Geronimus transformations of bivariate linear functionals

Journal Pre-proof Geronimus transformations of bivariate linear functionals

Francisco Marcellán, Misael E. Marriaga, Teresa E. Pérez, Miguel A. Piñar

PII:

S0022-247X(19)31004-2

DOI:

https://doi.org/10.1016/j.jmaa.2019.123736

Reference:

YJMAA 123736

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

9 July 2019

Please cite this article as: F. Marcellán et al., Geronimus transformations of bivariate linear functionals, J. Math. Anal. Appl. (2020), 123736, doi: https://doi.org/10.1016/j.jmaa.2019.123736.

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GERONIMUS TRANSFORMATIONS OF BIVARIATE LINEAR FUNCTIONALS ´ ´ FRANCISCO MARCELLAN, MISAEL E. MARRIAGA, TERESA E. PEREZ, ˜ AND MIGUEL A. PINAR Abstract. Given a linear functional u in the linear space of polynomials in two variables with real coefficients and a polynomial λ(x, y), in this contribution we deal with Geronimus transformations of u, i.e., those linear functionals v such that u = λ(x, y)v. The connection formulas between the sequences of bivariate orthogonal polynomials with respect to u and v are given. A matrix interpretation of such transformations by using LU and U L factorizations of the block Jacobi matrices associated with such polynomials is given. Finally, some illustrative examples of Geronimus transformations of weight function supported in domains of R2 are discussed.

1. Introduction The analysis of perturbations of a linear functional u defined on the linear space of polynomials with real coefficients is being deeply studied in the theory of orthogonal polynomials on the real line. In particular, when dealing with a positive definite case, i.e., the linear functional has an integral representation in terms of a probability measure supported in an infinite subset of the real line, such perturbations provide interesting information in the framework of Gaussian quadrature rules. Taking into account the perturbation yields new nodes and Christoffel numbers. Among the perturbations of linear functionals, spectral linear perturbations have attracted the interest of the researchers (see [37]). Such perturbations are generated by two particular families, the so called Christoffel and Geronimus transformations.  = Christoffel perturbations, that appear when considering a new functional u p(x)u, where p(x) is a polynomial, were studied in 1858 by the German mathematician E. B. Christoffel in the framework of Gaussian quadrature rules. He found explicit formulas relating the corresponding sequences of orthogonal polynomials with respect to two measures, the Lebesgue measure dμ supported in the Date: November 28, 2019. 2010 Mathematics Subject Classification. Primary: 42C05; 33C50. Key words and phrases. Bivariate orthogonal polynomials, classical and semiclassical orthogonal polynomials. The work of the first and second authors (FM and MEM) has been supported by Ministerio de Ciencia, Innovaci´ on y Universidades (MICINN) grant PGC2018-096504-B-C33. Third and fourth authors (TEP and MAP) thank FEDER/Ministerio de Ciencia, Innovaci´ on y Universidades – Agencia Estatal de Investigaci´ on/PGC2018-094932-B-I00 and Research Group FQM-384 by Junta de Andaluc´ıa. 1

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´ ´ ˜ F. MARCELLAN, M. E. MARRIAGA, T. E. PEREZ, AND M. A. PINAR

interval (−1, 1) and dˆ μ(x) = p(x)dμ(x), with p(x) = (x − q1 ) · · · (x − qN ) a signed polynomial in the support of dμ, as well as the distribution of their zeros as nodes in such quadrature rules. Nowadays, these are called Christoffel formulas, and can be considered a classical result in the theory of orthogonal polynomials which can be found in a number of monographs, see for example [11], [32]. Explicit relations between the corresponding sequences of orthogonal polynomials and the coefficients of the corresponding three term recurrence relations have been extensively studied, see [20], as well as the connection between the corresponding Jacobi matrices in the framework of the so-called Darboux transformations based on the LU factorization of such matrices (see [9] and [36], among others). Connection formulas between two families of orthogonal polynomials related by the Christoffel transformation allow to express any polynomial of a given degree n of the second family times the polynomial p(x) of degree h as a linear combination of all polynomials of degree less than or equal to n + h and greater than or equal to n in the first family. Geronimus transformations appear when dealing with perturbed functionals v defined by p(x)v = u, where p(x) is a polynomial. Such a kind of transformations were used by the Russian mathematician J. L. Geronimus, see [21], in order to have a nice proof of a result by W. Hahn [22] concerning the characterization of classical orthogonal polynomials (Hermite, Laguerre, Jacobi, and Bessel) as those orthogonal polynomials whose first derivatives are also orthogonal polynomials. Again, as happened for the Christoffel transformation, within the Geronimus transformation one can find the expression of the orthogonal polynomials of the second family in terms of a linear combination of the polynomials of degree less than or equal to n and greater than or equal to n−h, where h is the degree of the polynomial p(x). See for example the work of P. Maroni [29] for a perturbation of the type p(x) = x − a, [4] and [6] for a quadratic case, and [30] for the cubic case. As an inverse problem, we can deal with the fact when a linear combination of a fixed number of consecutive polynomials of a sequence of orthogonal polynomials constitutes a sequence of orthogonal polynomials, or, equivalently, when a sequence of quasi-orthogonal polynomials is again a sequence of orthogonal polynomials. This question was analyzed for a particular situation when we have a linear combination of three orthogonal polynomials, the so called quasi-orthogonality of order 2 in [8]. Later on, the general case has been studied in [3] as well as in [7]. The corresponding linear functionals are related by a Geronimus transformation. Some applications to the analysis of the corresponding Gaussian quadrature rules are also given in [7]. Analytic properties of quasi-orthogonal polynomials of order 1, i.e., linear combinations of two consecutive polynomials of a sequence of monic orthogonal polynomials with respect to a probability measure supported on an infinite subset of the real line (see [11]) have been studied in [24]. In particular, the electrostatic interpretation of their zeros from the second order differential equation that such polynomials satisfy, is deduced.

GERONIMUS TRANSFORMATIONS OF BIVARIATE LINEAR FUNCTIONALS

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Geronimus transformations also appear in the framework of Krall polynomials, i.e., orthogonal polynomials satisfying higher order differential (difference) equations. The corresponding measures are Geronimus transformations of classical ones (see [18, 19]). Notice that for a probability measure dμ supported on the real line, a perturbation of such a measure by a signed rational function p(x) q(x) can be considered as the composition of a Christoffel and a Geronimus transformation. The connection between the corresponding sequences of monic orthogonal polynomials has been studied in [33]. Geronimus transformations appear as a particular choice of symmetric bilinear forms B such that for a given linear functional u and a polynomial s(x), we get B(s(x) p(x), q(x)) = B(p(x), s(x) q(x)) = u, p(x) q(x). This problem has been solved in [16]. Among such bilinear forms we have the so called Sobolev type inner products. In such a paper, the connection between the Hessenberg matrix associated with the multiplication operator with respect to the monic orthogonal polynomials associated with B and the Jacobi matrix associated with the sequence of monic orthogonal polynomials associated with u is stated again by using LU and U L factorizations. The particular case s(x) = x2 has been analyzed in [15]. A connection with polynomial mappings has been pointed out in [14]. For a more general framework concerning polynomial perturbations of bilinear forms, in [10] the authors deal with a perturbation of Hermitian bilinear form φ in ˜ the linear space of polynomials defined by φ(p(z), q(z)) = φ((z−α)p(z), (z−α)q(z)). In such a situation the Hessenberg matrices associated with the multiplication operator in terms of the sequences of orthogonal polynomials with respect to φ and φ˜ are related by using the QR and RQ factorizations. In this manuscript we focus our attention on the study of Geronimus transformations for linear functionals defined in the linear space of polynomials in two variables with real coefficients. As in the univariate case Geronimus and Christoffel transformation are intimately connected. But, contrary to what happens when we deal with one variable, the existing literature on this subject in several variables is quite reduced. To our best knowledge, Christoffel modifications in several variables were first considered in [2], where the left multiplication of a moment functional times a polynomial of degree 1 was studied. There, using Favard’s theorem in several variables ([17]), necessary and sufficient conditions about the existence of orthogonality properties for each one of the corresponding orthogonal families were given in terms of the coefficients in the three term relations. Similar results were also obtained in [13] for a Christoffel modification by means of a second degree polynomial. We must point out that this was not a trivial extension of the case when the degree of the polynomial is 1, since in several variables not every polynomial of higher degree factorizes as a product of polynomials of degree 1.

