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(3) Author's personal copy Mediterr. J. Math. 6 (2010) 139–150 DOI 10.1007/s00009-012-0223-3 © 2012 Springer Basel AG. Mediterranean Journal of Mathematics. Cubic Decomposition of 2-Orthogonal Polynomial Sequences Pascal Maroni and Teresa A. Mesquita∗ Abstract. In the present work, we study the general cubic decomposition (CD) of a 2-orthogonal polynomial sequence, beginning with a characterization of all the elements involved in such CD. The recurrence coefficients of the 2-orthogonal sequences which admit a diagonal CD are described and we prove that the correspondent principal components are also 2-orthogonal. Finally, we analyse the CD of a 2-symmetric and 2-orthogonal sequence. Mathematics Subject Classification (2010). Primary 42C05; Secondary 33C45. Keywords. 2-orthogonal polynomials, finite-type relations, cubic decomposition.. Introduction In 1992, while studying the classical orthogonal polynomials of dimension d, Douak and Maroni [7] obtained a natural and complete cubic decomposition of a d-symmetric and d-orthogonal sequence (where d is a positive integer), proving some properties fulfilled by the polynomial sequences involved. More precisely, given such sequence {Wn }n≥0 , there are three other polynomial sequences {Pn }n≥0 , {Qn }n≥0 and {Rn }n≥0 , so that W3n (x) = Pn (x3 ), W3n+1 (x) = xQn (x3 ) and W3n+2 (x) = x2 Rn (x3 ), and which are also dorthogonal. The relation between W3n (x) and Pn (x) reminds us the problem first proposed by Chihara in 1964 [6], of finding a pair of orthogonal polynomial sequences {Wn }n≥0 and {Pn }n≥0 so that W3n (x) = Pn (z), where z is a cubic in x. The first answer was presented in 1966 by Barrucand and Dickinson [1]. Further investigations, regarding this same problem, were carried out trough the last decades, namely, in 1993 by Marcellán and Sansigre [14], in 2000 and 2001 by Marcellán and Petronilho ∗ Corresponding. author..

(4) Author's personal copy 2. P. Maroni and T. A. Mesquita. [17, 18]. All these works deal with a pair of orthogonal sequences fulfilling an identity analogous to one of the three indicated above, and the remaining two of the three subsequences {W3n (x)}n≥0 , {W3n+1 (x)}n≥0 and {W3n+2 (x)}n≥0 , are written as rational fractions. Wider researches, concerning polynomial transformations of measures, sieved polynomials, polynomial mappings and positive-definite linear functionals touched this theme, as, for instance, [3, 11, 9, 20, 4, 5, 16, 15, 13, 12]. Outside the context of orthogonality, the single definition of a 2-symmetric sequence raises a special case of a full polynomial cubic decomposition (CD) which was introduced and studied for orthogonal sequences in 1989 by Rodriguez and Tasis [10], and more recently, completely generalized by Maroni et al. [23]. With respect to this last general CD, a 2-symmetric sequence has the most simple CD, which we will call diagonal CD. With regard to d-symmetric sequences, we must refer that, in 2001 [2], Ben Cheikh proved several results concerning orthogonal (n − 1)-symmetric curves achieving a unified treatment of some problems related to the symmetrization of sequences of orthogonal polynomials on a real line or on a unit circle. For this matter, and as expected, some of the functional relations which characterize a diagonal CD are already proved in this work of 2001, although under the orthogonality hypothesis. The aim of the present work is twofold. On one hand, we present basic results concerning the CD of a 2-orthogonal polynomial sequences, and, on the other hand, we study the diagonal CD (considering all parameters) for this kind of polynomial sequence. In the first section we present the basic definitions and results needed in the sequel, giving a special attention to the presentation of the CD in study and recalling an important theorem which characterizes the nine sequences that constitute such CD. The second section is reserved to the proof of a characterization of all the elements involved in a CD of a 2-orthogonal polynomial sequence and a corollary from which we can establish sufficient conditions for the 2-orthogonality of the principal components. Moreover, the recurrence coefficients of the 2-orthogonal sequences which admit a diagonal CD are described and we prove that the correspondent principal components are also 2-orthogonal, generalizing the result obtained by Douak and Maroni. In the third section, we focus our attention in 2-orthogonal and 2-symmetric sequences.. 1. Preliminaries and notations 1.1. Basic definitions and results Let P be the vector space of polynomials with coefficients in C and let P " be its dual. We denote by "u, p# the action of the form or linear functional u ∈ P " on p ∈ P. In particular, (u)n = "u, xn #, n ≥ 0, are called the moments of u. In the following, we will call polynomial sequence (PS) to any sequence {Wn }n≥0 such that deg Wn = n, ∀n ≥ 0. We will also call monic polynomial sequence.

(5) Author's personal copy Vol. 6 (2010). Cubic Decomposition of 2-Orthogonal Polynomial Sequences. 3. (MPS) a PS so that in each polynomial the leading coefficient is equal to one. Given a MPS {Wn }n≥0 , there are complex sequences, {χn,ν }0≤ν≤n, n≥0 and {βn }n≥0 such that W0 (x) = 1, W1 (x) = x − β0 , n ! Wn+2 (x) = (x − βn+1 )Wn+1 (x) − χn,ν Wν (x).. (1.2). βn = "wn , xWn (x)#, χn,ν = "wν , xWn+1 (x)#, 0 ≤ ν ≤ n, n ≥ 0.. (1.3). (1.1). ν=0. This relation is called the structure relation of {Wn }n≥0 , and {βn }n≥0 and {χn,ν }0≤ν≤n, n≥0 are called the structure coefficients. Also, there exists a unique sequence {wn }n≥0 , wn ∈ P " , called the dual sequence of {Wn }n≥0 , such that "wn , Wm # = δn,m , n, m ≥ 0, where δn,m denotes the Kronecker symbol. Moreover, we can prove that [21]. Lemma 1.1. [21] For each u ∈ P " and each m ≥ 1, the two following propositions are equivalent. a. "u, Wm−1 # = ( 0, "u, Wn # = 0, n ≥ m. "m−1 b. ∃λν ∈ C, 0 ≤ ν ≤ m − 1, λm−1 (= 0 such that u = ν=0 λν wν .. Definition 1.2. [7, 8, 19, 25] Given Γ1 , Γ2 , . . . , Γd ∈ P " , d ≥ 1, the polynomial sequence {Wn }n≥0 is called d-orthogonal polynomial sequence (d-OPS) with respect to Γ = (Γ1 , . . . , Γd ) if it fulfils "Γα , Wm Wn # = 0, n ≥ md + α, m ≥ 0, α. ( 0, m ≥ 0, "Γ , Wm Wmd+α−1 # =. (1.4) (1.5). for each integer α = 1, . . . , d. The conditions (1.4) are called the d-orthogonality conditions and the conditions (1.5) are called the regularity conditions. In this case, the functional Γ, of dimension d, is said regular. Remark 1.3. If d = 1, then we meet again the notion of regular orthogonality. As a further matter, the d-dimensional"functional Γ is not unique. Nevertheα−1 α less, from Lemma 1.1, we have: Γα = ν=0 λα ν uν , λα−1 (= 0, 1 ≤ α ≤ d. Therefore, since w = (w0 , . . . , wd−1 ) is unique, from now on, we will only consider the canonical functional of dimension d, w = (w0 , . . . , wd−1 ), saying that {Wn }n≥0 is d-OPS (d ≥ 1) with respect to w = (w0 , . . . , wd−1 ) if ( 0, m ≥ 0, "wν , Wm Wn # = 0, n ≥ md + ν + 1, m ≥ 0, "wν , Wm Wmd+ν # = for each integer ν = 0, 1, . . . , d − 1. A d-MOPS satisfies also a recurrence relation of order (d + 1) as the following result establishes. Theorem 1.4. [19] Let {Wn }n≥0 be a MPS. The following assertions are equivalent: a. {Wn }n≥0 is d-orthogonal with respect to w = (w0 , . . . , wd−1 )..

