Recurrences for the coefficients of series expansions with respect to classical orthogonal polynomials (original) (raw)
Numerical Algorithms, 2000
Let {P k } be a sequence of the semi-classical orthogonal polynomials. Given a function f satisfying a linear second-order differential equation with polynomial coefficients, we describe an algorithm to construct a recurrence relation satisfied by the coefficients a k [f ] in f = k a k [f ]P k. A systematic use of basic properties (including some nonstandard ones) of the polynomials {P k } results in obtaining a recurrence of possibly low order. Recurrences for connection or linearization coefficients related to the first associated generalized Gegenbauer, Bessel-type and Laguerre-type polynomials are given explicitly.
Integral Transforms and Special Functions, 2006
A formula expressing explicitly the derivatives of Bessel polynomials of any degree and for any order in terms of the Bessel polynomials themselves is proved. Another explicit formula, which expresses the Bessel expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of its original Bessel coefficients, is also given. A formula for the Bessel coefficients of the moments of one single Bessel polynomial of certain degree is proved. A formula for the Bessel coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its Bessel coefficients is also obtained. Application of these formulae for solving ordinary differential equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica) in order to build and solve recursively for the connection coefficients between Bessel-Bessel polynomials is described. An explicit formula for these coefficients between Jacobi and Bessel polynomials is given, of which the ultraspherical polynomial and its consequences are important special cases. Two analytical formulae for the connection coefficients between Laguerre-Bessel and Hermite-Bessel are also developed.
Recurrence equations involving different orthogonal polynomial sequences and applications
2021
Consider {pn}n=0, a sequence of polynomials orthogonal with respect to w(x) > 0 on (a, b), and polynomials {gn,k}n=0, k ∈ N0, orthogonal with respect to ck(x)w(x) > 0 on (a, b), where ck(x) is a polynomial of degree k in x. We show how Christoffel’s formula can be used to obtain mixed three-term recurrence equations involving the polynomials pn, pn−1 and gn−m,k,m ∈ {2, 3, . . . , n−1}. In order for the zeros of pn and Gm−1gn−m,k to interlace (assuming pn and gn−m,k are co-prime), the coefficient of pn−1, namely Gm−1, should be of exact degree m − 1, in which case restrictions on the parameter k are necessary. The zeros of Gm−1 can be considered to be inner bounds for the extreme zeros of the (classical or q-classical) orthogonal polynomial pn and we give examples to illustrate the accuracy of these bounds. Because of the complexity the mixed three-term recurrence equations in each case, algorithmic tools, mainly Zeilberger’s algorithm and its q-analogue, are used to obtain them.
Recurrence Relation and Differential Equation for a Class of Orthogonal Polynomials
Results in Mathematics, 2018
Given real number s > -1/2 and the second degree monic Chebyshev polynomial of the first kind T2(x), we consider the polynomial system {p 2,s k } "induced" by the modified measure dσ 2,s (x) = | T2(x)| 2s dσ(x), where dσ(x) = 1/ √ 1x 2 dx is the Chebyshev measure of the first kind. We determine the coefficients of the three-term recurrence relation for the polynomials p 2,s k (x) in an analytic form and derive a differential equality, as well as the differential equation for these orthogonal polynomials. Assuming a logarithmic potential, we also give an electrostatic interpretation of the zeros of p 2,s 4ν (x)(ν ∈ N).
Nonlinear equations for the recurrence coefficients of discrete orthogonal polynomials
Journal of Mathematical Analysis and Applications, 2011
We study polynomials orthogonal on a uniform grid. We show that each weight function gives two potentials and each potential leads to a structure relation (lowering operator). These results are applied to derive second order difference equations satisfied by the orthogonal polynomials and nonlinear difference equations satisfied by the recursion coefficients in the three-term recurrence relations.
