypergeometric sequences of symmetric orthogonal polynomials of a discrete variable are introduced and their standard properties, such as second-order difference equations, explicit forms of the polynomials and three term recurrence relations are obtained. As a consequence of two specific Sturm–Liouville problems, it is proved that these polynomials are finitely orthogonal with respect to two symmetric weight functions.

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.

Recurrence Relations for Wronskian Hermite Polynomials

We consider polynomials that are defined as Wronskians of certain sets of Hermite polynomials. Our main result is a recurrence relation for these polynomials in terms of those of one or two degrees smaller, which generalizes the well-known three term recurrence relation for Hermite polynomials. The polynomials are defined using partitions of natural numbers, and the coefficients in the recurrence relation can be expressed in terms of the number of standard Young tableaux of these partitions. Using the recurrence relation, we provide another recurrence relation and show that the average of the considered polynomials with respect to the Plancherel measure is very simple. Furthermore, we show that some existing results in the literature are easy corollaries of the recurrence relation.

A q-generalization of the para-Racah polynomials

New bispectral orthogonal polynomials are obtained from an unconventional truncation of the Askey–Wilson polynomials. In the limit q→1, they reduce to the para-Racah polynomials which are orthogonal with respect to a quadratic bi-lattice. The three term recurrence relation and q-difference equation are obtained through limits of those of the Askey–Wilson polynomials. An explicit expression in terms of hypergeometric series and the orthogonality relation are provided. A q-generalization of the para-Krawtchouk polynomials is obtained as a special case. Connections with the q-Racah and dual-Hahn polynomials are also presented.

On global fluctuations for non-colliding processes

We study the global fluctuations for a class of determinantal point processes coming from large systems of non-colliding processes and non-intersecting paths. Our main assumption is that the point processes are constructed by biorthogonal families that satisfy finite term recurrence relations. The central observation of the paper is that the fluctuations of multi-time or multi-layer linear statistics can be efficiently expressed in terms of the associated recurrence matrices. As a consequence, we prove that different models that share the same asymptotic behavior of the recurrence matrices, also share the same asymptotic behavior for the global fluctuations. An important special case is when the recurrence matrices have limits along the diagonals, in which case we prove Central Limit Theorems for the linear statistics. We then show that these results prove Gaussian Free Field fluctuations for the random surfaces associated to these systems. To illustrate the results, several examples will be discussed, including non-colliding processes for which the invariant measures are the classical orthogonal polynomial ensembles and random lozenge tilings of a hexagon.

Bivariate Legendre approximation

Spectral methods are among the numerical methods commonly used for approximating solutions of boundary value problems. In this paper we propose, a generalization of the spectral Tau method in dimension 2, this method is generalized by the use of a new two-dimensional polynomial basis constructed by a three terms recurrence relation. We also present an estimation of error committed by the proposed method.

A class of symmetric q-orthogonal polynomials with four free parameters

By using a generalization of Sturm–Liouville problems in q-difference spaces, a class of symmetric q-orthogonal polynomials with four free parameters is introduced. The standard properties of these polynomials, such as a second order q-difference equation, the explicit form of the polynomials in terms of basic hypergeometric series, a three term recurrence relation and a general orthogonality relation are presented. Some particular examples are then studied in detail.

Numerically efficient algorithms for anisotropic scale and translation Tchebichef moment invariants

Anisotropic scale and translation invariants (ASTI) for Tchebichef moments have been proposed by Zhu et al. [27]. Since these invariants are derived via the decomposition of Tchebichef polynomials, it is unavoidable that the invariant algorithms inherit the complexities from the Tchebichef polynomials defined in terms of hypergeometric functions. Furthermore, in order to achieve anisotropic scale and translation invariance, the computation of translation invariants and scale invariants need to be performed sequentially. These have turned out to be the bottleneck for the invariant algorithms. Experimental results show that some of the computed ASTI features from symmetric patterns are less accurate. Thus, we would like to extend the work of Zhu et al.[27] to simplify the complexity of the algorithms and further improve the accuracy of the computed features. The three terms recurrence relation of the Tchebichef polynomials has been used to simplify and improve the computational efficiency of the invariant algorithms. Skew transformations are deployed to enhance the numerical accuracy of ASTI for Tchebhcief moments. Our studies show that the skewed features are less sensitive to noise and significantly enhance the accuracy of pattern recognition systems. This has been verified by the experiments on recognition of printed English letters and leaf patterns corrupted by noise and scaled and translated deformations. The simplification of the algorithms using the orthogonal property of basis functions also can be used to simplify more complex invariants like affine invariants of discrete Tchebichief moments. It can also be extended to derive invariants for other orthogonal based moments like Legendre moments, Krawtchouk moments, Hahn moments, etc.

Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials

The multi-indexed Laguerre and Jacobi polynomials form a complete set of orthogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the 'holes' in their degrees. The multi-indexed Laguerre and Jacobi polynomials have Wronskian expressions originating from multiple Darboux transformations. For the ease of applications, two different forms of simplified expressions of the multi-indexed Laguerre and Jacobi polynomials are derived based on various identities. The parity transformation property of the multi-indexed Jacobi polynomials is derived based on that of the Jacobi polynomial.

Three Term Relations for a Class of Bivariate Orthogonal Polynomials

We study matrix three term relations for orthogonal polynomials in two variables constructed from orthogonal polynomials in one variable. Using the three term recurrence relation for the involved univariate orthogonal polynomials, the explicit expression for the matrix coefficients in these three term relations are deduced. These matrices are diagonal or tridiagonal with entries computable from the one variable coefficients in the respective three term recurrence relation. Moreover, some interesting particular cases are considered.

Coherent pairs of measures and Markov–Bernstein inequalities

All the coherent pairs of measures associated to linear functionals c0 and c1, introduced by Iserles et al. in 1991, have been given by Meijer in 1997. There exist seven kinds of coherent pairs. All these cases are explored in order to give three term recurrence relations satisfied by polynomials. The smallest zero μ1,n of each of them of degree n has a link with the Markov–Bernstein constant Mn appearing in the following Markov–Bernstein inequalities:c1((p′)2)≤Mn2c0(p2),∀p∈Pn, where Mn=1μ1,n. The seven kinds of three term recurrence relations are given. In the case where c0=e−xdx+δ(0) and c1=e−xdx, explicit upper and lower bounds are given for μ1,n, and the asymptotic behavior of the corresponding Markov–Bernstein constant is stated. Except in a part of one case, limn→∞⁡μ1,n=0 is proved in all the cases.