Related Polynomials Research Articles

Recurrence Relations of Exceptional Laurent Biorthogonal Polynomials

Recurrence Relations of Exceptional Laurent Biorthogonal Polynomials

Another type of forward and backward shift relations for orthogonal polynomials in the Askey scheme

The forward and backward shift relations are basic properties of the (basic) hypergeometric orthogonal polynomials in the Askey scheme (Jacobi, Askey-Wilson, q-Racah, big q-Jacobi etc.) and they are related to the factorization of the differential or difference operators. Based on other factorizations, we obtain another type of forward and backward shift relations. Essentially, these shift relations shift only the parameters.

Ultracool Dwarf Absolute Magnitude Versus Spectral Type Relations for Euclid and Roman Near-infrared Filters

We synthesize Euclid Near Infrared Spectrometer and Photometer photometry for the Y E J E H E filters and Roman Wide Field Instrument photometry for the F106, F129, F146, F158, F184 and F213 filters using SpeX prism spectra and parallaxes of 688 field-age and 151 young (≲200 Myr) ultracool dwarfs (spectral types M6–T9). For the above filters, we derive empirical absolute magnitude-spectral type polynomial relations that enable the calculation of photometric distances for ultracool dwarfs to be observed with Euclid and Roman, in the absence of parallax measurements. The synthesized photometry can also be used to generate color–color figures to distinguish high-redshift galaxies from brown dwarf interlopers.

Mumford-Tate groups of 1-motives and Weil pairing

We show how the geometry of a 1-motive M (that is existence of endomorphisms and relations between the points defining it) determines the dimension of its motivic Galois group Galmot(M). Fixing periods matrices ΠM and ΠM⁎ associated respectively to a 1-motive M and to its Cartier dual M⁎, we describe the action of the Mumford-Tate group of M on these matrices. In the semi-elliptic case, according to the geometry of M we classify polynomial relations between the periods of M and we compute exhaustively the matrices representing the Mumford-Tate group of M. This representation brings new light on Grothendieck periods conjecture in the case of 1-motives.

Transforming Radical Differential Equations to Algebraic Differential Equations

In this paper, we present an algorithmic procedure that transforms, if possible, a given system of ordinary or partial differential equations with radical dependencies in the unknown function and its derivatives into a system with polynomial relations among them by means of a rational change of variables. The solutions of the given equation and its transformation correspond one-to-one. This work can be seen as a generalization of previous work on reparametrization of ODEs and PDEs with radical coefficients.

Linear integral operators on Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_p$$\\end{document} spaces representing polynomial covariance type commutation relations

In this work, we present methods for constructing representations of polynomial covariance type commutation relations AB=BF(A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$AB=BF(A)$$\\end{document} by linear integral operators in Banach spaces Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_p$$\\end{document}. We derive necessary and sufficient conditions on the kernel functions for the integral operators to satisfy the covariance type commutation relation for general polynomials F, as well as for important cases, when F is arbitrary affine or quadratic polynomial, or arbitrary monomial of any degree. Using the obtained general conditions on the kernels, we construct concrete examples of representations of the covariance type commutation relations by integral operators on Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_p$$\\end{document}. Also, we derive useful general reordering formulas for the integral operators representing the covariance type commutation relations, in terms of the kernel functions.

New Mixed Recurrence Relations of Two-Variable Orthogonal Polynomials via Differential Operators

New Mixed Recurrence Relations of Two-Variable Orthogonal Polynomials via Differential Operators

Evaluation of and period polynomial relations

Abstract In studying the depth filtration on multiple zeta values, difficulties quickly arise due to a disparity between it and the coradical filtration [9]. In particular, there are additional relations in the depth graded algebra coming from period polynomials of cusp forms for $\operatorname {\mathrm {SL}}_2({\mathbb {Z}})$ . In contrast, a simple combinatorial filtration, the block filtration [13, 28] is known to agree with the coradical filtration, and so there is no similar defect in the associated graded. However, via an explicit evaluation of $\zeta (2,\ldots ,2,4,2,\ldots ,2)$ as a polynomial in double zeta values, we derive these period polynomial relations as a consequence of an intrinsic symmetry of block graded multiple zeta values in block degree 2. In deriving this evaluation, we find a Galois descent of certain alternating double zeta values to classical double zeta values, which we then apply to give an evaluation of the multiple t values [22] $t(2\ell ,2k)$ in terms of classical double zeta values.

Polynomial algebras from commutants: Classical and Quantum aspects of 𝒜3

We review some aspects of the Racah algebra R(n), including the closure relations, pointing out their role in superintegrability, as well as in the description of the symmetry algebra for several models with coalgebra symmetry. The connection includes the generic model on the (n − 1) sphere. We discuss an algebraic scheme of constructing Hamiltonians, integrals of the motion and symmetry algebras. This scheme reduces to the Racah algebra R(n) and the model on the (n − 1) sphere only for the case of specific differential operator realizations. We review the method, which allows us to obtain the commutant defined in the enveloping algebra of 𝔰𝔩(n) in the classical setting. The related 𝒜3 polynomial algebra is presented for the case 𝔰𝔩(3). An explicit construction of the quantization of the scheme for 𝒜3 by symmetrization of the polynomial and the replacement of the Berezin bracket by commutator and symmetrization of the polynomial relations is presented. We obtain the additional quantum terms. These explicit relations are of interest not only for superintegrability, but also for other applications in mathematical physics.