Journal of Computational and Applied Mathematics, 1997
We present a simple approach in order to compute recursively the connection coefficients between two families of classical (discrete) orthogonal polynomials (Charlier, Meixner, Kravchuk, Hahn), i.e., the coefficients Cm(n) in the expression Pn(x) = ~=0 Cm(n)Qm(x), where {Pn(x)} and {Qm(x)} belong to the aforementioned class of polynomials. This is done by adapting a general and systematic algorithm, recently developed by the authors, to the discrete classical situation. Moreover, extensions of this method allow to give new addition formulae and to estimate Cm(n)-asymptotics in limit relations between some families.
Derivation of New Class of Orthogonal Polynomials with Recurrence Relation
FUOYE Journal of Engineering and Technology, 2016
This paper presents the derivation of a new class of orthogonal polynomials named ADEM-B orthogonal polynomials, q n (x) valid in the interval [-1, 1] with respect to weight function w(x) = x 2-1. The analysis of some basic properties of the polynomials shows that the polynomials are symmetrical depending on whether index n in q n (x) is even or odd. The recurrence relation of the class of the polynomials is presented and a brief review of the formulation of existing scheme is considered to test the applicability of the polynomials. Findings reveal that these polynomials produce the same results as in zeros of Chebyshev and Legendre polynomials.
Journal of Computational and Applied Mathematics, 1997
We present a simple approach in order to compute recursively the connection coefficients between two families of classical (discrete) orthogonal polynomials (Charlier, Meixner, Kravchuk, Hahn), i.e., the coefficients Cm(n) in the expression Pn(x) = ~=0 Cm(n)Qm(x), where {Pn(x)} and {Qm(x)} belong to the aforementioned class of polynomials. This is done by adapting a general and systematic algorithm, recently developed by the authors, to the discrete classical situation. Moreover, extensions of this method allow to give new addition formulae and to estimate Cm(n)-asymptotics in limit relations between some families.
Journal of Advanced Research, 2010
Formulae expressing explicitly the q-difference derivatives and the moments of the polynomials P n (x ; q) ∈ T (T ={P n (x ; q) ∈ Askey-Wilson polynomials: Al-Salam-Carlitz I, Discrete q-Hermite I, Little (Big) q-Laguerre, Little (Big) q-Jacobi, q-Hahn, Alternative q-Charlier) of any degree and for any order in terms of P i (x ; q) themselves are proved. We will also provide two other interesting formulae to expand the coefficients of general-order q-difference derivatives D p q f (x), and for the moments x D p q f (x), of an arbitrary function f(x) in terms of its original expansion coefficients. We used the underlying formulae to relate the coefficients of two different polynomial systems of basic hypergeometric orthogonal polynomials, belonging to the Askey-Wilson polynomials and P n (x ; q) ∈ T. These formulae are useful in setting up the algebraic systems in the unknown coefficients, when applying the spectral methods for solving q-difference equations of any order. stressed that, even when explicit forms for these coefficients are available, it is often useful to have a linear recurrence relation satisfied by these coefficients. This recurrence relation may serve as a tool for detection of certain properties of the expansion coefficients of the given function, and for numerical evaluation of these quantities, using a judiciously chosen algorithm . The construction of such recurrences attracted much interest in the last few years. Special emphasis has been given to the classical continuous orthogonal polynomials (Hermite, Laguerre, Jacobi and Bessel), the discrete cases (Hahn, Meixner, Krawtchouk and Charlier) and the basic hypergeometric orthogonal polynomials, belonging to the Askey-Wilson polynomials.
Matrix Calculus-Based Approach to Orthogonal Polynomial Sequences
Mediterranean Journal of Mathematics, 2020
In this paper, an approach to orthogonal polynomials based on matrix calculus is proposed. Known and new basic results are given, such as recurrence relations and determinant forms. New algorithms, similar, but not identical, to the Chebyshev one, for practical calculation of the polynomials are presented. The cases of monic and symmetric orthogonal polynomial sequences and the case of orthonormal polynomial sequences have been considered. Some classical and non-classical examples are given. The work is framed in a broader perspective, already started by the authors. It provides for the determination of properties of a general sequence of polynomials and, therefore, their applicability to special classes of the most important polynomials.