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´ ´ ˜ F. MARCELLAN, M. E. MARRIAGA, T. E. PEREZ, AND M. A. PINAR

In [5], the authors show a connection of multivariate Christoffel and more general linear spectral transformations of moment functionals with the Darboux factorization of the Jacobi block matrix associated with the three term recurrence relations for multivariate orthogonal polynomials. The structure of the manuscript is as follows. In Section 2 we overview the basic background concerning multivariate orthogonal polynomials with a special attention to the three term relations they satisfy following mainly [17]. In Section 3, an algebraic approach to Geronimus transformation u = λ(x, y)v in terms of the dual basis of the basis on the linear space of polynomials Bλ = {xh y k λn : n  0, (h, k) ∈ A(λ)} is given, where A(λ) will be defined in (3.1). In fact, Geronimus transformations of a moment functional associated to a bivariate polynomial λ(x, y) are characterized in terms of the kernel of Tλ : Π → Π the linear transformation associated with the left multiplication of a functional times the polynomial λ(x, y). Section 4 is focused on polynomial perturbations of bivariate linear functionals. An analogue of the Christoffel formula of the univariate case is deduced. Furthermore, a matrix interpretation of Geronimus and Christoffel transformations in terms of the LU and UL factorizations of the block Jacobi matrices associated with the three term recurrence relations is obtained. In Section 5 we show some examples of Geronimus transformations. The first one deals with a tensor product of two Geronimus transformations of linear functionals in one variable. The second one is related to a Geronimus transformation of the classical weight on the unit ball. The third one focus the attention on a Geronimus transformation for a bivariate measure supported on the deltoid. In this last example the polynomial connecting two given moment functionals on the deltoid is obtained, providing in this way an answer to a question raised by Y. Xu in [35] in relation with some cubature formulae. Finally, as an Appendix we have included the proofs of the main results in Section 4. 2. Polynomials in two variables For each n  0, Πn denotes the linear space ofbivariate polynomials with real coefficients of total degree at most n, and Π = n0 Πn the linear space of all bivariate polynomials with real coefficients. Throughout this work, we fix the order of the monomials as follows, 1 < y < x < · · · < y n < x y n−1 < · · · < xn−1 y < xn < · · · by using the graded lexicographical order. Let n  m  am,k xk y m−k , p(x, y) = m=0 k=0

be a polynomial of total degree n, that is,

n k=0

|an,k | > 0, and let

m = max{ k : 0  k  n, an,k = 0}. We say that LC(p) := an,m is the leading coefficient of p, LM(p) := xm y n−m is the leading monomial of p, and LT(p) := LC(p)·LM(p)=an,m xm y n−m is the leading term of p. We use the following division algorithm in Π:

GERONIMUS TRANSFORMATIONS OF BIVARIATE LINEAR FUNCTIONALS

5

Division Algorithm: Let f ∈ Π be a polynomial. For every polynomial p ∈ Π, there exist unique polynomials q, r ∈ Π such that: (1) p = qf + r. (2) LM(qf ) < LM(p). (3) Either r = 0 or r is a linear combination, with real coefficients, of monomials, none of which is divisible by LM(f ). We call r the remainder of p on division by f . A more general polynomial division by an ordered s-tuple of polynomial divisors F = (f1 , f2 , . . . , fs ) can be found in [12, Chapter 2, Section 3, Theorem 3]. In this work, division by one divisor will be enough. From now on, following [12], for any polynomial p ∈ Π, we will denote the remainder of p on division by f by pf . A useful observation is that, using the division algorithm, the polynomial p can be written uniquely as a sum of an element of the principal ideal If generated by f , namely qf , and a polynomial not in If , namely pf . Moreover, the polynomial q such that p = qf + pf can be expressed as q =

p − pf . f

A useful tool throughout this work is the representation of a basis of Π as column vectors. Definition 2.1. A polynomial system (PS) is a sequence {Pn }n0 of polynomial vectors of increasing size with Pn = (Pn,0 (x, y), Pn,1 (x, y), . . . , Pn,n (x, y)) , where {Pn,k (x, y) : 0  k  n, n  0} is a basis of Π such that, for a fixed n  0, deg Pn,k (x, y) = n, and {Pn,k (x, y) : 0  k  n} are n + 1 linearly independent polynomials. An example of a PS is the canonical basis of Π, {Xn }n0 , where, for each n  0, Xn = (xn−k y k )0kn = (xn , xn−1 y, . . . , xy n−1 , y n ) , is a column vector of size (n + 1). The superscript  denotes the transpose. For each n  0, the vector Pn can be written as Xn−1 + · · · + G0n X0 , Pn = Gnn Xn + Gn−1 n n where each Gm n is a (n + 1) × (m + 1) matrix of real numbers, such that Gn is a non-singular square matrix, called the (matrix) leading coefficient of the polynomial ˆ n }n0 with P ˆ n = (Gn )−1 Pn is said to be a monic PS, since every Pn . The PS {P n entry of ˆ n = (Pˆn,0 (x, y), Pˆn,1 (x, y), . . . , Pˆn,n (x, y)) , P

is a monic PS of the form Pˆn,k (x, y) = xn−k y k + lower total degree terms,

0  k  n.

Let Π denote the algebraic dual space of Π, that is, the set of all linear functionals u : Π → R, and let u, p be the image of a polynomial p ∈ Π by u ∈ Π . We denote by 0 ∈ Π the null linear functional, that is, the linear functional such that 0, p = 0 for every polynomial p ∈ Π.

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´ ´ ˜ F. MARCELLAN, M. E. MARRIAGA, T. E. PEREZ, AND M. A. PINAR

As usual, we define the left product of a polynomial q ≡ q(x, y) ∈ Π times u as a new functional q u satisfying q u, p = u, q p,

∀p ∈ Π.

In general, given any sequence of real numbers {μn,m : n  0, 0  m  n}, we can define a linear functional u ∈ Π by means of its action u, xn−m y m  = μn,m ,

n  0,

0  m  n,

and extended by linearity to all bivariate polynomials. In this case, μn,m = u, xn−m y m  are called moments, and we usually say that u is a moment functional. Definition 2.2. A PS {Pn }n0 is an orthogonal polynomial system (OPS) with respect to a linear functional u if  0, n = m,  u, Pn Pm  = (2.1) Hn , n = m, where 0 is the matrix with zeros as entries of adequate size, and Hn is a symmetric and non-singular (n + 1) × (n + 1) matrix. A moment functional u defined on Π is quasi-definite if and only if there exists an OPS associated with u. In particular, if u is quasi-definite, then μ0,0 = u, 1 = 2 u, P0 P 0  = H0 = 0. Moreover, u is positive definite [17, p. 63] if u, p  > 0 for every p ∈ Π, p ≡ 0. If u is positive definite, then it is quasi-definite, and the non-singular matrix Hn is positive definite for n  0. Notice that in the multivariate case, an OPS associated with a quasi-definite linear functional u is not unique but there exists a unique monic OPS. Following [27], if the PS {Pn }n0 is orthogonal with respect to a quasi-definite linear functional u, then the row functionals −1 Un = P n Hn u,

satisfy the duality conditions Un , P m

 =

0, In+1 ,

n  0,

(2.2)

n = m, n = m.