(6) Author's personal copy 4. P. Maroni and T. A. Mesquita. b. {Wn }n≥0 satisfies a (d + 1)-order recurrence relation (d ≥ 1): Wm+d+1 (x) = (x − βm+d )Wm+d (x) −. d−1 !. ν=0. d−1−ν γm+d−ν Wm+d−1−ν (x), m ≥ 0,. with initial conditions W0 (x) = 1, W1 (x) = x − β0 and if d ≥ 2 : n−2 ! d−1−ν Wn (x) = (x − βn−1 )Wn−1 (x) − γn−1−ν Wn−2−ν (x), 2 ≤ n ≤ d, ν=0. 0 (= 0, m ≥ 0. and regularity conditions: γm+1 c. For each (n, ν), n ≥ 0, 0 ≤ ν ≤ d − 1, there are d polynomials Λµ (n, ν), 0 ≤ µ ≤ d − 1 such that. wnd+ν =. d−1 !. µ=0. Λµ (n, ν)wµ , n ≥ 0, 0 ≤ ν ≤ d − 1, and also fulfilling. deg Λµ (n, ν) = n, 0 ≤ ν ≤ d − 1,. deg Λµ (n, ν) ≤ n, 0 ≤ µ ≤ ν − 1, if 1 ≤ ν ≤ d − 1,. deg Λµ (n, ν) ≤ n − 1, ν + 1 ≤ µ ≤ d − 1, if 0 ≤ ν ≤ d − 2.. Definition 1.5. [7] A PS {Wn }n≥0 is d-symmetric if it fulfils. Wn (ξk x) = ξkn Wn (x), n ≥ 0, k = 1, 2, . . . , d, $ # d+1 = 1. where ξk = exp 2ikπ d+1 , k = 1, . . . , d, ξk If d = 1, then ξ1 = −1 and we meet the definition of a symmetric PS in which we have the following property Wn (−x) = (−1)n Wn (x), n ≥ 0. Lemma 1.6. [7] A PS {Wn }n≥0 is d-symmetric if and only if it fulfils Wm (x) = xµ. n !. am,(d+1)p+µ x(d+1)p ,. p=0. where m = (d + 1)n + µ, 0 ≤ µ ≤ d, n ≥ 0.. Definition 1.7. [7] The functional (Γ1 , . . . , Γd ) is d-symmetric if, for n ≥ 0,. (Γν )(d+1)n+µ−1 = "Γν , x(d+1)n+µ−1 # = 0, 1 ≤ ν ≤ d, 1 ≤ µ ≤ d + 1, ν (= µ.. In particular, the functional w = (w0 , . . . , wd−1 ) is d-symmetric if for each integer 0 ≤ j ≤ d − 1: (wj )(d+1)n+i = 0, i = 0, 1, . . . , d, i (= j, n ≥ 0. If d = 1, then we meet the definition of a symmetric form in which we have the following property (Γ)(2n+1) = 0, n ≥ 0. Theorem 1.8. [7] Let {Wn }n≥0 be a d-orthogonal MPS with respect to the functional w = (w0 , . . . , wd−1 ). The following statements are equivalent: a. The functional Γ is d-symmetric. b. {Wn }n≥0 is d-symmetric. c. {Wn }n≥0 fulfils the following recurrence relation: 0 Wn (x), n ≥ 0, Wn+d+1 (x) = xWn+d (x) − γn+1. Wn (x) = xn , 0 ≤ n ≤ d..

(7) Author's personal copy Vol. 6 (2010). Cubic Decomposition of 2-Orthogonal Polynomial Sequences. 5. Definition 1.9. [22] We say that a PS {Wn }n≥0 is compatible with Φ if Φwn (= 0, n ≥ 0. Remark 1.10. Any OPS is compatible with any monic polynomial and any MPS is compatible with Φ = 1. Definition 1.11. [22] Given two MPSs {Qn }n≥0 and {Rn }n≥0 , if there is an integer s ≥ 0 such that Φ(x)Qn (x) =. n+t !. ν=n−s. λn,ν Rν (x), n ≥ s,. (1.6). ∃r ≥ s : λr,r−s (= 0, (1.7) we say that (1.6)-(1.7) is a finite-type relation between sequences {Rn }n≥0 and {Qn }n≥0 , with respect to Φ(x). When, instead of (1.7), we take λn,n−s (= 0, n ≥ s,. (1.8). we shall say that (1.6)-(1.8) is a strictly finite-type relation. Theorem 1.12. [22] Let {Qn }n≥0 and {Rn }n≥0 be two MPS and {vn }n≥0 and {rn }n≥0 their dual sequences, respectively. Let {Rn }n≥0 be compatible with a polynomial Φ. The following properties are equivalent. i. There is an integer s ≥ 0 such that Φ(x)Qn (x) =. n+t !. ν=n−s. λn,ν Rν (x), n ≥ s,. ∃r ≥ s : λr,r−s (= 0.. ii. There are an integer s ≥ 0 and an application from N into N : m *→ µm satisfying: max(0, m−t) ≤ µm ≤ m+s, m ≥ 0, ∃m0 ≥ 0 : µm0 = m0 +s, and such that µm ! λν,m vν , m ≥ t, λµm ,m (= 0, m ≥ 0. Φrm = ν=m−t. Remark 1.13. When the relation between {Rn }n≥0 and {Qn }n≥0 is strictly of finite-type, we have µm = m + s, m ≥ 0. 1.2. Cubic decomposition of a monic polynomial sequence Choosing a cubic polynomial )(x) = x3 + px2 + qx + r; p, q, r ∈ C. (1.9). and three constants a, b and c, for any MPS {Wn }n≥0 , we obtain by Euclidean division [23], three MPSs {Pn }n≥0 , {Qn }n≥0 and {Rn }n≥0 , and further six sequences in P, so that W3n (x) = Pn ()(x)) + (x − a)a1n−1 ()(x)) + (x − b)(x − c)a2n−1 ()(x)), (1.10) W3n+1 (x) = b1n ()(x)) + (x − a)Qn ()(x)) + (x − b)(x − c)b2n−1 ()(x)), (1.11) W3n+2 (x) = c1n ()(x)) + (x − a)c2n ()(x)) + (x − b)(x − c)Rn ()(x)), (1.12).

(8) Author's personal copy 6. P. Maroni and T. A. Mesquita. with deg a1n−1 ≤ n − 1, deg a2n−1 ≤ n − 1, deg b1n ≤ n, deg b2n−1 ≤ n − 1, deg c1n ≤ n and deg c2n ≤ n. Organizing the nine component sequences in a matrix, we introduce the following notation, as have been done previously in [23].   Pn (x) a1n−1 (x) a2n−1 (x) (1.13) Mn (x) =  b1n (x) Qn (x) b2n−1 (x)  c2n (x) Rn (x) c1n (x) Indeed, (1.10-1.12) is the most general cubic decomposition (CD) of a given MPS {Wn }n≥0 and it was presented in [23]. In this CD of {Wn }n≥0 , the sequences {Pn }n≥0 , {Qn }n≥0 , {Rn }n≥0 are called the principal components, and the remaining six {a1n−1 }n≥0 , {a2n−1 }n≥0 , {b1n }n≥0 , {b2n−1 }n≥0 , {c1n }n≥0 , {c2n }n≥0 are called the secondary components. Notice that these latest are nor necessarily (free) polynomial sequences, neither monic. We will denote by {wn }n≥0 , {un }n≥0 , {vn }n≥0 and {rn }n≥0 the dual sequences of {Wn }n≥0 , {Pn }n≥0 , {Qn }n≥0 and {Rn }n≥0 , respectively. The following result characterizes the nine component sequences of a CD, for any given MPS, that is, for any set of structure coefficients. Theorem 1.14. [23] A MPS {Wn }n≥0 , with structure coefficients (1.1) and (1.2), admits the CD (1.10)–(1.12) if and only if the following relations are fulfilled for n ≥ 0, (Z0 ) b10 (x) = a − β0 , "n−1 (Z1 ) c1n (x) = − ν=0 χ3n,3ν+1 b1ν (x) − (β3n+1 − a)b1n (x) + Θ(x)b2n−1 (x) "n−1 "n − ν=0 χ3n,3ν+2 c1ν (x) − ν=0 χ3n,3ν Pν (x) − (a − b)(a − c)Qn (x), " n (Z2 ) c2n (x) = − ν=0 χ3n,3ν a1ν−1 (x) + b1n (x) + Lb2n−1 (x) "n−1 "n−1 − ν=0 χ3n,3ν+2 c2ν (x)− ν=0 χ3n,3ν+1 Qν (x)−(β3n+1 +a−b−c)Qn (x), "n "n−1 (Z3 ) Rn (x) = − ν=0 χ3n,3ν a2ν−1 (x) − ν=0 χ3n,3ν+1 b2ν−1 (x) "n−1 −(β3n+1 + b +" c + p)b2n−1 (x) + Qn (x) − ν=0 χ3n,3ν+2 Rν (x), n (Z4 ) Pn+1 (x) = − ν=0 χ3n+1,3ν Pν (x)−(β3n+2 −a)c1n (x)−(a−b)(a−c)c2n (x) "n "n−1 − ν=0 χ3n+1,3ν+2 c1ν (x) − ν=0 χ3n+1,3ν+1 b1ν (x) + Θ(x)Rn (x), " "n−1 n (Z5 ) a1n (x) = − ν=0 χ3n+1,3ν a1ν−1 (x) + c1n (x) − ν=0 χ3n+1,3ν+2 c2ν (x) "n a − b − c)c2n (x) − ν=0 χ3n+1,3ν+1 Qν (x) + LRn (x), −(β3n+2 + " "n n (Z6 ) a2n (x) = − ν=0 χ3n+1,3ν a2ν−1 (x) − ν=0 χ3n+1,3ν+1 b2ν−1 (x) "n−1 +c2n (x) − ν=0 χ3n+1,3ν+2 Rν (x) − (β3n+2 + b +"c + p)Rn (x), n (Z7 ) b1n+1 (x) = −(a − b)(a − c)a1n (x) + Θ(x)a2n (x) − ν=0 χ3n+2,3ν+1 b1ν (x) "n "n 1 − ν=0 χ3n+2,3ν+2 "n cν (x) − ν=0 χ3n+2,3ν Pν (x) − (β3n+3 − a)Pn+1 (x), (Z8 ) Qn+1 (x) = − ν=0 χ3n+2,3ν a1ν−1 (x) − (β3n+3 + a − b − c)a1n (x) "n "n χ3n+2,3ν+2 c2ν (x) + Pn+1 (x) − ν=0 χ3n+2,3ν+1 Qν (x), +La2n (x) − ν=0" n (Z9 ) b2n (x) = a1n (x) − ν=0 χ3n+2,3ν a2ν−1 (x) − (β3n+3 + b + c + p)a2n (x) "n "n − ν=0 χ3n+2,3ν+1 b2ν−1 (x) − ν=0 χ3n+2,3ν+2 Rν (x), "−1 where by convention ν=0 . = 0 and Θ(x) = x − r + aL + bc(b + c + p), L = bc − q − (b + c + p)(b + c).. (1.14). (1.15).

