A CHARACTERIZATION OF THE FOUR CHEBYSHEV ORTHOGONAL FAMILIES

We obtain a property which characterizes the Chebyshev orthogonal polynomials of first, second, third, and fourth kind. Indeed, we prove that the four Chebyshev sequences are the unique classical orthogonal polynomial families such that their linear combinations, with fixed length and constant coefficients, can be orthogonal polynomial sequences.


Introduction
The classical orthogonal polynomials (OP) on the real line, which are the most useful and important families, can be characterized by different conditions (see [3]).In the case of bounded support, the classical model is the family of Jacobi polynomials, which has been extensively studied.Very well-known families of Jacobi polynomials are the so-called Chebyshev polynomials of first, second, third, and fourth kind (see [9]).
The orthogonal polynomials with respect to rational modifications of these Chebyshev weights are the Bernstein polynomials (see [3,9]).It is well known that the Bernstein polynomials can be expressed as linear combinations of Chebyshev polynomials, with fixed length and constant coefficients (see [4,5]).This property of orthogonality satisfied by the linear finite combinations of sequences of orthogonal polynomials is quite general.Indeed, taking into account the result of Uvarov (see [10]), the orthogonal polynomial sequences with respect to a rational modification of a positive measure on an interval can be written as a linear combination of the orthogonal polynomials with respect to the original measure, with fixed length and coefficients depending on the degrees.Although the representation of the Bernstein polynomials is a consequence of Uvarov's result, it is important to note that in this first case, the coefficients are constant.
The study of linear combinations of orthogonal polynomials is an interesting subject for different reasons.For example, several quadrature formulas have been obtained choosing the nodal points as the zeros of some linear combinations.Another reason could be the important role that these linear combinations of OP play in the problem of linearization of products of OP, as well as, in the problem of approximation of the polynomial solution for differential equations.For a detailed presentation of these problems and some others, see [6].
The general problem of the orthogonality for combinations of length three was studied in [2].Indeed, there necessary and sufficient conditions on the sequences of coefficients of the linear combinations with fixed length three were obtained in order that the new sequences be orthogonal.Very interesting examples appear in [2] and also in [1].Another interesting contribution in the same direction is [7], in which the authors study the orthogonality of linear combinations of two orthogonal polynomial sequences.Applying the results in [2,7], we have studied in [4] the following inverse problem, that is, we have determined that the four Chebyshev sequences are the unique Jacobi sequences such that their linear combinations with length 2 or 3 and constant coefficients can be orthogonal.
Our aim in this paper is to prove that the four Chebyshev families are the unique families among all the classical orthogonal polynomials such that linear finite combinations with arbitrary fixed length and constant coefficients can be orthogonal sequences.However, this property can be satisfied by nonclassical orthogonal polynomial sequences.Indeed, we give examples of nonclassical orthogonal polynomial sequences such that linear combinations, with fixed length and constant coefficients, are orthogonal.
The structure of the paper is the following.In Section 2, first we recall some properties concerning the classical orthogonal polynomials and we present the Bernstein orthogonal polynomials.Moreover, we construct several examples of finite linear combinations, with constant coefficients, of nonclassical orthogonal polynomials, such that they are again orthogonal sequences.Finally, in Section 3, we prove our main result, that is, the property that characterizes the four Chebyshev orthogonal polynomial sequences.
By virtue of their orthogonality, the classical monic polynomials satisfy the following three-term recurrence relations: where , with P 0 (x) = 1 and P −1 (x) = 0 (β 0 = −α in case α + β = 0); E. Berriochoa et al. 2073 where with where with H 0 (x) = 1 and Notice that the unique classical families with constant coefficients in the three-term recurrence relation are those Jacobi families corresponding to α = ±1/2 and β = ±1/2, that is, the Chebyshev families.Indeed, they satisfy for n ≥ 2 the same recurrence relation The Bernstein polynomials generalize the classical Chebyshev polynomials.They are orthogonal with respect to positive measures obtained by introducing a positive polynomial divisor into the Chebyshev weight function (see [3,5,9]).Next, we recall a useful representation for these Bernstein sequences.
Let q k (x) be a polynomial with real coefficients and positive in [−1 ,1].By Féjer-Riesz representation, there exists a polynomial If dµ 1 (x) is the rational modification of the Chebyshev measure of first kind dµ 1 (x) = dx/(πq k (x) √ 1 − x 2 ), and {P n (x,µ 1 )} is the monic orthogonal polynomial sequence with respect to µ 1 (MOPS(µ 1 )), then for n ≥ k, where { Ṫn (x)} is the sequence of monic Chebyshev polynomials of first kind.If dµ 2 (x) is the rational modification of the Chebyshev measure of second kind where { Un (x)} is the sequence of monic Chebyshev polynomials of second kind.If dµ 3 (x) is the rational modification of the Chebyshev measure of third kind where { Ẇn (x)} is the sequence of monic Chebyshev polynomials of third kind.
2074 Chebyshev orthogonal polynomials If dµ 4 (x) is the rational modification of the Chebyshev measure of fourth kind dµ 4 (x) = 1/(πq k (x)) (1 − x)/(1 + x)dx, and {P n (x,µ 4 )} is the (MOPS(µ 4 )), then for n≥k, where { Vn (x)} is the sequence of monic Chebyshev polynomials of fourth kind.Hence, it is clear that these types of linear combinations of Chebyshev polynomials are again orthogonal sequences.Although our aim is to prove that this property characterizes these Chebyshev families among all classical families, nevertheless, this property can be satisfied by nonclassical families.Next we present examples of nonclassical sequences such that they satisfy the same above property as the Chebyshev families.
Example 2.1.Let {P n (x)} be the MOPS satisfying the recurrence relation (2.11) with P −1 (x) = 0 and P 0 (x) = 1, and such that η n = a n , with a ∈ R, a = 0, and n = 0, for all n.Then the sequence of monic polynomials {Q n (x)} defined by is the MOPS with respect to a regular linear functional.
It is clear that the sequence {P n (x)} is different from the Chebyshev families.Besides, it is easy to see that the sequence {Q n (x)} satisfies the following three-term recurrence relation: with c n = a n−1 and b n = n .Indeed, (2.14) Therefore, by Favard's theorem (see [3]), the sequence {Q n (x)} is the MOPS with respect to a regular linear functional or a positive definite functional.
In order to obtain that the sequence {Q n (x)} be a MOPS with respect to a positve Borel measure on [−1,1], we can take, for example, n = 1/4 and a = 1/16.Then b n = 1/4 and c n = 1/64, and therefore the measure of orthogonality of the sequence {Q n (x)} belongs to the class M(1/4,1/4), see [8].Hence, the support of the measure is a subset of [−1,1].E. Berriochoa et al. 2075

