Recurrence relations for the coefficients of the Fourier series expansions with respect to q-classical orthogonal polynomials (original) (raw)
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.
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.
Applicationes Mathematicae, 2002
Let {P k } be any sequence of classical orthogonal polynomials. Further, let f be a function satisfying a linear differential equation with polynomial coefficients. We give an algorithm to construct, in a compact form, a recurrence relation satisfied by the coefficients a k in f = k a k P k. A systematic use of the basic properties (including some nonstandard ones) of the polynomials {P k } results in obtaining a low order of the recurrence.
Expansion of analytic functions in q-orthogonal polynomials
The Ramanujan Journal, 2009
A classical result on expansion of an analytic function in a series of Jacobi polynomials is extended to a class of qorthogonal polynomials containing the fundamental Askey-Wilson polynomials and their special cases. The function to be expanded has to be analytic inside an ellipse in the complex plane with foci at ±1. Some examples of explicit expansions are discussed.
q-Hermite Polynomials and Classical Orthogonal Polynomials
Canadian Journal of Mathematics, 1996
We use generating functions to express orthogonality relations in the form of q-beta integrals. The integrand of such a q-beta integral is then used as a weight function for a new set of orthogonal or biorthogonal functions. This method is applied to the continuous q-Hermite polynomials, the Al-Salam-Carlitz polynomials, and the polynomials of Szegő and leads naturally to the Al-Salam-Chihara polynomials then to the Askey-Wilson polynomials, the big q-Jacobi polynomials and the biorthogonal rational functions of Al-Salam and Verma, and some recent biorthogonal functions of Al-Salam and Ismail.
Representations of q-orthogonal polynomials
Journal of Symbolic Computation, 2012
The linearization problem is the problem of finding the coefficients C k (m, n) in the expansion of the product P n (x)Q m (x) of two polynomial systems in terms of a third sequence of polynomials R k (x), P n (x)Q m (x) = n+m k=0 C k (m, n)R k (x). Note that, in this setting, the polynomials P n , Q m and R k may belong to three different polynomial families. If Q m (x) = 1, we are faced with the so-called connection problem, which for P n (x) = x n is known as the inversion problem for the family R k (x). In this paper we use an algorithmic approach to compute the connection and linearization coefficients between orthogonal polynomials of the q-Hahn tableau. These polynomial systems are solution of a q-differential equation of the type σ(x)D q D 1/q P n (x) + τ(x)D q P n (x) + λ n P n (x) = 0, where the q-differential operator D q is defined by D q f (x) = f (q x) − f (x) (q − 1)x .
$q$-Hermite polynomials and classical orthogonal polynomials
Canadian Journal of Mathematics, 1996
We use generating functions to express orthogonality relations in the form of q-beta. integrals. The integrand of such a q-beta. integral is then used as a weight function for a new set of orthogonal or biorthogonal functions. This method is applied to the continuous q-Hermite polynomials, the Al-Salam-Carlitz polynomials, and the polynomials of Szegö and leads naturally to the Al-Salam-Chihara polynomials then to the Askey-Wilson polynomials, the big q-Jacobi polynomials and the biorthogonal rational functions of Al-Salam and Verma, and some recent biorthogonal functions of Al-Salam and Ismail.
2013
We review properties of the q-Hermite polynomials and indicate their links with the Chebyshev, Rogers-Szegö, Al-Salam-Chihara, continuous q-utraspherical polynomials. In particular, we recall the connection coefficients between these families of polynomials. We also present some useful and important finite and infinite expansions involving polynomials of these families including symmetric and non-symmetric kernels. In the paper, we collect scattered throughout literature useful but not widely known facts concerning these polynomials. It is based on 43 positions of predominantly recent literature.