(9) Author's personal copy Vol. 6 (2010). Cubic Decomposition of 2-Orthogonal Polynomial Sequences. 7. Finally, we present a result with several equivalent characterizations of the most simple CD, where the six secondary components vanish, that is, where the matrix Mn (x) is a diagonal matrix. This situation will be referred both as a diagonal CD and according to the next definition. Definition 1.15. A MPS for which the CD ) (1.10)-(1.12) * has all the secondary a b c components equal to zero will be called - symmetric. p q r ) * 0 0 0 Remark 1.16. Regarding Lemma 1.6, a - symmetric MPS is in 0 0 0 fact a 2-symmetric MPS. Theorem 1.17. [23] Let {Wn }n≥0 be a MPS. The following assertions are equivalent, where L is defined by (1.15) and n ≥ 0. 2 a. a1m = a2m = b1m = b)2m = c1m = c* m = 0, m ≥ 0, a b c or {Wn }n≥0 is - symmetric. p q r. b. σ$ (w3n+1 ) = 0, σ$ ((x − a)w3n ) = 0, σ$ ((x − b)(x − c)w3n ) = 0, σ$ (w3n+2 ) = 0, σ$ ((x − a)w3n+2 ) = 0, σ$ ((x − b)(x − c)w3n+1 ) = 0. c. The content of b. and un = σ$ (w3n ), vn = σ$ ((x − a)w3n+1 ), rn = σ$ ((x − b)(x − c)w3n+2 ). d. (d.1) β3n = a, (d.2) β3n+1 = b + c − a, (d.3); β3n+2 = −(b + c + p), (d.4) χ3n,3ν = −(a − b)(a − c)"uν , Qn #, 0 ≤ ν ≤ n, (d.5) χ3n,3ν+1 = 0, (d.6) χ3n,3ν+2 = "rν , Qn #, 0 ≤ ν < n, (d.7) χ3n+1,3ν = "uν , xRn (x)# + [aL + bc(b + c + p) − r]"uν , Rn (x)#, (d.8) χ3n+1,3ν+1 = L"vν , Rn (x)#, 0 ≤ ν ≤ n, (d.9) χ3n+1,3ν+2 = 0, 0 ≤ ν < n, (d.10) χ3n+2,3ν = 0, 0 ≤ ν ≤ n, (d.11) χ3n+2,3ν+1 = "vν , Pn+1 (x)#, (d.12) χ3n+2,3ν+2 = 0, 0 ≤ ν ≤ n.. 2. Cubic decomposition of a 2-orthogonal sequence Given a 2-orthogonal MPS {Wn }n≥0 , we know that it fulfils a third order recurrence relation, which can be written as follows, where the correspondent 1 0 0 , γn+1 , n ≥ 0, with γn+1 (= 0, n ≥ 0. recurrence coefficients are βn , γn+1 1 0 Wn+3 (x) = (x − βn+2 )Wn+2 (x) − γn+2 Wn+1 (x) − γn+1 Wn (x), n ≥ 0,. W0 (x) = 1, W1 (x) = x − β0 , W2 (x) = (x − β1 )W1 (x) − γ11 .. Taking into consideration Theorem 1.14, we begin to give necessary and sufficient relations concerning the decomposed MPS 2-orthogonality. Theorem 2.1. A MPS defined by (1.10)–(1.12) is 2-orthogonal if and only if the following relations are met, for n ≥ 0, where Θ(x) and L are defined by (1.14) and (1.15), and c1−1 (x) = c2−1 (x) = R−1 (x) = 0. (B0 ) b10 (x) = a − β0 ,.

(10) Author's personal copy P. Maroni and T. A. Mesquita. 8. 0 1 1 (B1 ) c1n (x) = −(β3n+1 − a)b1n (x) + Θ(x)b2n−1 (x) − γ3n cn−1 (x) − γ3n+1 Pn (x). (B2 ). c2n (x). − (a − b)(a − c)Qn (x),. 1 0 2 = −γ3n+1 a1n−1 (x) + b1n (x) + Lb2n−1 (x) − γ3n cn−1 (x). − (β3n+1 + a − b − c)Qn (x),. 1 a2n−1 (x) − (β3n+1 + b + c + p)b2n−1 (x) + Qn (x) (B3 ) Rn (x) = −γ3n+1 0 Rn−1 (x), − γ3n. 0 1 Pn (x) − (β3n+2 − a)c1n (x) − γ3n+2 b1n (x) (B4 ) Pn+1 (x) = −γ3n+1. − (a − b)(a − c)c2n (x) + Θ(x)Rn (x),. 0 a1n−1 (x) + c1n (x) − (β3n+2 + a − b − c)c2n (x) (B5 ) a1n (x) = −γ3n+1 1 Qn (x) + LRn (x), − γ3n+2. 0 1 a2n−1 (x) − γ3n+2 b2n−1 (x) + c2n (x) (B6 ) a2n (x) = −γ3n+1. (B7 ). b1n+1 (x). − (β3n+2 + b + c + p)Rn (x),. 0 = −(a − b)(a − c)a1n (x) + Θ(x)a2n (x) − γ3n+2 b1n (x) 1 c1n (x) − (β3n+3 − a)Pn+1 (x), − γ3n+3. 1 c2n (x) (B8 ) Qn+1 (x) = −(β3n+3 + a − b − c)a1n (x) + La2n (x) − γ3n+3 0 Qn (x), + Pn+1 (x) − γ3n+2. (B9 ) b2n (x) = a1n (x) − (β3n+3 + b + c + p)a2n (x) 0 1 b2n−1 (x) − γ3n+3 Rn (x). − γ3n+2. Proof. The sequence {Wn }n≥0 is 2-orthogonal if and only if its structure 1 0 , χn+1,n = γn+1 (= 0, and χn+1,ν = 0, coefficients are: χn+1,n+1 = γn+2 0 ≤ ν < n. Then, Theorem 1.14 concludes the proof. ! In the next corollary, we present three relations that begin as recurrence relations of third order (for each principal component) and after they are completed with a linear combination of elements of only two secondary component sequences. They establish conditions that assure the 2-orthogonality of the principal components. Corollary 2.2. A 2-orthogonal MPS with CD given by (1.10)–(1.12) fulfils the following relations, where Θ(x) and L are defined by (1.14) and (1.15). , + Pn+3 (x) = Θ(x) − Ā3n+3 Pn+2 (x) − B̄3n+3 Pn+1 (x) − C̄3n+3 Pn (x) (2.1) − M̄3n+3 b1n (x) − K̄3n+3 b1n+1 (x) − H̄3n+3 b1n+2 (x) − N̄3n+3 c1n (x) − V̄3n+3 c1n+1 (x) − S̄3n+3 c1n+2 (x),. , + Qn+3 (x) = Θ(x) − Ā3n+4 Qn+2 (x) − B̄3n+4 Qn+1 (x) − C̄3n+4 Qn (x) (2.2) − M̄3n+4 c2n (x) − K̄3n+4 c2n+1 (x) − H̄3n+4 c2n+2 (x). − N̄3n+4 a1n (x) − V̄3n+4 a1n+1 (x) − S̄3n+4 a1n+2 (x),.