For orthogonal polynomials in two variables a three term relation with respect to the multiplication by each variable holds. These three term relations are written in a vector form and have matrix coefficients. Proposition 2.3. [17]. Let {Pn }n0 be an arbitrary monic PS. There exists a quasi-definite linear functional u such that {Pn }n0 is the corresponding monic OPS if and only if, for k = 1, 2, there exist matrices of real numbers as entries Bn,k and Cn,k of size (n + 1) × (n + 1) and (n + 1) × n, respectively, such that (1) For n  0,



x Pn = Ln,1 Pn+1 + Bn,1 Pn + Cn,1 Pn−1 , y Pn = Ln,2 Pn+1 + Bn,2 Pn + Cn,2 Pn−1 ,

(2.3)

where P−1 = 0, Ln,1 = (In+1 |0), and Ln,2 = (0|In+1 ). Here, In+1 denotes the identity matrix of size (n + 1) × (n + 1) and 0 denotes the column of zeros of appropriate size.

GERONIMUS TRANSFORMATIONS OF BIVARIATE LINEAR FUNCTIONALS

7

(2) The rank conditions rank Cn,k = n,

k = 1, 2,

rank Cn = n + 1,

(2.4)

where Cn = (Cn,1 , Cn,2 ) is the so-called joint matrix of size (n + 1) × 2 n, hold. 3. Geronimus transformations of bivariate linear functionals In this section, we will extend to the bivariate framework the theory of Geronimus transformations of linear functionals presented in [16] for the univariate case in the framework of bilinear forms, while we give a functional approach. We will see that the Geronimus transformation of a bivariate linear functional involves an infinite sum of elements of a basis of linear functionals, as opposed to the univariate case where a finite sum is involved. Let λ ≡ λ(x, y) be any fixed polynomial. Using λ, we will construct an appropriate basis of Π for the study of Geronimus transformations. We introduce the set   A(λ) =

(h, k) : h, k  0, xh y k

LM(λ)

⊂ N2 .

= xh y k

(3.1)

That is, (h, k) ∈ A(λ) if and only if xh y k does not belong to the principal ideal generated by LM(λ). Notice that in the univariate case, if ϕ(t) =

s 

(t − ai )di = tN + lower degree terms,

N=

i=1

s 

di ,

di ∈ N,

i=1

then the set {0, 1, 2, . . . , N − 1}, plays the same role as A(λ). Nevertheless, for a bivariate polynomial λ, A(λ) is always infinite. Let LM(λ) = xm y  , where m,  ∈ N, and, for each n  0, let ILM(λn ) be the principal ideal generated by LM(λn ). Then, every monomial belonging to ILM(λ) has the form xm+i y +j with i, j  0. Hence, we have a one-to-one correspondance between the monomials in ILM(λ) and the set (m, ) + N2 . In fact, for each n  0, there exists a one-to-one mapping between the monomials in ILM(λn ) and the set (nm, n) + N2 . Therefore, every monomial belonging to ILM(λn ) but not belonging to ILM(λn+1 ) is xnm+h y n+k

with

(h, k) ∈ A(λ).

Proposition 3.1. Let λ ≡ λ(x, y) be any polynomial. The set Bλ = {xh y k λn : n  0, (h, k) ∈ A(λ)} is a basis of Π.

(3.2)

´ ´ ˜ F. MARCELLAN, M. E. MARRIAGA, T. E. PEREZ, AND M. A. PINAR

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Figure 1. LM(λ) = xm y  . All monomials not divisible by LM(λ) have powers in the white region, all monomials divisible by LM(λ) but not divisible by LM(λ2 ) have powers in the light gray region, and so on. Proof. Let xm y  , m,  ∈ N, be a monomial. By the division algorithm, there exist unique polynomials q, r ∈ Π such that xm y  = q(x, y) λ(x, y) + r(x, y), where all the monomials in r(x, y) are not divisible by LM(λ), that is,  r(x, y) = r(h,k) xh y k , r(h,k) ∈ R. (h,k)∈A(λ)

We apply this observation as follows. Write  (h,k) h k q(x, y) = q0 x y +

(h,k )

q1





xh y k ,

(h,k )∈A(λ)

(h,k)∈A(λ) (h,k)



(h,k )

and q1 are real numbers. By the first part of the proof, for every where q0   xh y k with (h, k  ) ∈ A(λ), there exist unique polynomials ϑ1 , ρ1 ∈ Π such that 



xh y k = ϑ1 (x, y) λ(x, y) + ρ1 (x, y), where all the monomials in ρ1 (x, y) are not divisible by LM(λ), that is,  (h,k) h k (h,k) ρ1 (x, y) = ρ1 x y , ρ1 ∈ R. (h,k)∈A(λ)

Then, where q2 (x, y) =

xm y  = q2 (x, y) λ(x, y)2 + q1 (x, y) λ(x, y) + r(x, y), 

(h,k )

q1

(h,k )∈A(λ)

q1 (x, y) =

ϑ1 (x, y)





(h,k)∈A(λ)

(h,k )∈A(λ)

(h,k ) (h,k) ρ1

q1

xh y k +



(h,k)

q0

xh y k .

(h,k)∈A(λ) m 

Notice that LM(ϑ1 ) < LM(q) < x y . Continuing in this way with the term multiplied by λ2 , we see that the process must stop in a finite number of steps, and every monomial can be written as a unique linear combination of the elements of  Bλ . Hence, Bλ is a basis of polynomials.

GERONIMUS TRANSFORMATIONS OF BIVARIATE LINEAR FUNCTIONALS

9

Given a polynomial λ(x, y), we order the basis Bλ defined in (3.2) as follows. If    deg(xh y k λn ) < deg(xm y h λn ), then 





x h y k λ n < xm y k λ n . 





If deg(xh y k λn ) = deg(xh y k λn ), then 



x h y k λn < xh y k λ n



whenever







LM(xh y k λn ) < LM(xm y k λn ).

