Recurrences for the coefficients of series expansions with respect to classical orthogonal polynomials (original) (raw)
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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.
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.