(11) Author's personal copy Vol. 6 (2010). Cubic Decomposition of 2-Orthogonal Polynomial Sequences. 9. , + Rn+3 (x) = Θ(x) − Ā3n+5 Rn+2 (x) − B̄3n+5 Rn+1 (x) − C̄3n+5 Rn (x) (2.3) − M̄3n+5 a2n (x) − K̄3n+5 a2n+1 (x) − H̄3n+5 a2n+2 (x). − N̄3n+5 b2n (x) − V̄3n+5 b2n+1 (x) − S̄3n+5 b2n+2 (x), where. 0 0 0 1 Ān = γn+2 + γn+3 + γn+4 + γn+3 (βn+2 + 2βn+3 + p) 1 (2βn+3 + βn+4 + p) + γn+4. + (βn+3 − a)(βn+3 + a − b − c)(βn+3 + b + c + p). − (βn+3 − a)L + (a − b)(a − c)(βn+3 + b + c + p);. 1 1 1 0 0 0 0 B̄n = γn+1 γn+2 γn+3 + γn0 γn+2 + γn+1 (γn+2 + γn+3 ). 0 1 0 1 γn+3 (βn + βn+2 + βn+3 + p) + γn+2 γn+1 (βn + βn+1 + βn+3 + p); + γn+1 0 0 C̄n = γn−2 γn0 γn+2 ; 0 0 1 0 1 0 1 M̄n = γn−1 (γn+1 γn+3 + γn+2 γn+1 ) + γn0 γn+2 γn−1 ; 0 1 0 1 1 1 K̄n = γn+1 γn+3 + γn+3 γn+2 + γn+2 γn+3 (βn+1 + βn+2 + βn+3 + p) # 0 1 1 1 1 + γn+2 + γn+2 + γn+3 + γn+4 + (a − b)(a − c) − L γn+1. + (βn+3 − a)(βn+1 + βn+3 + a + p). $ + (βn+1 + a − b − c)(βn+1 + b + c + p) ;. 1 1 1 H̄n = γn+3 + γn+4 + γn+5 − L + (a − b)(a − c). + (βn+4 − a)(βn+3 + βn+4 + a + p). + (βn+3 + a − b − c)(βn+3 + b + c + p);. 0 1 0 1 1 1 N̄n = γn1 (γn+1 γn+3 + γn+2 γn+1 ) + γn0 γn+2 γn+3 0 + γn0 γn+2 (βn−1 + βn+1 + βn+3 + p); 0 0 V̄n = γn+2 (βn+1 + βn+2 + βn+3 + p) + γn+3 (βn+2 + βn+3 + βn+4 + p) # 1 1 1 1 + γn+3 γn+2 + γn+3 + γn+4 + (a − b)(a − c) − L. + (βn+2 + a − b − c)(βn+2 + b + c + p) $ + (βn+3 − a)(βn+2 + βn+3 + p + a) ;. S̄n = βn+3 + βn+4 + βn+5 + p.. Remark 2.3. The initial conditions of relations (2.1)-(2.3) can be written explicitly through relations (B0 )-(B9 ) of Theorem 2.1 with n = 0, 1, 2. This will not be done here due to their extensive expressions. Proof. In order to obtain the indicated relations, we apply the elimination method to the list of identities of Theorem 2.1. With respect to relation (2.1), we begin to consider (B4 ), with n → n + 1, 1 Pn+2 (x) = Θ(x)Rn+1 (x) − (a − b)(a − c)c2n+1 (x) − γ3n+5 b1n+1 (x) 0 Pn+1 (x), − (β3n+5 − a)c1n+1 (x) − γ3n+4.

(12) Author's personal copy P. Maroni and T. A. Mesquita. 10. and we need to substitute Θ(x)Rn+1 (x) − (a − b)(a − c)c2n+1 (x) by an equivalent expression written in terms of elements of the sequences {Pn (x)}n≥0 , {b1n (x)}n≥0 and {c1n (x)}n≥0 . Having the purpose of deducing such expression, let us consider relation (B3 ), with n → n + 1, where Qn+1 (x) is replaced by the expression given by (B8 ), yielding: 0 Rn+1 (x) = Pn+1 (x) − γ3n+2 Qn (x) − (β3n+3 + a − b − c)a1n (x). 1 1 0 )a2n (x) − (β3n+4 + b + c + p)b2n (x) − γ3n+3 c2n (x) − γ3n+3 Rn (x). + (L − γ3n+4. Replacing, also, Qn (x) by the expression given by (B3 ), we get: 0 1 1 Rn+1 (x) = Pn+1 (x) − γ3n+2 γ3n+1 a2n−1 (x) + (L − γ3n+4 )a2n (x). −. −. 0 (β3n+1 + b + c + p)b2n−1 (x) − (β3n+4 + γ3n+2 0 0 0 0 γ3n+2 Rn−1 (x) − (γ3n+2 + γ3n+3 )Rn (x) γ3n. b+c+. (2.4). p)b2n (x). 1 c2n (x). − (β3n+3 + a − b − c)a1n (x) − γ3n+3. To obtain a suitable expression for Θ(x)Rn+1 (x) as explained above, we will have to take identity (2.4) multiplied by Θ(x), therefore, it is time to write Θ(x)a2n (x), Θ(x)b2n−1 (x), Θ(x)Rn (x), Θ(x)a1n (x) and Θ(x)c2n (x) as linear combinations of elements of the sequences {Pn (x)}n≥0 , {b1n (x)}n≥0 and {c1n (x)}n≥0 plus an additional term with coefficient −(a − b)(a − c). In fact, by (B7 ), (B1 ) and (B4 ) we have (respectively): 0 Θ(x)a2n (x) = (a − b)(a − c)a1n (x) + γ3n+2 b1n (x) + b1n+1 (x). Θ(x)b2n−1 (x). 1 c1n (x) + (β3n+3 − a)Pn+1 (x), + γ3n+3 0 1 = (β3n+1 − a)b1n (x) + γ3n cn−1 (x) + c1n (x). (2.5). 1 + γ3n+1 Pn (x) (2.6). + (a − b)(a − c)Qn (x),. 1 b1n (x) + (β3n+2 − a)c1n (x) + (a − b)(a − c)c2n (x) (2.7) Θ(x)Rn (x) = γ3n+2 0 Pn (x) + Pn+1 (x). + γ3n+1. These identities permit to write, through (B9 ) multiplied by Θ(x), the following, for n ≥ 0: 0 Θ(x)b2n−1 (x) + Θ(x)b2n (x) Θ(x)a1n (x = (β3n+3 + b + c + p)Θ(x)a2n (x) + γ3n+2 1 Θ(x)Rn (x) + γ3n+3 # $ 1 0 1 1 ⇒ Θ(x)an (x) = γ3n+2 (β3n+1 + β3n+3 − a + b + c + p) + γ3n+2 γ3n+3 b1n (x). 0 0 γ3n+2 c1n−1 (x) + (β3n+3 + β3n+4 − a + b + c + p)b1n+1 (x) + γ3n $ # 1 0 0 c1n (x) + c1n+1 (x) + γ3n+3 (β3n+2 + β3n+3 − a + b + c + p) + γ3n+2 + γ3n+3 # $ 1 1 + γ3n+3 + γ3n+4 + (β3n+3 − a)(β3n+3 + b + c + p) Pn+1 (x) (2.8) $ # 0 1 0 1 Pn (x) + (a − b)(a − c)φ1 (x), + γ3n+2 γ3n+1 + γ3n+1 γ3n+3.

(13) Author's personal copy Vol. 6 (2010). Cubic Decomposition of 2-Orthogonal Polynomial Sequences. 11. 1 0 with φ1 (x) = (β3n+3 + b + c + p)a1n (x) + γ3n+3 c2n (x) + γ3n+2 Qn (x) + Qn+1 (x). Using (B6 ) multiplied by Θ(x), we obtain the following, for n ≥ 1: 0 1 Θ(x)c2n (x) = γ3n+1 Θ(x)a2n−1 (x) + Θ(x)a2n (x) + γ3n+2 Θ(x)b2n−1 (x). + (β3n+2 + b + c + p)Θ(x)Rn (x) 0 0 γ3n+1 b1n−1 (x) ⇒ Θ(x)c2n (x) = γ3n−1 # $ 0 0 1 + γ3n+1 + γ3n+2 + γ3n+2 (β3n+1 + β3n+2 − a + b + c + p) b1n (x) # $ 0 1 0 1 + b1n+1 (x) + γ3n+1 γ3n + γ3n γ3n+2 (2.9) c1n−1 (x) # $ 1 1 + γ3n+2 + γ3n+3 + (β3n+2 − a)(β3n+2 + b + c + p) c1n (x) # $ 0 1 1 + γ3n+1 (β3n + β3n+2 − a + b + c + p) + γ3n+1 γ3n+2 Pn (x). + (β3n+2 + β3n+3 − a + b + c + p)Pn+1 (x) + (a − b)(a − c)φ2 (x),. 0 1 with φ2 (x) = γ3n+1 a1n−1 (x) + a1n (x) + (β3n+2 + b + c + p)c2n (x) + γ3n+2 Qn (x). Thus, we can now multiply (2.4) by Θ(x) and introduce (2.5)-(2.9). Afterwards, we add, in both members, the term −(a − b)(a − c)c2n+1 (x) and after several algebraic simplifications where the identities (B2 ), (B5 ), (B8 ) and (B9 ) are used, we obtain the following: , + 0 Pn+1 (x) Θ(x)Rn+1 (x) − (a − b)(a − c)c2n+1 (x) = Θ(x) − Ā3n + γ3n+4. − B̄3n Pn (x) − C̄3n Pn−1 (x) − M̄3n b1n−1 (x) − K̄3n b1n (x) , 1 + 1 bn+1 (x) − N̄3n c1n−1 (x) − V̄3n c1n (x) + − H̄3n + γ3n+5 , + + − S̄3n + β3n+5 − a c1n+1 (x), n ≥ 1.. (2.10). Finally, identity (2.1) is established when we insert (2.10) information in (B4 ). The relations (2.2) and (2.3) can be obtained by similar calculations. ! Remark 2.4. • b1n = c1n = 0, n ≥ 0 ⇒ {Pn }n≥0 is 2-orthogonal. • c2n = a1n = 0, n ≥ 0 ⇒ {Qn }n≥0 is 2-orthogonal. • a2n = b2n = 0, n ≥ 0 ⇒ {Rn }n≥0 is 2-orthogonal. • If M̄n = K̄n = H̄n = N̄n = V̄n = S̄n = 0, n ≥ 0, then the principal components are 2-orthogonal. • In [24] it was proved a converse of corollary 2.2. Example 2.5. Taking, as an example, the 2-Chebyshev MOPS defined by 0 1 = γ, γn+1 = α, n ≥ 0, with γ (= 0 [8], we obtain: βn = 0, γn+1 M̄n = 3αγ 2 , K̄n = pα2 + qγ + 6αγ, H̄n = q + 3α, + , N̄n = γ 3α2 + pγ , V̄n = qα + 3α2 + 2pγ, S̄n = p.. Thus, M̄n = K̄n = H̄n = N̄n = V̄n = S̄n = 0, n ≥ 0, if and only if p = q = 0 and α = 0, that is, the given 2-Chebyshev MOPS is 2-symmetric. In that case, the three principal components fulfill the following recurrence relation Bn+3 (x) = (x − r − 3γ)Bn+2 (x) − 3γ 2 Bn+1 (x) − γ 3 Bn (x), n ≥ 0, with the initial conditions: P0 (x) = Q0 (x) = R0 (x) = 1,.