Example. Let λ(x, y) = y(1 − x − y). Then, LM(λ) = xy and A(λ) = {(h, 0) : h  0} ∪ {(0, k) : k  0}. According to the order established for the basis Bλ , we have 1 < y < x < y 2 < λ < x2 < y 3 < yλ < xλ < x3 < · · · We can write xy = −λ − y 2 + y, x3 y = −x2 λ − λ2 − 2y 2 λ − y 4 − xλ + 3yλ + 3y 3 − λ − 3y 2 + y, where the expansion of the monomials in terms of the basis Bλ is written in decreasing order. We will denote the dual basis of Bλ by Bλ = {un,h,k : n  0, (h, k) ∈ A(λ)},

(3.3)

where un,h,k , x y λ  = δn,n δh,h δk,k . Every polynomial p ∈ Π can be represented as +∞   un,h,k , p xh y k λn . p(x, y) = h k n

n=0 (h,k)∈A(λ)

Clearly, the unique polynomials q, pλ ∈ Π such that p = q λ + pλ are +∞   p − pλ = un,h,k , p xh y k λn−1 , λ n=1 (h,k)∈A(λ)  λ p (x, y) = u0,h,k , p xh y k .

q(x, y) =

(h,k)∈A(λ)

Definition 3.2. Let λ(x, y) be any polynomial. Given any linear functional u ∈ Π , we define uλ ∈ Π as

p − pλ λ , u , p = u, λ for every polynomial p ∈ Π. We have the following result. Proposition 3.3. Let λ(x, y) be a polynomial. Given any linear functional u, we get +∞   uλ = u, xh y k λn−1  un,h,k , n=1 (h,k)∈A(λ)

where {un,h,k : n  0, (h, k) ∈ A(λ)} is defined in (3.3).

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´ ´ ˜ F. MARCELLAN, M. E. MARRIAGA, T. E. PEREZ, AND M. A. PINAR λ

Notice that for any polynomial p ∈ Π, λ p = 0 and for any linear functional u ∈ Π , λ(x, y)uλ = u. Nevertheless, λ

λ(x, y)u , p = u, p − u, pλ . Definition 3.4. The solutions of the equation u = λ(x, y)v are said to be Geronimus transformations of u associated with the polynomial λ(x, y). Theorem 3.5. Given any linear functional u, the solutions of the equation u = λ(x, y)v are  ah,k u0,h,k , (3.4) v = uλ + (h,k)∈A(λ)

where ah,k = v − uλ , xh y k . Proof. The linear functional uλ satisfies u = λ(x, y)uλ . Let v be another linear functional such that u = λ(x, y)v. Then, v−u = λ

+∞ 



an,h,k un,h,k ,

n=0 (h,k)∈A(λ)

where an,h,k = v − uλ , xh y k λn . But, for n  1, an,h,k = λ(x, y)(v − uλ ), xh y k λn−1  = 0, and, setting ah,k ≡ a0,h,k , the result follows. The coefficients ah,k in (3.4) are free parameters.  For every polynomial λ(x, y), let us define the linear operator Tλ : Π → Π

u → λ(x, y) u.

such that

The following result shows that {u0,h,k : (h, k) ∈ A(λ)} ⊂ Bλ , constitutes a basis for Ker(Tλ ) = {u ∈ Π : Tλ (u) = λ(x, y) u = 0}. Furthermore, Geronimus transformations of a linear functional u associated with λ(x, y) are characterized in terms of Ker(Tλ ). Corollary 3.6. Let λ(x, y) be a polynomial. The linear functional v is a Geronimus transformation of u associated with the polynomial λ(x, y) if and only if v − uλ ∈ Ker(Tλ ). Proof. The necessary condition follows from Theorem 3.5. To prove the sufficient condition, observe that for each (h, k) ∈ A(λ),       λ(x, y) u0,h,k , xh y k λn = u0,h,k , xh y k λn+1 = 0, n  0, (h , k  ) ∈ A(λ), that is, λ(x, y)u0,h,k = 0 on the basis Bλ . Then, Span{u0,h,k : (h, k) ∈ A(λ)} ⊆ Ker(Tλ ). On the other hand, let w ∈ Ker(Tλ ). Write w=

+∞ 



n=0 (h,k)∈A(λ)

cn,h,k un,h,k ,

GERONIMUS TRANSFORMATIONS OF BIVARIATE LINEAR FUNCTIONALS

11

where cn,h,k = w, xh y k λn . But, for n  1, cn,h,k = λ(x, y)w, xh y k λn−1  = 0. This means that w ∈ Span {u0,h,k : (h, k) ∈ A(λ)} and, in turn, Ker(Tλ ) ⊆ Span {u0,h,k : (h, k) ∈ A(λ)}. Thus Ker(Tλ ) = Span {u0,h,k : (h, k) ∈ A(λ)}. In particular, v − uλ ∈ Ker(Tλ ), hence λ(x, y)(v − uλ ) = 0. But λ(x, y)uλ = u and, therefore, u = λ(x, y)v.  If we denote by ≡λ the equivalence relation on Π defined as u ≡λ v

u − v ∈ Ker(Tλ ),

if and only if 

then Corollary 3.6 says that v ∈ Π is a Geronimus transformation of a linear functional u if and only if v ≡λ uλ , where we take uλ as the canonical representative of the equivalence class. A major difference between univariate and bivariate polynomials is that in the latter case, the zeros of a polynomial are algebraic varieties which may consist of discrete points as well as algebraic curves. Given a polynomial λ(x, y), we will show that linear functionals defined by some line integrals over the zeros of λ(x, y) belong to Ker(Tλ ). Let λ(x, y) be a polynomial with deg λ = N such that λ(x, y) =

s 

s 

λi (x, y)di ,

i=1

di = N,

di ∈ N,

i=1

where each λi (x, y) has no factors different than 1 and itself, that is, each λi is an irreducible polynomial. Theorem 3.7. For each λi (x, y) of the form λi (x, y) = x−ai , respectively, λi (x, y) = y − bi , the linear functionals

1 wi,j , p = ∂xj p(ai , y)dy, 0  j  di − 1, 0

respectively,

 i,j , p = w

1

0

∂yj p(x, bi )dx,

0  j  di − 1,

belong to Ker(Tλ ). Proof. Write wi,j =

+∞ 



(i,j)

wn,h,k un,h,k ,

n=0 (h,k)∈A(λ)

where

(i,j)

wn,h,k = wi,j , xh y k λn . Since λ(x, y)n = (x − ai )ndi qi (x, y), where qi (x, y) is not divisible by x − ai , for 0  j  di − 1, we have ∂xj λn =

j  l=0

j! (ndi )! (x − ai )ndi −l ∂xj−l qi . (j − l)!l! (ndi − l)!

Then, (i,j)

wn,h,k = 0,

n  1.

´ ´ ˜ F. MARCELLAN, M. E. MARRIAGA, T. E. PEREZ, AND M. A. PINAR

12

It follows that



wi,j =

(hj, k)∈A(λ)

Similarly,



 i,j = w

(h, kj)∈A(λ)

h! ah−j u0,h,k . (h − j)!(k + 1) i k! bk−j u0,h,k . (k − j)!(h + 1) i

As shown in the proof of Corollary 3.6, Ker(Tλ ) = Span {u0,h,k : (h, k) ∈ A(λ)}.  i,j , belong to Ker(Tλ ) since it can be written in terms Hence, wi,j , respectively w  of the basis {u0,h,k : (h, k) ∈ A(λ)}. We remark that the integrals defining the functionals presented in Theorem 3.7 can be defined over any bounded interval [a, b]. We have chosen the interval [0, 1] for simplicity. Theorem 3.8. For each λi (x, y) such that the algebraic variety defined by λi (x, y) = 0 is parametrizable by x(t) = fi (t), y(t) = gi (t) where a  t  b, the linear functionals

b j d vi,j , p = p(fi (t), gi (t))dt, 0  j  di − 1, j dt a belong to Ker(Tλ ). Proof. Write +∞ 

vi,j =



(i,j)

vn,h,k un,h,k ,

n=0 (h,k)∈A(λ)

where

(i,j)

vn,h,k = vi,j , xh y k λn . Since λ(x, y)n = λi (x, y)ndi q0,i (x, y), where q0,i (x, y) is not divisible by λi (x, y), for 0  j  di − 1 and 0  l  j, we have

and

∂y λn

=

∂y2 λn

= .. .