New mixed recurrence relations of two-variable orthogonal polynomials via differential operators
2020
In this paper, we derive new recurrence relations for two-variable orthogonal polynomials for example Jacobi polynomial, Bateman's polynomial and Legendre polynomial via two different differential operators Xi=left(fracpartialpartialz+sqrtwfracpartialpartialwright)\Xi =\left(\frac{\partial }{\partial z} +\sqrt{w} \frac{\partial }{\partial w} \right)Xi=left(fracpartialpartialz+sqrtwfracpartialpartialwright) and Delta=left(frac1wfracpartialpartialz+frac1zfracpartialpartialwright)\Delta =\left(\frac{1}{w} \frac{\partial }{\partial z} +\frac{1}{z} \frac{\partial }{\partial w} \right)Delta=left(frac1wfracpartialpartialz+frac1zfracpartialpartialwright). We also derive some special cases of our main results.
On the associated orthogonal polynomials
Journal of Computational and Applied Mathematics, 1990
By using the second-order recurrence relation this paper gives some new results on associated orthogonal polynomials without referring to the continued fractions' tool. Some results are very useful for obtaining the second-order differential equation satisfied by the semi-classical orthogonal polynomials (Hendriksen and van Rossum (1985) Maroni (1987)) (cf. Section 3). Also, the main formula derives from Proposition 2.6, by which the fourth-order differential equation, satisfied by some Laguerre-Hahn polynomials (Magnus (1984)), is obtained (cf. Behnehdi and Ronveaux (1989), Dini et al. (1989), Ronveaux et al. (1990)).
An Analysis of the Recurrence Coefficients for Symmetric Sobolev-Type Orthogonal Polynomials
Symmetry
In this contribution we obtain some algebraic properties associated with the sequence of polynomials orthogonal with respect to the Sobolev-type inner product:p,qs=∫Rp(x)q(x)dμ(x)+M0p(0)q(0)+M1p′(0)q′(0), where p,q are polynomials, M0, M1 are non-negative real numbers and μ is a symmetric positive measure. These include a five-term recurrence relation, a three-term recurrence relation with rational coefficients, and an explicit expression for its norms. Moreover, we use these results to deduce asymptotic properties for the recurrence coefficients and a nonlinear difference equation that they satisfy, in the particular case when dμ(x)=e−x4dx.
Journal of Physics A: Mathematical and General, 2005
Formulae expressing explicitly the q-difference derivatives and the moments of the polynomials P n (x ; q) ∈ T (T ={P n (x ; q) ∈ Askey-Wilson polynomials: Al-Salam-Carlitz I, Discrete q-Hermite I, Little (Big) q-Laguerre, Little (Big) q-Jacobi, q-Hahn, Alternative q-Charlier) of any degree and for any order in terms of P i (x ; q) themselves are proved. We will also provide two other interesting formulae to expand the coefficients of general-order q-difference derivatives D p q f (x), and for the moments x D p q f (x), of an arbitrary function f(x) in terms of its original expansion coefficients. We used the underlying formulae to relate the coefficients of two different polynomial systems of basic hypergeometric orthogonal polynomials, belonging to the Askey-Wilson polynomials and P n (x ; q) ∈ T. These formulae are useful in setting up the algebraic systems in the unknown coefficients, when applying the spectral methods for solving q-difference equations of any order.
Linear differential equations and orthogonal polynomials: a novel approach
2002
A novel method, connecting the space of solutions of a linear differential equation, of arbitrary order, to the space of monomials, is used for exploring the algebraic structure of the solution space. Apart from yielding new expressions for the solutions of the known differential equations, the procedure enables one to derive various properties of the orthogonal polynomials and functions, in a unified manner. The method of generalization of the present approach to the multi-variate case is pointed out and also its connection with the wellknown factorization technique. It is shown that, the generating functions and Rodriguez formulae emerge naturally in this method.