(14) Author's personal copy 12. P. Maroni and T. A. Mesquita. P1 (x) = x − r − γ, Q1 (x) = x − r − 2γ, R1 (x) = x − r − 3γ, P2 (x) = x2 +r2 −2x(r+2γ)+4rγ+γ 2 , Q2 (x) = x2 +r2 −x(2r+5γ)+5rγ+3γ 2 , R2 (x) = x2 + r2 − 2x(r + 3γ) + 6rγ + 6γ 2 . 2.1. The diagonal cubic decomposition of a 2-orthogonal sequence As mentioned before, in reference [7] we find a natural cubic decomposition for a d-orthogonal and d-symmetric MPS, when a = b = c = 0 and )(x) = x3 . It is in fact what we called above a diagonal cubic decomposition, since all the secondary components are trivial. Therefore, unlike the orthogonal case (see [23]), we know particular examples of diagonal CDs of 2-orthogonal MPSs, more specifically, of 2-symmetric sequences (with a = b = c = p = q = r = 0). The next result describes the 2-orthogonal sequences admitting a diagonal CD, as well as all the elements of that CD. Theorem 2.6. Let {Wn }n≥0 be a MPS defined by (1.10)–(1.12), such that a1n = a2n = b1n = b2n = c1n = c2n = 0, n ≥ 0.. Then {Wn }n≥0 is 2-orthogonal if and only if the following relations are met, where Θ(x) and L are defined by (1.14) and (1.15). (b2 ) (a − b)(a − c) = 0, (b1 ) L = 0, (b3 ) β3n = a, n ≥ 0, (b4 ) β3n+1 = b + c − a, n ≥ 0, (b6 ) γn1 = 0, n ≥ 1, (b5 ) β3n+2 = −(b + c + p), n ≥ 0, 0 (b7 ) Pn+1 (x) = Qn+1 (x) + γ3n+2 Qn (x), n ≥ 0, 0 (b8 ) Qn+1 (x) = Rn+1 (x) + γ3n+3 Rn (x), n ≥ 0, 0 (b9 ) Θ(x)Rn (x) = Pn+1 (x) + γ3n+1 Pn (x), n ≥ 0. Proof. Let us consider the relations of Theorem 2.1, with a1n = a2n = b1n = b2n = c1n = c2n = 0, n ≥ 0. 1 Pn (x) + (a − b)(a − c)Qn (x) = 0, (B̃0 ) β0 = a, (B̃1 ) γ3n+1. 0 Rn−1 (x) + Rn (x), (B̃2 ) β3n+1 + a − b − c = 0, (B̃3 ) Qn (x) = γ3n. 0 1 Pn (x) = Θ(x)Rn (x), (B̃5 ) γ3n+2 Qn (x) = LRn (x), (B̃4 ) Pn+1 (x) + γ3n+1. (B̃6 ) β3n+2 + b + c + p = 0, (B̃7 ) β3n+3 = a, 0 1 Qn (x) + Qn+1 (x), (B̃9 ) γ3n+3 = 0, n ≥ 0. (B̃8 ) Pn+1 (x) = γ3n+2. Notice that the relations (B̃2 ), (B̃3 ), (B̃4 ), (B̃6 ) and (B̃8 ) correspond to identities (b4 ), (b8 ), (b9 ), (b5 ) and (b7 ). Also, (B̃7 ) and (B̃0 ) correspond to (b3 ). 1 + (a − b)(a − c) = 0, n ≥ 0. Let us consider (B̃1 ), From (B̃1 ), we obtain γ3n+1 with n → n + 1, and let us substitute Pn+1 (x) by the expression given by (B̃8 ), as follows. $ # 1 0 Qn (x) + Qn+1 (x) + (a − b)(a − c)Qn+1 (x) = 0. γ3n+4 γ3n+2. 1 We conclude immediately that γ3n+4 = 0 and thus (a − b)(a − c) = 0, which 1 = 0, n ≥ 0, due to (B̃1 ). Also, corresponds to (b2 ). Moreover, we have γ3n+1 replacing the term Q + n1(x) of (B̃ , 5 ) by the expression given 1by (B̃3 ), we get: 1 0 γ3n Rn−1 (x) + γ3n+2 − L Rn (x) = 0, which implies γ3n+2 − L = 0 and γ3n+2.

(15) Author's personal copy Vol. 6 (2010). Cubic Decomposition of 2-Orthogonal Polynomial Sequences. 13. 1 1 γ3n+2 = 0, n ≥ 1. Hence, L = 0 = γ3n+2 , n ≥ 0. Therefore, we have obtained (b1 ) and taking into consideration (B̃9 ) we get (b6 ). Conversely, if we suppose the relations (b0 )-(b9 ) and a1n = a2n = b1n = 2 bn = c1n = c2n = 0, n ≥ 0, we can easily prove that (B̃0 )-(B̃9 ) are fulfilled, which correspond to the list of relations of Theorem 2.1 for a diagonal CD. !. When a 2-orthogonal MPS admits a diagonal CD, it fulfils the list of conditions of Theorem 2.6, which imply that the coefficients M̄n , K̄n , H̄n , N̄n , V̄n , S̄n of Corollary 2.2 vanish for n ≥ 0. Hence, we know that the three principal component sequences fulfil the following recurrence relations of third order, where n ≥ 0. 0 0 0 Pn+3 (x) = {Θ(x) − γ3n+5 − γ3n+6 − γ3n+7 }Pn+2 (x). −. P0 (x) = P2 (x) =. 0 0 0 0 0 0 + γ3n+4 (γ3n+5 + γ3n+6 )}Pn+1 (x) − γ3n+1 γ3n+3 γ3n+5 Pn (x), 0 1, P1 (x) = Θ(x) − γ1 , {Θ(x) − γ20 − γ30 − γ40 }P1 (x) − γ10 (γ20 + γ30 );. 0 0 0 Qn+3 (x) = {Θ(x) − γ3n+6 − γ3n+7 − γ3n+8 }Qn+2 (x). −. 0 0 γ3n+6 {γ3n+4. Q0 (x) = Q2 (x) =. 0 0 0 0 + + γ3n+7 )}Qn+1 (x) − γ3n+2 γ3n+4 γ3n+6 Qn (x), 1, Q1 (x) = Θ(x) − γ10 − γ20 , {Θ(x) − γ30 − γ40 − γ50 }Q1 (x) − γ10 γ30 − γ20 (γ30 + γ40 );. 0 0 γ3n+7 {γ3n+5. R0 (x) =. R2 (x) =. (2.12). 0 0 γ3n+5 (γ3n+6. 0 0 0 Rn+3 (x) = {Θ(x) − γ3n+7 − γ3n+8 − γ3n+9 }Rn+2 (x). −. (2.11). 0 0 γ3n+5 {γ3n+3. (2.13). 0 0 γ3n+6 (γ3n+7. 0 0 0 0 + + γ3n+8 )}Rn+1 (x) − γ3n+3 γ3n+5 γ3n+7 Rn (x), 0 0 0 1, R1 (x) = Θ(x) − γ1 − γ2 − γ3 , {Θ(x) − γ40 − γ50 − γ60 }R1 (x) − γ20 γ40 − γ30 (γ40 + γ50 ).. In brief, given a 2-orthogonal MPS {Wn }n≥0 , with respect to (w0 , w1 ), defined by (1.10)-(1.12), such that a1n = a2n = b1n = b2n = c1n = c2n = 0, n ≥ 0, the principal components are also 2-orthogonal, with respect to (u0 , u1 ), (v0 , v1 ) and (r0 , r1 ), respectively. Furthermore, each one of the following features: 2-orthogonality, diagonal CD and the fact that the principal components fulfil finite-type relations, allow us to achieve relations between the forms w0 , w1 ,u0 , u1 , v0 , v1 , r0 and r1 , as the following result announces. Theorem 2.7. Given a 2-orthogonal MPS {Wn }n≥0 , with respect to (w0 , w1 ), defined by (1.10)–(1.12), so that a1n = a2n = b1n = b2n = c1n = c2n = 0, n ≥ 0, we have the following relations involving w0 , w1 and elements of the dual sequences of the principal components. u0 = σ$ (w0 ), u1 = σ$ (w3 ),. (2.14). v0 = σ$ ((x − a)w1 ), v1 = σ$ ((x − a)w4 ),. (2.15). r0 = σ$ ((x − b)(x − c)w2 ), r1 = σ$ ((x − b)(x − c)w5 ),. (2.16).