∂xj−l ∂yl λn

=

i −1 i i −1 n di λnd q0,i ∂y λi + λnd ∂y q0,i = λnd q1,i , i i i i −2 i −1 i −2 (n di − 1) λnd q1,i ∂y λi + λnd ∂y q1,i = λnd q2,i , i i i

i −j λnd qj,i , i

j    j  j−l  l j−l l dj n f (t) gi (t) ∂x ∂y λ(fi (t), gi (t))n , λ(fi (t), gi (t)) = l i dtj l=0 j    j  j−l  l f (t) gi (t) λi (fi (t), gi (t))nβi −j qj,i (fi (t), gi (t)). = l i l=0

Then, (i,j)

vn,h,k = 0, and vi,j =

 (hj,kj)∈A(λ)

n  1, (i,j)

v0,h,k u0,h,k ,

GERONIMUS TRANSFORMATIONS OF BIVARIATE LINEAR FUNCTIONALS

13

where (i,j)

v0,h,k =

j    j l=0

h! k! l (h + l − j)!(k − l)!

b a

fi (t)h+l−j gi (t)k−l fi (t)j−l gi (t)l dt.

From the proof of Corollary 3.6, we know that {u0,h,k : (h, k) ∈ A(λ)} constitutes a basis for Ker(Tλ ) and, hence, vi,j belongs to Ker(Tλ ).  4. Polynomial perturbations of functionals In this section, we discuss linear relations between two orthogonal polynomial bases when their associated linear functionals satisfy the equation u = λ(x, y)v where λ(x, y) is a polynomial of total degree N . The results obtained in this section extend those deduced in [2, 13] to longer linear relations. The proofs are given in Appendix A. Definition 4.1. For a fixed N  1, two monic PS {Pn }n0 and {Qn }n0 satisfy a linear relation of order N if Qn = Pn + Mn,1 Pn−1 + Mn,2 Pn−2 + · · · + Mn,N Pn−N , Q0 = P0 ,

n  1,

(4.1)

where Mn,h , n  1, 1  h  N , are (n + 1) × (n − h + 1) matrices with real numbers as entries, such that Mn,h ≡ 0 when n < h. By convention, we set P−k = 0 for k  1. Using a similar argument as provided in [28], it can be shown that (4.1) is invariant under a change of polynomial basis. Relation (4.1) can be translated to the corresponding dual bases as follows. Proposition 4.2. Let {Pn }n0 and {Qn }n0 be two monic PS satisfying a connection relation of order N as (4.1), and let {Un }n0 and {Vn }n0 be the corresponding dual bases. Then, Un = Vn+N Mn+N,N + · · · + Vn+1 Mn+1,1 + Vn ,

n  0.

(4.2)

When both monic PS are orthogonal, using (2.2), the above proposition can be stated as follows. Corollary 4.3. Let {Pn }n0 and {Qn }n0 be two monic OPS related by (4.1) and let u and v be their corresponding quasi-definite linear functionals. Then, Kn u = Ln+N v,

n  0,

(4.3)

where Kn Ln+N

= =

−1 P n Hn ,    −1  −1 Mn+N,N + · · · + Q H  −1 Qn+N H n+1 n+1 Mn+1,1 + Qn Hn , n+N

are 1 × (n + 1) row vectors of polynomials of total degree n and at most n + N , respectively. Here, for n  0, Hn = u, Pn P n ,

 n = v, Qn Q . H n

These matrices of size (n + 1) × (n + 1) are symmetric and non-singular.

14

´ ´ ˜ F. MARCELLAN, M. E. MARRIAGA, T. E. PEREZ, AND M. A. PINAR

In the following result, we show that two monic OPS {Pn }n0 and {Qn }n0 satisfying a linear relation of order N  1 such as (4.1) are characterized in terms of their associated quasi-definite linear functionals. Furthermore, we show that there exists a fixed polynomial λ(x, y) such that, for each n  0, λ(x, y) Pn is a linear combination of at most N consecutive terms of {Qn }n0 . The proof is shown in Appendix A. Theorem 4.4. Let {Pn }n0 and {Qn }n0 be two monic OPS and u and v their corresponding quasi-definite linear functionals. Then the following statements are equivalent: (i) There exist matrices with real entries Mn,h , 1  h ≤ N, n  1, of size (n + 1) × (n − h + 1), with Mn,h ≡ 0 when n < h, and MN,N ≡ 0 , such that {Pn }n0 and {Qn }n0 are related by (4.1). (ii) There exists a polynomial λ(x, y) of degree exactly N such that u = λ(x, y) v. (iii) There exist a polynomial λ(x, y) of degree exactly N and matrices of real numbers Kn,h , 0  h  N, n  0, of size (n + 1) × (n + N − h + 1), with  −1 , such that {Pn }n0 and {Qn }n0 Kn,N non-singular and K0,N = H0 H 0 satisfy λ(x, y) Pn = Kn,0 Qn+N + · · · + Kn,N −1 Qn+1 + Kn,N Qn ,

n  0.

(4.4)

Observe that (4.4) is the bivariate analogue of the well known Christoffel formula for univariate polynomials (see [20] as well as [37] and the references therein). We can deduce some properties about the rank of all matrices Mn,N in (4.1) in terms of the rank of Mh,h for 1 ≤ h ≤ N . See Appendix A for the proof. Proposition 4.5. Let {Pn }n0 and {Qn }n0 be two monic OPS related by (4.1). Then, (1) If rank Mh,h = 0 for every 1  h  N , then rank Mn,h = 0 for 1  h  N and n  1, (2) If rank MN,N = 1, then rank Mn,N = n − m + 1 for every n  N . Notice that Proposition 4.5 implies that if two OPS {Pn }n0 and {Qn }n0 satisfy (4.1) and rank MN,N = 1, then Qn = Pn for n  N . 4.1. Inverse problem. Now, let us analyze the following problem: assuming that one of the two PS {Pn }n0 and {Qn }n0 related by (4.1) is orthogonal, characterize when the other one is also orthogonal. Taking into account Proposition 4.5, we will only consider the case when rank MN,N = 1. Whenever the PS {Qn }n0 is orthogonal, we will use the tilde notation in its corresponding three term relations:  n,1 Qn + C n,1 Qn−1 , x Qn = Ln,1 Qn+1 + B n  0, (4.5)   y Qn = Ln,2 Qn+1 + Bn,2 Qn + Cn,2 Qn−1 , n,k , k = 1, 2, and C n = (C n,1 , C n,2 ) and assume that the rank conditions (2.4) for C are satisfied. We obtain the following two characterizations. As above, see Appendix A for the proof.

