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Recursive approach to connection and linearization coefficients between polynomials
... The study of such a problem has attracted lot of interest in the last few years. Special emphasis was given to the classical continuous (Hermite, Laguerre, Jacobi and Bessel) [8,9,10,11,15,19202124,262728 and the discrete cases (Charlier, Meixner, Kravchuk and Hahn) [7,9,12,14,27]. The main aim of the present paper is to show that the ideas given in [14,26] can be extended in a very easy way to the q-polynomials on the exponential lattice x(s) = c 1 q s + c 3 . ...
... The study of such a problem has attracted lot of interest in the last few years. Special emphasis was given to the classical continuous (Hermite, Laguerre, Jacobi and Bessel) [8,9,10,11,15,19202124,262728 and the discrete cases (Charlier, Meixner, Kravchuk and Hahn) [7,9,12,14,27]. The main aim of the present paper is to show that the ideas given in [14,26] can be extended in a very easy way to the q-polynomials on the exponential lattice x(s) = c 1 q s + c 3 . If fact, if P n (x), Q m (x) and R j (x) are polynomials in x(s) = c 1 q s + c 3 , then it is possible to ÿnd a recurrence relation for the linearization coeecients L mjn , which is an alternative approach to the one given in [5]. ...
In this paper we present a simple recurrent algorithm for solving the linearization problem involving some families of q-polynomials in the exponential lattice x(s)=c1qs+c3. Some simple examples are worked out in detail.
... Usually, the coefficients are given recurrently or in an algebraic manner by means of terminating hypergeometric functions. But, generally speaking, methods proposed so far are strongly tailored to the specific families of polynomials involved and use several of their characterizing properties: recurrence relations, generating functions, orthogonality weights, etc. (see, e.g., [8, 9, 14]). The aim of this paper is to set up an alternative approach to solve the general linearization problem (3) for hypergeometric polynomials, that is, to determine the analytic expression for c(n,m,r) directly in terms of the coefficients of the second-order differential equations (1) corresponding to the hypergeometric polynomials p,, q, and h,,. ...
... The examples above illustrate our method to solve both the standard linearization and connection problems, which makes use of the coefficients of the second order differential equations corresponding to the hypergeometric polynomials involved. This approach is complementary to that described in [9, 14], which supplies the expansion coefficients recursively but needs two or more characterization properties of the involved polynomials. Furthermore, the method described here may be extended in a straightforward way to polynomials with a discrete orthogonality [3] , as well as to q-poly- nomials [2]. ...
We consider the problem of finding closed analytical formulas for both the linearization and connection coefficients for hypergeometric-type polynomials, directly in terms of the corresponding differential equations. We illustrate the method by producing explicit formulas for Hermite polynomials.
... It will also be seen that we can give closed-form expressions for the connection coefficients themselves, at least so far, as it is easy to give closed-form expressions for the Riordan arrays concerned. Thus, in the special case of orthogonal polynomials that are defined by Riordan arrays, we can achieve two of the main goals involved in the study of connection coefficients for orthogonal polynomials1234567. If í µí± í µí± (í µí±¥) and í µí± í µí± (í µí±¥) are two families of orthogonal polynomials , the connection coefficients í µí± í µí±,í µí± of í µí± í µí± in terms of í µí± í µí± , defined by í µí± í µí± = í µí± ∑ ...
The Chebyshev-Boubaker polynomials are the orthogonal polynomials whose coefficient arrays are defined by ordinary Riordan arrays. Examples include the Chebyshev polynomials of the second kind and the Boubaker polynomials. We study the connection coefficients of this class of orthogonal polynomials, indicating how Riordan array techniques can lead to closed-form expressions for these connection coefficients as well as recurrence relations that define them.
... Several other authors have contributed to the study of Bessel polynomials, among them, Agarwal [1], Artés et al. [3], Carlitz [6], Evans et al. [15], Godoy et al. [16], Han and Kwon [19], Lewanowicz [24] [25], Lewanowicz and Wo´zny [26], Luke [27, vol. 2], Ronveaux et al. [30] [31] and Zarzo et al. [35] ...
A new formula expressing explicitly the integrals of Bessel polynomials of any degree and for any order in terms of the Bessel polynomials themselves is proved. Another new explicit formula relating the Bessel coefficients of an expansion for infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is also established. An application of these formulae for solving ordinary differential equations with varying coefficients is discussed.
... This is particularly true for the connection problem between any two discrete polynomials. In the last decade, many algorithms [4,17,18], have developed a recurrent method, called the Na ViMa algorithm, for solving this problem. Lewanowicz111213, Lewanowicz and Wo´znyWo´zny [14] and Wo´znyWo´zny [20] have presented a very similar algorithm for finding the recurrence relation for both connection and linearization coefficients. ...
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 Δ q f(x), and for the moments x ℓΔ 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.