(16) Author's personal copy P. Maroni and T. A. Mesquita. 14. #. $ 0 0 0 0 rm = um + γ3m+2 + γ3m+3 γ3m+5 um+2 , um+1 + γ3m+3 0 rm = vm + γ3m+3 vm+1 , m ≥ 0;. (2.17) (2.18). and, there are polynomials Λµ (n, ν), 0 ≤ µ ≤ 1, such that w2 = Λ0 (1, 0)w0 + Λ1 (1, 0)w1 ,. w3 = Λ0 (1, 1)w0 + Λ1 (1, 1)w1 , (2.19). w4 = Λ0 (2, 0)w0 + Λ1 (2, 0)w1 ,. w5 = Λ0 (2, 1)w0 + Λ1 (2, 1)w1 , (2.20). and deg Λ0 (1, 0) = 1, deg Λ1 (1, 0) = 0, deg Λ0 (1, 1) ≤ 1, deg Λ1 (1, 1) = 1, deg Λ0 (2, 0) = 2, deg Λ1 (2, 0) ≤ 1, deg Λ0 (2, 1) ≤ 2, deg Λ1 (2, 1) = 2. Proof. Relations (2.14-2.16) are directly obtained from Theorem 1.17. Furthermore, applying Theorem 1.4, there are polynomials Λµ (n, ν), 0 ≤ µ ≤ 1, fulfilling relations (2.19-2.20) and whose degrees are characterized as announced. Finally, Theorem 2.6 puts in evidence two relations of finite type. More precisely, identities (b7 ) and (b8 ) yield the following strictly finitetype relation between sequences {Rn }n≥0 and {Pn }n≥0 , with respect to "n Φ(x) = 1 (see definition 1.11): Pn (x) = ν=n−2 λn,ν Rν (x), n ≥ 2, where 0 0 0 0 γ3n−1 (= 0, λn,n−1 = γ3n−1 + γ3n and λn,n = 1. λn,n−2 = γ3n−3 Similarly, identity (b8 ) is the following strictly finite-type relation between sequences {Rn }n≥0 and {Qn }n≥0 , with respect to Φ(x) = 1. Qn (x) =. n !. ν=n−1. 0 λn,ν Rν (x), n ≥ 1, where λn,n−1 = γ3n (= 0, λn,n = 1.. Thus, Theorem 1.12 establishes relations (2.17-2.18).. !. 3. The cubic decomposition of a 2-orthogonal and 2-symmetric sequence Motivated by the structure of a 2-symmetric sequence and its simple cubic decomposition, we will now focus our attention in that type of sequences, aiming to fully understand what further information can our general CD bring to the already known results. The next result describes any CD of a 2-orthogonal and 2-symmetric MPS. Theorem 3.1. A MPS {Wn }n≥0 , defined by (1.10)–(1.12) is 2-orthogonal and 2-symmetric if and only if the following relations are fulfilled for n ≥ 0. (IC1 ) a20 (x) = −p,. (IC2 ) b20 (x) = p2 − q,. 0 b2n−1 (x) + b2n (x), (i1 ) a1n (x) = (b + c + p)a2n (x) + γ3n+2 , 2 + 0 0 2 0 0 an−1 (x) + a2n (x) γ3n an−2 (x) + γ3n + γ3n+1 (i2 ) b1n (x) = γ3n−2 # $ 0 + (a − b − c)(b + c + p) − L b2n−1 (x) + (a + p)γ3n Rn−1 (x) + (a + p)Rn (x),.

(17) Author's personal copy Vol. 6 (2010). Cubic Decomposition of 2-Orthogonal Polynomial Sequences. 15. 0 0 0 (i3 ) c1n (x) = γ3n+1 (a + p)a2n−1 (x) + (a + p)a2n (x) + γ3n−1 γ3n+1 b2n−2 (x) + 0 , 0 + γ3n+1 + γ3n+2 b2n−1 (x) + b2n (x) # $ + (a − b − c)(b + c + p) − L Rn (x),. 0 a2n−1 (x) + a2n (x) + (b + c + p)Rn (x), (i4 ) c2n (x) = γ3n+1 # $ 0 (i5 ) Pn (x) = (a − b − c)(b + c + p) − L a2n−1 (x) + γ3n−1 (a + p)b2n−2 (x) 0 0 γ3n−1 Rn−2 (x) + (a + p)b2n−1 (x) + γ3n−3 , + 0 0 + γ3n−1 + γ3n Rn−1 (x) + Rn (x),. 0 Rn−1 (x) + Rn (x), (i6 ) Qn (x) = (b + c + p)b2n−1 (x) + γ3n , 2 + 0 0 0 0 an (x) + pa2n+1 (x) γ3n+3 a2n−1 (x) + p γ3n+3 + γ3n+4 (i7 ) pγ3n+1 0 0 0 γ3n+1 γ3n+3 b2n−2 (x) + γ3n−1 # $ + 0 , 0 0 0 0 + γ3n+3 γ3n+1 + γ3n+2 + γ3n+2 γ3n+4 b2n−1 (x) , + 0 0 0 b2n (x) + b2n+1 (x) + γ3n+4 + γ3n+5 + r − x + γ3n+3 0 Rn (x) + qRn+1 (x) = 0, + qγ3n+3. 0 0 0 a2n−1 (x) + qa2n (x) + pγ3n−1 γ3n+1 b2n−2 (x) (i8 ) qγ3n+1 , + 0 0 0 0 0 b2n−1 (x) + pb2n (x) + γ3n−3 γ3n−1 γ3n+1 Rn−2 (x) + p γ3n+1 + γ3n+2 # $ + , 0 0 0 0 0 + γ3n+1 γ3n−1 + γ3n + γ3n γ3n+2 Rn−1 (x) , + 0 0 0 Rn (x) + Rn+1 (x) = 0, + γ3n+2 + γ3n+3 + r − x + γ3n+1 # $ + 0 , 0 0 0 2 0 0 0 0 (i9 ) γ3n−2 γ3n γ3n+2 an−2 (x) + γ3n+2 γ3n + γ3n+1 + γ3n+1 γ3n+3 a2n−1 (x) , 2 + 0 0 0 0 an (x) + a2n+1 (x) + qγ3n+2 + γ3n+3 + γ3n+4 b2n−1 (x) + r − x + γ3n+2 + 0 0 0 γ3n+2 Rn−1 (x) + p γ3n+2 + qb2n (x) + pγ3n , 0 Rn (x) + pRn+1 (x) = 0, +γ3n+3. where, for any component sequence {ζn }n≥0 , ζi = 0, i = −2, −1.. Proof. Let us consider the relations of Theorem 2.1, with βn = 0, n ≥ 0, and γn1 = 0. (B̃0 ) b10 (x) = a, 0 1 cn−1 (x) − (a − b)(a − c)Qn (x), (B̃1 ) c1n (x) = a b1n (x) + Θ(x)b2n−1 (x) − γ3n 2 1 2 0 2 (B̃2 ) cn (x) = bn (x) + Lbn−1 (x) − γ3n cn−1 (x) − (a − b − c)Qn (x), 0 Rn−1 (x), (B̃3 ) Rn (x) = −(b + c + p)b2n−1 (x) + Qn (x) − γ3n 0 1 (B̃4 ) Pn+1 (x) = −γ3n+1 Pn (x) + a cn (x) − (a − b)(a − c)c2n (x) + Θ(x)Rn (x), 0 a1n−1 (x) + c1n (x) − (a − b − c)c2n (x) + LRn (x), (B̃5 ) a1n (x) = −γ3n+1 2 0 (B̃6 ) an (x) = −γ3n+1 a2n−1 (x) + c2n (x) − (b + c + p)Rn (x), 0 (B̃7 ) b1n+1 (x) = −(a − b)(a − c)a1n (x) + Θ(x)a2n (x) − γ3n+2 b1n (x) + a Pn+1 (x), 0 Qn (x), (B̃8 ) Qn+1 (x) = −(a − b − c)a1n (x) + La2n (x) + Pn+1 (x) − γ3n+2 2 1 2 0 2 (B̃9 ) bn (x) = an (x) − (b + c + p)an (x) − γ3n+2 bn−1 (x)..