GERONIMUS TRANSFORMATIONS OF BIVARIATE LINEAR FUNCTIONALS

15

Theorem 4.6. Let {Qn }n0 be a monic OPS satisfying (4.5). Define recursively a PS {Pn }n0 by (4.1) with rank MN,N = 1. Then {Pn }n0 is a monic OPS satisfying the three term relation (2.3) if and only if, for k = 1, 2, and n  1, Mn,h Cn−h,k + Mn,h+1 Bn−h−1,k + Mn,h+2 Ln−h−2,k n,k Mn,h+1 + C n,k Mn−1,h , 1  h  N − 2, = Ln,k Mn+1,h+2 + B

(4.6)

n,k Mn,N + C n,k Mn−1,N −1 , Mn,N −1 Cn−N +1,k + Mn,N Bn−N,k = B

(4.7)

Mn,N Cn−N,k

n,k Mn−1,N , =C

(4.8)

and n,k − Mn,1 Ln−1,k + Ln,k Mn+1,1 , Bn,k = B

(4.9)

n,k − Mn,1 Bn−1,k + B n,k Mn,1 − Mn,2 Ln−2,k + Ln,k Mn+1,2 . Cn,k = C

(4.10)

Here, Mn,h ≡ 0 for n < h. Now, we study the case when the monic PS {Pn }n0 is orthogonal. We refer to Appendix A for the proof of the following result. Theorem 4.7. Let {Pn }n0 be a monic OPS satisfying the three term relation (2.3). Define the PS {Qn }n0 by means of (4.1) with rank MN,N = 1. Then {Qn }n0 is a monic OPS satisfying (4.5) if and only if (4.6), (4.7), and (4.8) hold, and, for k = 1, 2, n,k = n, rank C

n = n + 1, rank C

(4.11)

where n,k = Bn,k + Mn,1 Ln−1,k − Ln,k Mn+1,1 , B n,k Mn,1 + Mn,2 Ln−2,k − Ln,k Mn+1,2 . n,k = Cn,k + Mn,1 Bn−1,k − B C Here, Mn,h ≡ 0 for n < h. We point out an essential difference between Theorem 4.6 and Theorem 4.7. In Theorem 4.6, where {Qn }n0 is assumed to be an OPS, the conditions of full rank for the matrices Cn,k , k = 1, 2, and the joint matrices Cn are deduced from (4.8). Indeed, using the Sylvester inequality ([23, p. 13]), n,k + rank Mn−1,N − n  rank C n,k Mn−1,N n − N = rank C n,k , rank Mn−1,N } = n − N.  min{rank C n,k Mn−1,N = n − N , and, thus, Hence, rank Mn,N Cn−N,k = rank C n − N = rank Mn,N Cn−N,k  min{rank Mn,N , rank Cn−N,k } = rank Cn−N,k  n − N, that is, rank Cn−N,k = n − N and rank Cn−N = n − N + 1. However, the situation is quite different in Theorem 4.7 where we assume that {Pn }n0 is an OPS. In this case, although the condition which characterizes the orthogonality of {Qn }n0 is the same (4.8), it is not possible to deduce the requiren . n,k , k = 1, 2, and the matrix C ment about the full rank of the matrices C

16

´ ´ ˜ F. MARCELLAN, M. E. MARRIAGA, T. E. PEREZ, AND M. A. PINAR

Now, we study further relations between the coefficients in (2.3) and (4.5) when {Pn }n0 and {Qn }n0 are two OPS related by (4.1). Let ⎛ ⎞ ⎛ ⎞ B0,k L0,k Q0 ⎜Q1 ⎟ ⎜C ⎟   ⎜ ⎟ ⎜ 1,k B1,k L1,k ⎟ Jk = ⎜ Q = ⎜Q2 ⎟ , ⎟ , k = 1, 2,   C2,k B2,k L2,k ⎝ ⎠ ⎝ ⎠ .. . . . . . . . . . . ⎞ ⎛ ⎛ ⎞ B0,k L0,k P0 ⎟ ⎜C1,k B1,k L1,k ⎜P1 ⎟ ⎟ ⎜ ⎜ ⎟ Jk = ⎜ P = ⎜P2 ⎟ , ⎟ , k = 1, 2. C B L 2,k 2,k 2,k ⎠ ⎝ ⎝ ⎠ .. .. .. .. . . . . Moreover, for any polynomial λ(x, y) = a0,0 +

h N  

ah,j xh−j y j ,

h=1 j=0

we define the semi-infinite matrix λ(J1 , J2 ) = a0,0 I +

h N  

ah,j J1h−j J2j ,

h=1 j=0

where I is the semi-infinite identity matrix. We define λ(J1 , J2 ) similarly. Notice that λ(J1 , J2 ) and λ(J1 , J2 ) are well defined since Jk and Jk , k = 1, 2, are banded matrices. From (2.3) and (4.5), we get λ(x, y) Q = λ(J1 , J2 ) Q

and

λ(x, y) P = λ(J1 , J2 ) P.

The proof of the following result is given in Appendix A. Theorem 4.8. Let {Pn }n0 and {Qn }n0 be two monic OPS with respect to the quasi-definite moment functionals u and v, respectively. If there exists a polynomial λ(x, y) of degree N such that u = λ(x, y) v, then λ(J1 , J2 ) = L U

and

λ(J1 , J2 ) = U L,

where L is a lower triangular matrix with 1’s in its main diagonal and U is an upper block triangular matrix. 5. Examples We will use the standard representation for classical orthogonal polynomials considered in the literature (see for instance [1, 11, 32]). (α,β) As usual, univariate Jacobi polynomials will be denoted by {Pn }n0 . They are orthogonal in [−1, 1] with respect to the weight function w(α,β) (t) = (1 − t)α (1 + t)β ,

α, β > −1,

with the normalization (formula (4.1.1), [32, p. 58])   n+α (α,β) . (1) = Pn n

GERONIMUS TRANSFORMATIONS OF BIVARIATE LINEAR FUNCTIONALS

17

(μ)

We denote by {Cn (t)}n0 the classical Gegenbauer polynomials, orthogonal with respect to the weight function w(t) = (1 − t2 )μ−1/2 , μ > −1/2, t ∈ (−1, 1), such that ((4.7.1), [32, p. 80]) Cn(μ) (t) =

Γ(μ + 1/2) Γ(n + 2μ) P (μ−1/2,μ−1/2) (t). Γ(2μ) Γ(n + μ + 1/2) n

(5.1)

5.1. Example 1. Let λ1 (x) = (x − 1)3 , λ2 (y) = y − 1, and λ(x, y) = λ1 (x) λ2 (y). (x) (y) Consider the univariate quasi-definite moment functionals σ 1 and σ 2 acting on (x) (y) the variables x and y, respectively, such that λ1 (x) σ 1 and λ2 (y) σ 2 are quasidefinite. Define the bivariate moment functional u by   (x) (y) , u, p = σ 1 , σ 2 , p(x, y) for every polynomial p ∈ Π. In [31] it was shown that u is quasi-definite. In this case, LM(λ) = x3 y. Then, A(λ) = {(h, 0) : h  0} ∪ {(h, k) : h = 0, 1, 2, k  1}. A solution v of the equation u = λ(x, y)v is  v = uλ + ah,k u0,h,k . (h,k)∈A(λ)

In particular,

v, p = uλ , p + M1

1 0

p(x, 1)dx + M2

1

p(1, y)dy 0

+ M3

1

0

∂x p(1, y)dy + M4

1 0

∂x2 p(1, y)dy,

where Mi , i = 1, 2, 3, are real numbers, is a solution. We can take advantage of the definition of u to construct another solution of the equation u = λ(x, y)v. We know that (x)

τ1

(y)

τ2

= =

(x)

λ1

(y)

λ2

σ1 σ2

+ Λ1 δ(x − 1) + Λ2 δ  (x − 1) + Λ3 δ  (x − 1), + Λ4 δ(y − 1), (x)

(y)

are multiple Geronimus transformations of σ 1 and σ 2 , respectively ([16]). Here, for a given univariate functional σ and a polynomial ϕ(t), the functional σ ϕ is defined as

p − pϕ , σ ϕ , p = σ, ϕ where pϕ is the remainder of p on the division by ϕ. But according to the definition of uλ , for n  1, uλ , xm y k (x − 1)3n (y − 1)n  = u, xm y k (x − 1)3n−1 (y − 1)n−1    (x) (y) , = σ 1 , σ 2 , xm y k (x − 1)3n−1 (y − 1)n−1