... Usually, the determination of the expansion coefficients in these particular cases required an indepth knowledge of special functions [6, 7] and, at times, ingenious induction arguments based on the three-term recurrence relation of the involved orthogonal polynomials (see Álvarez-Nodarse [8] and the references contained therein). Ronveaux et al.91011, Godoy et al. [12] and Area et al. [13] have developed a recurrent method, called Na V iMa algorithm, for solving the connection problem for all families of classical orthogonal polynomials, as well as some special kind of linearization problem and used it for solving different problems related with the associated Sobolev-type polynomials, etc. [9, 14, 15]. Let us point out that there is a very similar algorithm for finding the recurrence relations for both connection and linearization coefficients due to Lewanowicz161718. ...
Two formulae expressing explicitly the difference derivatives and the moments of a 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 a general-order difference derivatives q f(x), and for the moments x ℓ 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 two families of Meixner, Kravchuk and Charlier, is described. Three analytical formulae for the connection coefficients between Hahn–Charlier, Hahn–Meixner and Hahn–Kravchuk are also developed.
... In the last decade, many algorithms for solving the connection problem for all families of classical orthogonal polynomials (continuous/discrete) are developed. Ronveaux et al.789, Godoy et al. [10] and Area et al. [11] developed a recurrent method, called Na ViMa algorithm, for solving this problem, as well as some special kind of linearization problem, and used it for solving different problems related to the associated Sobolev-type polynomials, [7,121314. Lewanowicz [15, 17], Lewanowicz and Wo´znyWo´zny [18] and Wo´znyWo´zny [19] presented a very similar algorithm for finding the recurrence relation for both connection and linearization coefficients. ...
A formula expressing explicitly the difference derivatives of Hahn polynomials of any degree and for any order in terms of Hahn polynomials themselves is proved. Another explicit formula, which expresses the Hahn expansion coefficients of a general-order difference derivative of an arbitrary polynomial of a discrete variable in terms of its original Hahn coefficients, is also given. A formula for the Hahn coefficients of the moments of one single Hahn polynomial of certain degree is proved. A formula for the Hahn coefficients of the moments of a general-order difference derivative of an arbitrary polynomial of a discrete variable in terms of its Hahn coefficients is also obtained. Application of these formulae for solving ordinary difference equations with varying polynomial 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–Hahn, Meixner–Hahn, Kravchuk–Hahn and Charlier–Hahn is also developed.
... For instance, the m = n case of the standard linearization formula (2) for the sequence {pn(x)} is often required to evaluate the logarithmic potentials of orthogonal polynomials /~vn which appear in the calculation of the position and momentum information entropies of quantum systems [2,3]. The literature on these topics is extremely vast (for a review, see [4]; see also5678910), and a wide variety of methods have been devised for computing the linearization and connection coefficients, either in closed form or by means of recurrence relations (usually in k). Very recently, it has been shown by Lewanowicz [1] that the connection problem between two families of orthogonal polynomials can at times be solved in a very simple way by taking advantage of known theorems from the theory of generalized hypergeometric functions. ...
Several linearization-like and connection-like formulae relating the classical Gegenbauer polynomials and their squares are obtained using a theorem of the theory of generalized hypergeometric functions.
... Linearization and connection problems are frequently encountered in applications; for instance, the m = n case of the standard linearization formula (2) is often required to evaluate the logarithmic potentials of orthogonal polynomials appearing in the calculation of the position and momentum information entropies of quantum systems (see [8] and references therein). The literature on these topics is extremely vast (see, e.g., [1,2,4,21,23,24]), and a wide variety of methods have been devised for computing the linearization and connection coeecients, either in closed form or by means of recurrence relations (usually in k). Very recently, it has been shown by Lewanowicz [16] that the connection problem (3) can at times be solved in a very simple way by taking advantage of two known results from the theory of generalized hypergeometric functions, derived by Fields and Wimp [11] (see also [17, Vol. ...
The connection problem is considered in a hypergeometric function framework for (i) the two most general families of polynomials belonging to the Askey scheme (Wilson and Racah), and (ii) some generalized Laguerre and Jacobi polynomials falling outside that scheme (Sister Celine, Cohen and Prabhakar–Jain), which are relevant to the study of quantum-mechanical systems and include as particular cases, the generalizations of the classical families with Sobolev-type orthogonality. In addition, using the same method three new linearization-like formulae for the Gegenbauer polynomials are also derived: a linearization formula that generalizes the m=n case of Dougall's formula, the analogue of the m=n case of Nielsen's inverse linearization formula for Hermite polynomials, and a connection formula for the squares. Closed analytical formulae for the corresponding connection and linearization coefficients are given in terms of hypergeometric functions of unit argument, which at times can be further simplified and expressed as single hypergeometric terms.