Linear combinations with constant coefficients of classical polynomials
Jacobi polynomials.Now we consider the sequence of monic polynomials with k a fixed integer, and a i (i = 1,...,k) real numbers such that a k = 0. We continue denoting by {P (α,β) n (x)} the MOPS with respect to the Jacobi weight with parameters α and β.Theorem 3.1.Let {Q n+k (x)} be the sequence defined in (3.1).If {Q n (x)} is the MOPS with respect to a linear regular functional or a positive finite measure on [−1,1], then the sequence Proof.If {Q n (x)} n≥1 is a sequence of monic orthogonal polynomials, then there exist two sequences of real numbers { n } and {η n }, such that the following three-term recurrence relation holds, see [9], Taking into account relation (3.1), we can rewrite (3.2) as follows: On the other hand, if we use the recurrence relation satisfied by the Jacobi polynomials, then (3.1) becomes 2076 Chebyshev orthogonal polynomials By equating the second members of expressions (3.3) and (3.4), we get the following relations: If we substitute n+k−1 by β n+k−1 and η n+k−1 by γ n−1 , we can rewrite the other above equations using only the coefficients of the recurrence relation of the sequence Now we can consider the following two cases.
(i) If β = α, the expression γ n+k−(i+1) = γ n−1 can be written as follows: from which we obtain, after some computations, It is easy to see that the coefficients of the powers n 4 , n 3 , and n 2 are equal in both members of the above formula.By equating the coefficients of n and the E. Berriochoa et al. 2077 independent terms, we get Since k − i = 0, the system is compatible only for α 2 = 1/4.Therefore, we obtain that from which we deduce Developing in powers of n, we see that the coefficients of n 4 , n 3 , and n 2 are the same in both members of the equality, and by equating the coefficients of n and the independent term, it yields from which it follows that α 2 = 1/4.Therefore, we obtain that If we take into account the expression of the coefficients in the recurrence relation for the Jacobi polynomials and we denote R = a k /a k−1 , we have or equivalently Finally, we compute the coefficients of n 6 and n 5 obtaining the following system: If we substitute the first equation in the second one, we obtain k = 0, which is impossible.Therefore, case (2) cannot be satisfied.
Laguerre and Hermite polynomials.We consider the sequence of monic polynomials with k a fixed integer, and a i (i = 1,...,k) real numbers such that a k = 0. We continue denoting by {L (α) n (x)} the MOPS with respect to the Laguerre weight with parameter α.We also consider the sequence of monic polynomials with k a fixed integer, and a i (i = 1,...,k) real numbers such that a k = 0. We continue denoting by {H n (x)} the MOPS with respect to the Hermite weight.
Theorem 3.2.Let {R n+k (x)} be the sequence defined in (3.17), and let {S n+k (x)} be the sequence defined in (3.18).Then the sequence {R n (x)} is not the MOPS with respect to any linear regular functional or positive finite measure on [0,+∞), and {S n (x)} is not the MOPS with respect to any linear regular functional or positive finite measure on (−∞,+∞).
Proof.The idea is to solve the system (3.6) that appears in the previous theorem with the recurrence coefficients of the Laguerre and Hermite polynomials respectively.In the Laguerre case, if β n+k−1 − β n−1 = 0, then we get k = 0, which is impossible, and if β n+k−1 − β n−1 = 0, we also get to a contradiction.In the Hermite case, since β n = 0, we obtain γ n+k−1 = γ n−1 , which implies that k = 0.
As a consequence we obtain the following corollary.
Corollary 3.3.Consider the sequence of monic polynomials

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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First
Round of ReviewsMarch 1, 2009 .19) E. Berriochoa et al. 2079with k a fixed integer, and a i (i = 1,...,k) real numbers such that a k = 0, and {P n (x)} a classical MOPS (Jacobi, Laguerre, or Hermite).If {Q n (x)} is the MOPS with respect to a linear regular functional, then {P n (x)} must be one of the four Chebyshev orthogonal polynomial sequences.