λ1 λ2 (x) (y) = σ1 , σ2 , xm y k (x − 1)3n (y − 1)n ,

18

and

´ ´ ˜ F. MARCELLAN, M. E. MARRIAGA, T. E. PEREZ, AND M. A. PINAR



λ1 λ2 (x) (y) m k . u , x y  = 0 = σ 1 , σ2 ,x y λ

m k

Therefore,   (x) (y)  v, p := τ 1 , τ 2 , p(x, y)

λ1 (x) λ = u , p + Λ4 σ 1 , p(x, 1)



λ2 λ2 λ2 (y) (y) (y) 2 + Λ1 σ 2 , p(1, y) − Λ2 σ 2 , ∂x p(1, y) + Λ3 σ 2 , ∂x p(1, y) + Λ1 Λ4 p(1, 1) − Λ2 Λ4 ∂x p(1, 1) + Λ3 Λ4 ∂x2 p(1, 1).  satisfies u = λ(x, y) Clearly, v v and, thus, it is a Geronimus transformation of u.  − uλ belongs to Ker(Tλ ) according Notice that since λ(x, y) ( v − uλ ) = 0, then v to Corollary 3.6. (x) (y) Denote by {pn (x)}n0 , {qn (y)}n0 the monic OPS with respect to σ 1 and σ 2 , (x) respectively, and { pn (x)}n0 , { qn (y)}n0 the monic OPS with respect to λ1 (x) σ 1 (y) and λ2 (y) σ 2 , respectively. Then, the polynomials defined by Qn,m (x, y) = pn−m (x) qm (y),

n  0,

0  m  n,

n  0,

0  m  n,

are orthogonal with respect to u, and Pn,m (x, y) = pn−m (x) qm (y),

are orthogonal with respect to λ(x, y) u. pn (x)}n0 are related by (see [16, 20]) The polynomials {pn (x)}n0 and { pn (x) = pn (x) + an,1 pn−1 (x) + an,2 pn−2 (x) + an,3 pn−3 (x), qn (x)}n0 are related by and {qn (x)}n0 and { qn (y) = qn (y) + bn,1 qn−1 (y). Since λ(x, y) is a polynomial of degree 4, {Qn,m (x, y) : n  0, 0  m  n} and {Pn,m (x, y) : n  0, 0  m  n} are related by a connection formula as follows Qn,m (x, y) =Pn,m (x, y) + bh,1 Pn−1,m−1 (x, y) + am,1 Pn−1,m (x, y) + an−m,1 bm,1 Pn−2,m−1 (x, y) + an−m,2 Pn−2,m (x, y) + an−m,2 bm,1 Pn−3,m−1 (x, y) + an−m,3 Pn−3,m (x, y)

(5.2)

+ an−m,3 bm,1 Pn−4,m−1 (x, y), where Pn,m (x, y) = 0 for n < m (see Figure 2). 5.2. Example 2. Let us consider dνμ , the classical Gegenbauer univariate measure dνμ (t) = (1 − t2 )μ−1/2 dt,

μ > −1/2,

and the measure dη obtained from dνμ by adding two symmetric mass points at the ends of the supporting interval [−1, 1] ⊂ R, dη(t) = (1 − t2 )μ−1/2 dt + M δ(t − 1) + M δ(t + 1),

M > 0.

GERONIMUS TRANSFORMATIONS OF BIVARIATE LINEAR FUNCTIONALS



n=0 

n=1 

n=2 

n=3 

n=4 

n=5 n=6

19



 

 



   

 

 

 

 



 



Figure 2. The configuration of solid dots corresponds to the elements of {Pn,m (x, y) : n  0, 0  m  n} involved in (5.2). Observe that this configuration remains fixed for every value of n  0 and 0  m  n. Here, we take n = 6, m = 2 (red dot). (μ,M )

We denote by {Cn to dη, such that

Cn(μ,M ) (t) =

(t)}n0 the univariate orthogonal polynomials with respect Γ(μ + 1/2) Γ(n + 2μ) P (μ−1/2,μ−1/2,M,M ) (t), Γ(2μ) Γ(n + μ + 1/2) n

(α,β,M,N )

}n0 , for α, β > −1, M, N  0, are the Jacobi-type polynomials where {Pn (μ,0) (μ) studied in [26]. Observe that Cn (t) = Cn (t), n  0. Then, the bivariate polynomials   y (μ+1/2+m) (μ) 2 m/2 (μ) √ , (5.3) (x) (1 − x ) Cm Pn,m (x, y) = Cn−m 1 − x2 are orthogonal with respect to the positive definite moment functional uμ

uμ , p = p(x, y) (1 − x2 − y 2 )μ−1/2 dxdy, ∀p ∈ Π, B2

and the bivariate polynomials

   m y (μ,M ) 2 √ = 1−x Cm , 1 − x2 0  m  n, n  0, are orthogonal with respect to the positive definite moment functional v

v, p(x, y) = p(x, y) (1 − x2 − y 2 )μ−1/2 dxdy ) Q(μ,M n,m (x, y)

(μ+m+1/2) Cn−m (x)

B2

1

+M

p(x,

−1 1

+M −1

 1 − x2 )(1 − x2 )μ−1/2 dx

p(x, −

 1 − x2 )(1 − x2 )μ−1/2 dx,

∀p ∈ Π.

Since uμ+1 and v satisfy uμ+1 = λ(x, y) v with λ(x, y) = 1 − x2 − y 2 , v is a Geronimus transformation of uμ+1 . We will show that v has the form (3.4) by proving that v − uμ+1 λ = v − uμ + uμ − uμ+1 λ belongs to Ker(Tλ ). In this case, A(λ) = {(h, k) : h = 0, 1, k  0}.

´ ´ ˜ F. MARCELLAN, M. E. MARRIAGA, T. E. PEREZ, AND M. A. PINAR

20

Clearly, λ(x, y) (v − uμ ) = 0. Thus, v − uμ ∈ Ker(Tλ ). It remains to show that uμ − uμ+1 λ ∈ Ker(Tλ ). Write λ(x, y) (uμ − uμ+1 λ ) =

+∞ 



an,h,k un,h,k .

n=0 (h,k)∈A(λ)

For n  1, an,h,k = λ(x, y) (uμ − uμ+1 λ ), xh y k λn  = uμ+1 − uμ+1 , xh y k λn  = 0, and the result follows. (μ,M ) (μ+1) It was shown in [28] that {Qn,m (x, y) : n  0, 0  m  n} and {Pn,m (x, y) : n  0, 0  m  n} are related by a connection formula as follows (m)

(m)

(μ+1)

) (μ+1) Q(μ,M a0,n Pn,m (x, y) + a ˜2,n Pn,m−2 (x, y) n,m (x, y) =˜ (m)

(μ+1)

(m)

(μ+1)

+ c˜0,n Pn−2,m (x, y) + c˜2,n Pn−2,m−2 (x, y). 5.3. Example 3. It was shown in [35] that the generalized characteristic polynomials of a family of banded Toeplitz matrices are orthogonal polynomials in two variables. The bivariate Chebyshev polynomials of the second kind on the deltoid constitute a special case. These orthogonal polynomials generate a family of Gaussian cubature rules in two variables since they possess a maximal number of real common zeros and were studied in the context of two conjugate complex variables given by z = x + iy and z¯ = x − iy. They satisfy some properties that help us to relate them to orthogonal polynomials in two real variables (see [34] as well as [35]). Chebyshev polynomials are orthogonal with respect to wα (z) = w(x + iy) = [−3(x2 + y 2 + 1)2 + 8(x3 − 3xy 2 ) + 4]α ,

α = ±1/2,

on the deltoid, the region bounded by the Steiner’s hypocycloid, that is, the curve x + iy = (2eiθ + e−2iθ )/3,

0  θ  2π.