(18) Author's personal copy P. Maroni and T. A. Mesquita. 16. Let us first notice that from (B̃3 ), (B̃6 ) and (B̃9 ), we have immediately (i6 ),(i4 ) and (i1 ), respectively. Considering (B̃2 ) with n = 0 and regarding (B̃0 ) we have: c20 (x) = b + c, (3.1) and after the transformation n → n + 1 we can introduce (i4 ) and (i6 ), yielding , 2 + 0 0 0 0 an + a2n+1 γ3n+3 a2n−1 (x) + γ3n+3 + γ3n+4 (3.2) b1n+1 (x) = γ3n+1 # $ 2 0 + (a − b − c)(b + c + p) − L bn (x) + (a + p)γ3n+3 Rn (x) + (a + p)Rn+1 (x).. With respect to the initial conditions, we obtain (IC1 ) from (B̃6 ) with n = 0 and using (3.1). Also, from (B̃1 ) with n = 0, we have: c10 (x) = a2 − (a − b)(a − c).. (3.3). Using (3.1) and (3.3), we get from (B̃5 ) with n = 0:. a10 (x) = a2 − (a − b)(a − c) − (a − b − c)(b + c) + L = −bp − cp − q. (3.4). Thus, we obtain (IC2 ) considering (B̃9 ) with n = 0 and introducing (IC1 ) and (3.4). With this particular information, we can rewrite (3.2) in order to justify identity (i2 ), just by decreasing n and taking into consideration (B̃0 ). Let us consider (B̃5 ) with n replaced by n + 1. Inserting (i1 ) and (i4 ) we get: 0 0 0 c1n+1 (x) = γ3n+4 (a + p)a2n (x) + (a + p)a2n+1 (x) + γ3n+2 γ3n+4 b2n−1 (3.5) # $ + 0 , 0 + γ3n+4 + γ3n+5 b2n + b2n+1 + (a − b − c)(b + c + p) − L Rn+1 .. Hence, we can establish identity (i3 ) attending to (IC1) and (IC2). Proceeding in a similar way, we get (i5 ) from (B̃8 ). The remaining three identities (i7 ), (i8 ) and (i9 ) are obtained as the previous ones from relations (B̃1 ), (B̃4 ) and (B̃7 ), where (B̃1 ) is taken with n replaced by n + 1. Finally, supposing the enunciated list of relations, we can perform a straightforward confirmation of Theorem 2.1 relations for a 2-symmetric sequence, which concludes the proof. ! Theorem 3.1 shows that in a CD of a 2-orthogonal and 2-symmetric sequence, the six component sequences of the first and second columns of Mn (x), defined in (1.13), are written in terms of the three component sequences of the third column, more precisely {a2n }n≥0 , {b2n }n≥0 and {Rn }n≥0 , where these latest fulfil relations (i7 ) − (i9 ). For that reason, we were driven to investigate the CD where a2n (x) or b2n (x) vanish. Indeed, as the next Proposition clarifies, those cases are strictly related to the conditions p = q = 0 where the resultant CD has a lower triangular matrix Mn (x), which can also be considered a diagonal CD for the additional choice of parameters a = b = c = 0. Proposition 3.2. Let {Wn }n≥0 be a MPS defined by (1.10)–(1.12), 2-orthogonal and 2-symmetric. Then, the following statements are equivalent..

(19) Author's personal copy Vol. 6 (2010). Cubic Decomposition of 2-Orthogonal Polynomial Sequences. 17. a. a1n (x) = 0, n ≥ 0; b. a2n (x) = 0, n ≥ 0; c. b2n (x) = 0, n ≥ 0; d. p = q = 0; e. a1n (x) = a2n (x) = b2n (x) = 0, n ≥ 0; f. The component sequences fulfil the following (where n ≥ 0 and Ri = 0, i = −2, −1):. a1n (x) = a2n (x) = b2n (x) = 0; c2n (x) = (b + c)Rn (x); # $ 0 0 0 0 Pn (x) = γ3n−3 γ3n−1 Rn−2 (x) + γ3n−1 + γ3n Rn−1 (x) + Rn (x);. Qn (x) =. (3.6). 0 γ3n Rn−1 (x). + Rn (x); (3.7) # $ b1n (x) = aQn (x); c1n (x) = a(b + c) − bc Rn (x); , + 0 0 0 Rn (x) − γ3n+2 − γ3n+3 (3.8) Rn+1 (x) = x − r − γ3n+1 # $ + , 0 0 0 0 0 0 0 0 − γ3n+1 γ3n−1 + γ3n + γ3n γ3n+2 γ3n−1 γ3n+1 Rn−2 (x), Rn−1 (x) − γ3n−3 g. W3n (x) = Pn ()(x)), W3n+1 (x) = aQ # n ()(x)) + (x $ − a)Qn ()(x)), W3n+2 (x) = a(b + c) − bc Rn ()(x)) + (x − a)(b + c)Rn ()(x)). +(x − b)(x − c)Rn ()(x)), where )(x) = x3 +r and {Pn }n≥0 , {Qn }n≥0 and {Rn }n≥0 fulfil relations (3.6), (3.7) and ) (3.8), respectively. * 0 0 0 h. {Wn }n≥0 is - symmetric. 0 0 r Proof. In the following arguments we will use the content of Theorem 3.1 following its notation. a. ⇒ d. Let us suppose that a1n (x) = 0, n ≥ 0. By (i1 ) this means that 0 b2n (x) = −(b + c + p)a2n (x) − γ3n+2 b2n−1 (x), n ≥ 0.. (3.9). Reading (3.9) with n = 0 and inserting (IC1) and (IC2), we obtain −q = p(b + c).. (3.10). Let us now consider (i7 )-(i9 ) with n = 0: , , + + p γ30 + γ40 a20 (x) + pa21 (x) + r − x + γ30 + γ40 + γ50 b20 (x). +b21 (x) + qγ30 + qR1 (x) = 0, , qa20 (x) + pb20 (x) + r − x + γ10 + γ20 + γ30 + R1 (x) = 0, + , , + r − x + γ20 + γ30 + γ40 a20 (x) + a21 (x) + qb20 (x) + p γ20 + γ30 +. +pR1 (x) = 0. a21 (x). (3.11) (3.12) (3.13). Using (IC1) and (IC2), we obtain R1 (x) from (3.12), then from (3.13) and finally b21 (x) from (3.11). The coefficient of x of the resultant b21 (x) is 3p2 − 2q. On the other hand, we can calculate b21 (x) from (3.9) with n = 1 and the correspondent coefficient of x is 2(b + c + p)p. The two equations.