Second structure relation for the Dunkl-classical orthogonal polynomials

Second structure relation for the Dunkl-classical orthogonal polynomials

Ultracool Dwarf Absolute Magnitude versus Spectral Type Relations for JWST NIRCam Filters

We synthesize JWST NIRCam photometry for a range of narrow, medium, and wide filters using SpeX prism spectra and parallaxes of 688 field-age and 151 young (≲200 Myr) ultracool dwarfs (spectral types M6–T9). We derive absolute magnitude-spectral type polynomial relations for the F164N, F187N, and F212N narrow filters; the F140M, F162M, F182M, and F210M medium filters; and the F115W, F150W, and F200W wide filters. Our relations enable the calculation of photometric distances for ultracool dwarfs observed with JWST in the absence of parallax measurements. Additionally, using the synthesized photometry to generate color–color figures can help distinguish high-redshift galaxies from brown dwarf interlopers.

On semi-classical weight functions on the unit circle

We consider orthogonal polynomials on the unit circle associated with certain semi-classical weight functions. This means that the Pearson-type differential equations satisfied by these weight functions involve two polynomials of degree at most 2. We determine all such semi-classical weight functions and this also includes an extension of the Jacobi weight function on the unit circle. General structure relations for the orthogonal polynomials and non-linear difference equations for the associated complex Verblunsky coefficients are established. As application, we present several new structure relations and non-linear difference equations associated with some of these semi-classical weight functions.

A Structure Relation for Some Orthogonal Polynomials

A Structure Relation for Some Orthogonal Polynomials

The bilateral birth–death chain generated by the associated Jacobi polynomials

AbstractWe give a probabilistic interpretation of the associated Jacobi polynomials, which can be constructed from the three‐term recurrence relation for the classical Jacobi polynomials by shifting the integer index n by a real number t. Under certain restrictions, this will give rise to a doubly infinite tridiagonal stochastic matrix, which can be interpreted as the one‐step transition probability matrix of a discrete‐time bilateral birth–death chain with state space on . We also study the unique UL and LU stochastic factorizations of the transition probability matrix, as well as the discrete Darboux transformations and corresponding spectral matrices. Finally, we use all these results to provide an urn model on the integers for the associated Jacobi polynomials.

Special transforms of the generalized bivariate Fibonacci and Lucas polynomials

This paper deals with the Catalan, Hankel, binomial transforms of the generalized bivariate Fibonacci and Lucas polynomials. Also, some useful results such as generating functions, Binet formulas, summations of transforms defined by using recurrence relations of these special polynomials are presented. Furthermore, certain important relations among these transforms are deduced by using obtained new formulas. Finally, the Catalan, Cassini, Vajda and d'Ocagne formulas for these transforms are also derived.

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.

Polynomial Relations Among Kappa Classes on the Moduli Space of Curves

Abstract We construct an infinite collection of universal—independent of $(g,n)$—polynomials in the Miller–Morita–Mumford classes $\kappa _m\in H^{2m}( \overline{\mathcal{M}}_{g,n},{\mathbb{Q}})$, defined over the moduli space of genus $g$ stable curves with $n$ labeled points. We conjecture vanishing of these polynomials in a range depending on $g$ and $n$.

New Orthogonality Relations for Super-Jack Polynomials and an Associated Lassalle–Nekrasov Correspondence

AbstractThe super-Jack polynomials, introduced by Kerov, Okounkov and Olshanski, are polynomials in$$n+m$$n+mvariables, which reduce to the Jack polynomials when$$n=0$$n=0or$$m=0$$m=0and provide joint eigenfunctions of the quantum integrals of the deformed trigonometric Calogero–Moser–Sutherland system. We prove that the super-Jack polynomials are orthogonal with respect to a bilinear form of the form$$(p,q)\mapsto (L_pq)(0)$$(p,q)↦(Lpq)(0), with$$L_p$$Lpquantum integrals of the deformed rational Calogero–Moser–Sutherland system. In addition, we provide a new proof of the Lassalle–Nekrasov correspondence between deformed trigonometric and rational harmonic Calogero–Moser–Sutherland systems and infer orthogonality of super-Hermite polynomials, which provide joint eigenfunctions of the latter system.

Two-dimensional contiguous relations for the linearization coefficients of classical orthogonal polynomials

ABSTRACT By using the three-term recurrence relation for orthogonal polynomials, we produce a collection of two-dimensional contiguous relations for certain generalized hypergeometric functions. These generalized hypergeometric functions arise through linearization coefficients for some classical orthogonal polynomials in the Askey-scheme, namely Gegenbauer (ultraspherical), Hermite, Jacobi and Laguerre polynomials.

A Generic and Strictly Banded Spectral Petrov–Galerkin Method for Differential Equations with Polynomial Coefficients

In this paper we describe a generic spectral Petrov–Galerkin method that is sparse and strictly banded for any linear ordinary differential equation with polynomial coefficients. The method applies to all subdivisions of Jacobi polynomials (e.g., Chebyshev and Legendre), utilizes well-known recurrence relations of orthogonal polynomials, and leads to almost exactly the same discretized system of equations as the integration preconditioners [E. A. Coutsias, T. Hagstrom, and D. Torres, Math. Comp., 65 (1996), pp. 611–635] if this method was redesigned to make use of trial functions that satisfy a given problem’s boundary conditions. A link between the new Petrov–Galerkin method and IP is revealed through a new recursion relation for Jacobi polynomials. Because of the strictly banded nature of all coefficient matrices, the new method extends easily and efficiently to multiple dimensions though the use of tensor product methods.