Recurrences for the coefficients of series expansions with respect to classical orthogonal polynomials

Lewanowicz, Stanisław

doi:10.4064/am29-1-9

Cited by 8 publications

(5 citation statements)

References 16 publications

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“…For a brief background, definitions for some terminology and most basic properties of these polynomials, please refer to Gasper and Rahman [ [24], p. [3][4][5][6] and Koekoek and Swarttouw [[53], p.113-114].…”

Section: Family σ(X) τ(X)mentioning

confidence: 99%

Efficient algorithms for construction of recurrence relations for the expansion and connection coefficients in series of quantum classical orthogonal polynomials

Doha

Ahmed

2010

Journal of Advanced Research

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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.

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Section: Family σ(X) τ(X)mentioning

confidence: 99%

Efficient algorithms for construction of recurrence relations for the expansion and connection coefficients in series of quantum classical orthogonal polynomials

Doha

Ahmed

2010

Journal of Advanced Research

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“…Lewanowicz [11][12][13], Lewanowicz and Woźny [14] and Woźny [20] have presented a very similar algorithm for finding the recurrence relation for both connection and linearization coefficients. Also, a different algorithm for constructing the recurrence relations for the expansion coefficients of an arbitrary function of a discrete variable as *Email: hany_195@frcu.eun.eg a series in any one of the four families of classical orthogonal polynomials of a discrete variable (Hahn, Meixner, Kravchuk and Charlier), and the explicit formulae for the connection coefficients between them, is given in Doha and Ahmed [7,8].…”

Section: Introductionmentioning

confidence: 97%

Recurrence relation approach for expansion and connection coefficients in series of classical discrete orthogonal polynomials

Ahmed

2009

Integral Transforms and Special Functions

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The two formulae expressing explicitly the difference derivatives and the moments of discrete orthogonal polynomials {P n (x) : Meixner, Kravchuk and Charlier} of any degree and for any order in terms of P n (x) themselves are proved. Two other formulae for the expansion coefficients of general-order difference derivatives q f (x), and for the moments x q f (x), of an arbitrary function f (x) of a discrete variable in terms of its original expansion coefficients are also obtained. Application of these formulae for solving ordinary difference 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 Hahn-Charlier, Hahn-Meixner and Hahn-Kravchuk are described.

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“…where a n , b n , c n are rational functions of n [8]; a differential-difference equation (1.2) π(x)p ′ n (x) = α n p n+1 + β n p n + γ n p n−1 , with π(x) a polynomial of degree at most 2 and α n , β n , γ n rational functions of n [4]; they also satisfy a second order linear differential equation with polynomial coefficients [11].…”

Section: Introductionmentioning

confidence: 99%

Differential-Difference Properties of Hypergeometric Series

Brisebarre,

Salvy

2022

Preprint

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Six families of generalized hypergeometric series in a variable x and an arbitrary number of parameters are considered. Each of them is indexed by an integer n. Linear recurrence relations in n relate these functions and their product by the variable x. We give explicit factorizations of these equations as products of first order recurrence operators. Related recurrences are also derived for the derivative with respect to x. These formulas generalize wellknown properties of the classical orthogonal polynomials.

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Recurrences for the coefficients of series expansions with respect to classical orthogonal polynomials

Cited by 8 publications

References 16 publications

Efficient algorithms for construction of recurrence relations for the expansion and connection coefficients in series of quantum classical orthogonal polynomials

Efficient algorithms for construction of recurrence relations for the expansion and connection coefficients in series of quantum classical orthogonal polynomials

Recurrence relation approach for expansion and connection coefficients in series of classical discrete orthogonal polynomials

Differential-Difference Properties of Hypergeometric Series

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