On A Method for the Construction of Some q-Orthogonal Polynomials
Integral Transforms and Special Functions, 2003
This paper fully relies on the results of our paper [12] where we considered the orthogonality of rational functions Wn(s)W_n(s)Wn(s) as the Laplace transform images of a set of orthoexponential functions, obtained from Jacobi polynomials, and as the Laplace transform images of Laguerre polynomials. In Ref. [12] we proved that the orthogonality of Jacobi and Laguerre polynomials is induced by
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.
On q-orthogonal polynomials, dual to little and big q-Jacobi polynomials
Journal of Mathematical Analysis and Applications, 2004
We derive discrete orthogonality relations for polynomials, dual to little and big q-Jacobi polynomials. This derivation essentially requires use of bases, consisting of eigenvectors of certain self-adjoint operators, which are representable by a Jacobi matrix. Recurrence relations for these polynomials are also given.
On q-Hermite polynomials and their relationship with some other families of orthogonal polynomials
2011
We review properties of the q−q-q−Hermite polynomials and indicate their links with the Chebyshev, Rogers--Szeg\"{o}, Al-Salam--Chihara, continuous q−q-q−% utraspherical polynomials. In particular we recall the connection coefficients between these families of polynomials. We also present some useful and important finite and infinite expansions involving polynomials of these families including symmetric and non-symmetric kernels. In the paper we collect scattered throughout literature useful but not widely known facts concerning these polynomials. It is based on 43 positions of predominantly recent literature.
On the q-Charlier Multiple Orthogonal Polynomials
Symmetry, Integrability and Geometry: Methods and Applications, 2015
We introduce a new family of special functions, namely q-Charlier multiple orthogonal polynomials. These polynomials are orthogonal with respect to q-analogues of Poisson distributions. We focus our attention on their structural properties. Raising and lowering operators as well as Rodrigues-type formulas are obtained. An explicit representation in terms of a q-analogue of the second of Appell's hypergeometric functions is given. A high-order linear q-difference equation with polynomial coefficients is deduced. Moreover, we show how to obtain the nearest neighbor recurrence relation from some difference operators involved in the Rodrigues-type formula.
Results for some inversion problems for classical continuous and discrete orthogonal polynomials
Journal of Physics A: Mathematical and General, 1997
Explicit expressions for the coefficients in the expansion of classical discrete orthogonal polynomials (Charlier, Meixner, Krawtchouck, Hahn, Hahn-Eberlein) into the falling factorial basis are given. The corresponding inversion problems are solved explicitly. This is done by using a general algorithm, recently developed by the authors, which is also applied to this kind of inversion problem but relating the x n basis and the classical (continuous) orthogonal polynomials of Jacobi, Laguerre, Hermite and Bessel.
$q$-Laguerre polynomials and big qqq-Bessel functions and their orthogonality relations
Methods and applications of analysis, 1999
The g-Laguerre polynomials correspond to an indeterminate moment problem. For explicit discrete non-N-extremal measures corresponding to Ramanujan's i^i-summation, we complement the orthogonal g-Laguerre polynomials to an explicit orthogonal basis for the corresponding L 2 -space. The dual orthogonal system consists of so-called big q-Bessel functions, which can be obtained as a rigorous limit of the orthogonal system of big g-Jacobi polynomials. Interpretations on the SU(1,1) and 12(2) quantum groups are discussed.
Integral and series representations of q-polynomials and functions: Part I
Analysis and Applications, 2018
By applying an integral representation for q k 2 we systematically derive a large number of new Fourier and Mellin transform pairs and establish new integral representations for a variety of q-functions and polynomials that naturally arise from combinatorics, analysis, and orthogonal polynomials corresponding to indeterminate moment problems. These functions include q-Bessel functions, the Ramanujan function, Stieltjes-Wigert polynomials, q-Hermite and q −1-Hermite polynomials, and the q-exponential functions eq, Eq and Eq. Their representations are in turn used to derive many new identities involving q-functions and polynomials. In this work we also present contour integral representations for the above mentioned functions and 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.