... In Section 4 the coeecients of the linearization formula (1.2) are found in a fully analytical way in terms of the polynomial coeecients of the second-order diierence hypergeometric equations satissed by the polynomials p n (x) and q j (x). Notice that fq j g is not necessarily an orthogonal set, neither r m (x) is obliged to have a hypergeometric character, what widely extends the linearization formulas considered in the literature 5, 8, 9, 11, 17, 23, 25, 30, 31, 37, 45, 46, 47, 48]. Indeed, most authors study linearizaton formulas between classical discrete polynomials, usually within the same family (see e.g. 10, 11, 14, 53] save some of them, who nd a few other formulas which either involve polynomials of diferent classical families 17, 45] or include one of the aforementioned non-orthogonal families together with polynomials of the same classical system 11]. ...
Starting from the second-order difference hypergeometric equation satisfied by the set of discrete orthogonal polynomials , we find the analytical expressions of the expansion coefficients of any polynomial rm(x) and of the product rm(x)qj(x) in series of the set . These coefficients are given in terms of the polynomial coefficients of the second-order difference equations satisfied by the involved discrete hypergeometric polynomials. Here qj(x) denotes an arbitrary discrete hypergeometric polynomial of degree j. The particular cases in which corresponds to the non-orthogonal families ∗xm∗, the rising factorials or Pochhammer polynomials and the falling factorial or Stirling polynomials ∗x[m]∗ are considered in detail. The connection problem between discrete hypergeometric polynomials, which here corresponds to the product case with m = 0, is also studied and its complete solution for all the classical discrete orthogonal hypergeometric (CDOH) polynomials is given. Also, the inversion problems of CDOH polynomials associated to the three aforementioned nonorthogonal families are solved.
... We remark the fact that the literature on this subject is extremely vast and a wide variety of methods have been developed using several techniques like, recursion, hypergeometric approach, inversion and other combinatorial formulas, lowering operators, etc, to treat the connection coefficients for continuous, discrete and q−polynomials. See, among others, the publications [1, 2, 6, 7, 10,12131415161718242526. Here, we proceed with the simplest method based only on the recurrence relation fulfilled by any orthogonal sequence, which leads to a recurrence relation satisfied by the connection coefficients. ...
We deal with the problem of obtaining closed formulas for the connection coefficients between orthogonal polynomials and the
canonical sequence. We use a recurrence relation fulfilled by these coefficients and symbolic computation with the Mathematica language. We treat the cases of Gegenbauer, Jacobi and a new semi-classical sequence.
... The latter was a situation not solved up to now in the literature save for some particular cases. Furthermore, our method may be readily used to connect not only classical polynomials of nonsimilar character (e.g., Laguerre polynomials and Hermite polynomials), a problem which deserves further work [20] because of its many applications in a variety of fields in mathematics and physics, but also to obtain connection formulas between classical and non-standard orthogonal polynomials of hypergeometric type as well as between hypergeometric polynomials, not necessarily orthogonal, and a classical or non-standard orthogonal set. Finally, let us mention that the method described here may be extended in a straightforward manner to polynomials with a discrete orthogonality [3] as well as to q-polynomials [2] . ...
16 pages, no figures.-- MSC1991 codes: 33C45; 42C05. MR#: MR1625951 (99i:33012) Zbl#: Zbl 0944.33011 Let us consider an arbitrary hypergeometric polynomial $q_j(x)$ and a set of orthogonal hypergeometric polynomials $\{p_n(x)\}$ in the domain of orthogonality $\Gamma$. Here the expansion coefficients of $x $ and $x q_j(x),\ m\in{\bf N}_0$, in series of the set $\{p_n(x)\}$ are found in terms of the polynomials $\sigma(x)$ and $\tau(x)$ characterizing the second-order differential equations satisfied by the hypergeometric polynomials involved. The resulting general expressions, which are given in an explicit and compact form, are used to produce known (for checking) and unknown expansions for various concrete classical orthogonal polynomials. This work has been partially supported by the European project INTAS-93-219-ext. The first author also acknowledges the partial financial support of the Fundació Aula (Barcelona, Spain). The second author has been also partially supported by the Dirección General de Enseñanza Superior (DGES) of Spain under grant PB 95-1205 and by the Junta de Andalucía FQM207. Publicado
Let {P_k } and {Q_k} be any two sequences of classical orthogonal polynomials. Using theorems of the theory of generalized hypergeometric functions, we give closed-form expressions as well as recurrence relations for the coefficients a_{n,k} in the connection equation Q_n = \sum^n_{k=0} a_{n,k} P_k (n ∈ N).
In this paper, a concise method to express by induction the so-called extended connection coefficients between two orthogonal polynomial sequences in a general sense is highlighted. Some illustrative examples arising from the semi-classical case are studied.
Let {P k } and Q k be any two sequences of classical orthogonal polynomials. Using theorems of the theory of generalized hypergeometric functions, we give closed-form expressions as well as recurrence relations for the coefficients a n,k in the connection equation Qn = n k=0 a n,k P k (n ∈ N).
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