(20) Author's personal copy P. Maroni and T. A. Mesquita. 18. 3p2 − 2q = 2(b + c + p)p and (3.10) yield p = q = 0. b. ⇒ d. If a2n (x)+= 0, n ≥ 0, then (IC1) , implies p = 0. Furthermore, relation 0 b2n−1 (x) = 0. If we suppose that q (= 0, then (i9 ) becomes q b2n (x) + γ3n+2 we get n 0 b2n (x) = (−1)n+1 q γ3k+2 , n ≥ 1. +. b2n (x). ,. k=1. In particular, deg = 0, n ≥ 0. Considering (i7 ) with n = 0: , + 0 0 r − x + γ3 + γ4 + γ50 b20 (x) + b21 (x) + qγ30 + qR1 (x) = 0, , + and since b20 (x) = −q and b21 (x) = qγ50 , we obtain qR1 (x) = x − r − γ40 (−q). Taking (i7 ) with n = 1: $ # + , , + γ60 γ40 + γ50 + γ50 γ70 b20 (x) + r − x + γ60 + γ70 + γ80 b21 (x) + b22 (x) +qγ60 R1 (x) + qR2 (x) = 0,. and inserting the information previously obtained for b20 (x), b21 (x), b22 (x) and qR we get the following identity which is impossible if q (= 0. $ # 1 (x), + 0 , , + 0 0 0 0 γ6 γ4 + γ5 + γ5 γ7 (−q) + r − x + γ60 + γ70 + γ80 qγ50 − qγ80 γ50 + , +γ60 x − r − γ40 (−q) = −qR2 (x). c. ⇒ d. If b2n (x) = 0, n ≥ 0, then (IC1) says that q = p2 and (i7 ) and (i9 ) correspond to the following two identities: , 2 + 0 0 0 0 an (x) + pa2n+1 (x) γ3n+3 a2n−1 (x) + p γ3n+3 + γ3n+4 pγ3n+1 0 Rn (x) + p2 Rn+1 (x) = 0, (3.14) + p2 γ3n+3 # $ + 0 , 0 0 0 0 0 0 0 γ3n + γ3n+1 + γ3n+1 γ3n−2 γ3n γ3n+2 a2n−2 (x) + γ3n+2 γ3n+3 a2n−1 (x) , 2 + 0 0 0 0 0 an (x) + a2n+1 (x) + pγ3n + γ3n+3 + γ3n+4 γ3n+2 Rn−1 (x) + r − x + γ3n+2 , + 0 0 (3.15) + p γ3n+2 + γ3n+3 Rn (x) + pRn+1 (x) = 0.. If we suppose that p (= 0, then (3.14) yields. 0 0 0 pγ3n+3 Rn (x) + pRn+1 (x) = −γ3n+1 γ3n+3 a2n−1 (x) , 2 + 0 0 an (x) − a2n+1 (x), + γ3n+4 − γ3n+3. (3.16). 0 Rn (x) + pRn+1 (x) of (3.15) with this last expression, we and replacing pγ3n+3 obtain: + 0 , 2 0 0 0 0 0 γ3n + γ3n+1 an−1 (x) γ3n γ3n+2 a2n−2 (x) + γ3n+2 γ3n−2 , 2 + 0 0 0 0 Rn (x) = 0. + r − x + γ3n+2 an (x) + pγ3n γ3n+2 Rn−1 (x) + pγ3n+2 + 2 , In particular, we deduce that deg an (x) < n, which implies that the right hand of (3.16) has degree less than n + 1. But indeed, this is impossible since the left hand of (3.16) has degree n + 1. Hence, we must conclude that p = 0 and from (IC2) we have also q = 0. d. ⇒ f. and d. ⇒ g. If we suppose p = q = 0, (IC1) and (IC2) tell us that a20 (x) = b20 (x) = 0. Thus, identities (i7 ) and (i9 ) imply a2n (x) = b2n (x) =.

(21) Author's personal copy Vol. 6 (2010). Cubic Decomposition of 2-Orthogonal Polynomial Sequences. 19. 0, n ≥ 0. Moreover, the remaining list of relations of Theorem 3.1 establish the CD described in item f. which corresponds entirely to the CD of item g. To finalize the proof of the equivalence between a. − g., we remark that obviously we have: f. ⇒ e. ⇒ a.; e. ⇒ b.; e. ⇒ c.; e. ⇒ d. and e. ⇔ f. If we consider the CD presented in g., with )(x) = x3 + r, and we choose a = b = c = 0, then we obtain a diagonal CD where the principal components are the ones given by g. Conversely, item h. signifies that for a = b = c = 0 and p = q = 0, the correspondent CD of {Wn }n≥0 is given by the following: W3n (x) = Pn ()(x)), W3n+1 (x) = xQn ()(x)), W3n+2 (x) = x2 Rn ()(x)), where, due to the conditions p = q = 0, the principal components are defined by relations (3.6), (3.7) and (3.8), respectively. Altering the auxiliary polynomials x and x2 to x − a and (x − b)(x − c), for arbitrary constants a, b, c, we deduce the CD indicated in g. ! Corollary 3.3. Let {Wn }n≥0 be a 2-orthogonal and 2-symmetric MPS defined ) * a b c by (1.10)–(1.12). Then {Wn }n≥0 is - symmetric if and only if p q r a = b = c = p = q = 0. Proof. Let us suppose that we have a diagonal CD. Attending to the relations of Theorem 2.6 and since βn = 0, n ≥ 0, we conclude that a = b = c = p = q = 0. Conversely, if we choose a = b = c = p = q = 0, then Proposition 3.2 ! provides that b1n (x) = c1n (x) = c2n (x) = 0, n ≥ 0.. References [1] P. Barrucand and D. Dickinson, On cubic transformation of orthogonal polynomials, Proc. Amer. Math. Soc. 17 (1966), 810-814. [2] Y. Ben Cheikh, On some (n − 1)-symmetric linear functionals J. Comput. Appl. Math. 133 (2001), 207-218. [3] D. Bessis and P. Moussa, Orthogonality properties of iterated polynomial mappings, Comm. Math. Phys. 88 (1983), no. 4, 503-529. [4] J. A. Charris and M. E. H. Ismail, Sieved orthogonal polynomials. VII: Generalized polynomial mappings, Trans. Am. Math. Soc., Vol. 340, No. 1 (Nov., 1993), 71-93. [5] J. A. Charris, M. E. H. Ismail and S. Monsalve, On sieved orthogonal polynomials. X. General blocks of recurrence relations, Pacific J. Math. 163 (1994), no. 2, 237-267. [6] T. S. Chihara, On kernel polynomials and related systems, Boll. Un. Mat. Ital. 19 (3) (1964), 451-459. [7] K. Douak and P. Maroni, Les polynômes orthogonaux ”classiques” de dimension deux, Analysis 12 (1992), 71-107. [8] K. Douak and P. Maroni, On d-orthogonal Tchebyshev polynomials, I , Appl. Numer. Math. 24 (1997), 23-53. [9] J. S. Geronimo and W. Van Assche, Orthogonal polynomials on several intervals via a polynomial mapping, Trans. Am. Math. Soc., Vol. 308, No. 2 (Aug., 1988), 559-581..

(22) Author's personal copy 20. P. Maroni and T. A. Mesquita. [10] I. Rodriguez González and C. Tasis Montes, Descomposition cubica general de una sucesion de polinomios ortogonales, Actas del Simposium Polinomios Ortogonales y Aplicaciones, Gijón, 1989, 259-265. [11] M. E. H. Ismail, On sieved orthogonal polynomials. III: Orthogonality on several intervals, Trans. Am. Math. Soc., Vol. 294, No. 1 (Mar., 1986), 89-111. [12] M. N. de Jesus and J. Petronilho, On orthogonal polynomials obtained via polynomial mappings, J. Approx. Theory 162 (2010), 2243-2277. [13] Â. Macedo and P. Maroni, General quadratic decomposition, J. Difference Equ. Appl. 16, no. 11 (2010), 1309 -1329. [14] F. Marcellán and G. Sansigre, Orthogonal polynomials and cubic transformations, J. Comput. Appl. Math. 49 (1993), 161-168. [15] F. Marcellán and J. Petronilho, Eigenproblems for tridiagonal 2-Toeplitz matrices and quadratic polynomial mappings, Linear Algebra Appl. 260 (1997), 169-208. [16] F. Marcellán and J. Petronilho, Orthogonal polynomials and quadratic transformations, Port. Math. (N.S.) 56 (1999), 81-113. [17] F. Marcellán and J. Petronilho, Orthogonal polynomials and cubic polynomial mappings. I, Commun. Anal. Theory Contin. Fract. 8 (2000), 88-116. [18] F. Marcellán and J. Petronilho, Orthogonal polynomials and cubic polynomial mappings. II. The positive-definite case, Commun. Anal. Theory Contin. Fract. 9 (2001), 11-20. [19] P. Maroni, L’orthogonalité et les récurrences de polynômes d’ordre supérieur à deux, Ann. Fac. Sci. Toulouse 10 (1), (1989), 105-139. [20] P. Maroni, Sur la décomposition quadratique d´une suite de polynômes orthogonaux I, Rivista di Mat. Pura ed Appl. 6 (1990), pp. 19-53. [21] P. Maroni, Variations around classical orthogonal polynomials. Connected problems, J. Comput. Appl. Math. 48 (1993), 133-155. [22] P. Maroni, Semi-classical character and finite-type relations between polynomial sequences, Appl. Numer. Math. 31 (1999), 295-330. [23] P. Maroni, T. A. Mesquita and Z. da Rocha, On the general cubic decomposition of polynomial sequences, J. Difference Equ. Appl., 17, no. 9 (2011), 1303-1332. [24] T. A. Mesquita, Polynomial Cubic Decomposition. Ph.D. Thesis, Universidade do Porto, Faculdade de Ciências, Departamento de Matemática, 2010. [25] J. Van Iseghem, Approximants de Padé vectoriels, Thèse d" état, Univ. des Sciences et techiques de Lille-Flandre-Artois, 1987. Pascal Maroni CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris France and UPMC Univ. Paris 06, UMR 7598 Laboratoire Jacques-Louis Lions, F-75005, Paris France e-mail: maroni@ann.jussieu.fr.

(23) Author's personal copy Vol. 6 (2010). Cubic Decomposition of 2-Orthogonal Polynomial Sequences. Teresa A. Mesquita Instituto Superior Politécnico de Viana do Castelo Avenida do Atlântico 4900-348 Viana do Castelo Portugal and Centro de Matemática da Universidade do Porto Rua do Campo Alegre, 687 4169-007 Porto Portugal e-mail: teresam@portugalmail.pt Received: January 3, 2012. Accepted: May 25, 2012.. 21.

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