These polynomials were first studied in [25]. Let {Ukn (z, z¯) : n  0, 0  k  n} be the bivariate analogues of the Chebyshev polynomials of the second kind defined by the three term relation ⎧ n+1 n−1 n (z, z¯) − Uk−1 (z, z¯), 0  k  n, n  1, ⎨ Uk (z, z¯) = 3zUkn (z, z¯) − Uk+1 (5.4) ⎩ 0 z, U0 (z, z¯) = 1, U01 (z, z¯) = 3z, U11 (z, z¯) = 3¯ n (z, z¯) = 0 and Unn−1 (z, z¯) = 0. These polynomials satisfy where U−1 n (z, z Ukn (z, z¯) = Un−k ¯),

0  k  n,

n  0.

Hence, taking complex conjugate in (5.4), we obtain n+1 n−1 n n (z, z¯) = 3¯ z Un−k (z, z¯) − Un−k−1 (z, z¯) − Un−k (z, z¯), 0  k  n, n  1. Un+1−k

Chebyshev polynomials of the second kind are mutually orthogonal with respect to w1/2 . In [35], two families of generalized characteristic polynomials of banded Toeplitz matrices were studied. The polynomials of the first family {Pkn (z, z¯) : n  0, 0 

GERONIMUS TRANSFORMATIONS OF BIVARIATE LINEAR FUNCTIONALS

21

k  n} are orthogonal polynomials and satisfy the three term relation  zPn = Ln,1 Pn+1 + βn Pn + γn Pn−1 , n  0, P−1 = 0, P1 = 1, where Pn = (P0n (z, z¯), P1n (z, z¯), . . . , Pnn (z, z¯)) , ⎛ ⎞ ⎛ 0 c 0 2 ⎜ ⎟ ⎜ . . |c| .. .. ⎜ ⎟ ⎜ γn = ⎜ βn = ⎜ ⎟, ⎝ ⎝ 0 c⎠ 0

... ..

0

.

⎞ ⎟ ⎟ ⎟, ⎠

|c|2

and c is a non-zero complex number. Furthermore, n (z, z Pkn (z, z¯) = Pn−k ¯),

0  k  n,

n  0.

¯ −k Pkn (3α2 z, 3¯ α2 z¯), 0  k  n. Then the Let c = α ¯ 3 /|α|2 and Ukn (z, z¯) = αk−n α n polynomials Uk (z, z¯) are the Chebyshev polynomials of the second kind defined in (5.4). The polynomials of the second family of polynomials {Qnk (z, z¯) : n  0, 0  k  n} studied in [35] are orthogonal polynomials and satisfy the three term relation  zQn = Ln,1 Qn+1 + β˜n Qn + γ˜n Qn−1 , n  0, Q−1 = 0, Q1 = 1, where Qn = (Qn0 (z, z¯), Qn1 (z, z¯), . . . , Qnn (z, z¯)) , ⎛ ⎛ ⎞ 0 0 c  ⎜|c|2 ⎜ ⎟ ⎜ .. .. ⎜ ⎟ ⎜ . . γ˜n = ⎜ β˜n = ⎜ ⎟, ⎜ ⎝ ⎠ 0 c ⎝  c¯ − a ¯ 0 0 

c(c − a)

0 |c|



2

..

.

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎠

|c|2

where c and a are non-zero complex numbers. These polyomials also satisfy Qnk (z, z¯) = Qnn−k (z, z¯),

0  k  n,

n  0.

It was shown in [35] that {Pn }n≥0 and {Qn }n≥0 are related by a − c¯)P0n−3 − c2 |a − c|2 P1n−4 , Qn0 (z, z¯) = P0n − (a − c)P1n−1 + c2 (¯ Qn1 (z, z¯)

= .. .

P1n − c¯(a − c)P0n−2 + c2 (¯ a − c¯)P1n−3 − c|c|2 |a − c|2 P0n−5 ,

n−3 n−6 + c2 (¯ a − c¯)Pkn−3 + |c|4 |a − c|2 Pk−3 , Qnk (z, z¯) = Pkn + c¯2 (a − c)Pk−3 .. . n−2 n−3 n−5 n − c(¯ a − c¯)Pn−2 + c¯2 (a − c)Pn−4 − c¯|c|2 |a − c|2 Pn−5 , Qnn−1 (z, z¯) = Pn−1 n−1 n−3 n−4 a − c¯)Pn−2 + c¯2 (a − c)Pn−3 − c¯2 |a − c|2 Pn−5 , Qnn (z, z¯) = Pnn − (¯ n = 0, where 2  k  n − 2, and for a sake of simplicity we write Pkn ≡ Pkn (z, z¯), P−1 n−1 and Pn = 0.

´ ´ ˜ F. MARCELLAN, M. E. MARRIAGA, T. E. PEREZ, AND M. A. PINAR

22

Moreover,

Rkn (x, y)

=

 1  n √ Pkn (z, z¯) + Pn−k (z, z¯) , 2

0k<

Rkn (x, y)

=

 1  n √ Pkn (z, z¯) − Pn−k (z, z¯) , i 2

n 2

Rnn (x, y)

=

P nn (z, z¯),

n even,

Skn (x, y)

=

 1  √ Qnk (z, z¯) + Qnn−k (z, z¯) , 2

0k<

Skn (x, y)

=

 1  √ Qnk (z, z¯) − Qnn−k (z, z¯) , i 2

n 2

S nn (x, y)

=

Qnn (z, z¯),

n even,

2

2

n 2,

< k  n,

and

2

2

n 2,

< k  n,

are orthogonal polynomials in two real variables x = (z + z¯)/2, y = (z − z¯)/2i, with respect to some real valued quasi-definite linear functionals u and v, respectively ([34]). We deduce the following connection relations

n−1 n−3 n−4 S0n (x, y) = R0n − d1 R1n−1 − d2 Rn−2 + e1 R0n−3 + e2 Rn−3 − f1 R1n−4 − f2 Rn−5 , n−2 n−3 n−5 + e1 R1n−3 + e2 Rn−4 − h1 R0n−5 − h2 Rn−5 , S1n (x, y) = R1n − g1 R0n−2 + g2 Rn−2 .. . n−3 n−3 n−3 n−6 + g1 Rn−k−3 − g2 Rkn−3 − g2 Rn−k + |c|4 |a − c|2 Rn−k−3 , Skn (x, y) = Rkn + g1 Rk−3 .. . n−3 n−3 n−3 n−6 n n n−3 (x, y) = Rm + g1 Rn−m−3 − g2 Rm−3 − g1 Rn−m + g 2 Rm + |c|4 |a − c|2 Rm−3 , Sm .. . n−2 n−3 n−5 n n (x, y) = Rn−1 − g2 R0n−2 − g1 Rn−2 − e2 R1n−3 + e1 Rn−4 + h2 R0n−5 − h1 Rn−5 , Sn−1 n−1 n−3 n−4 − e2 R0n−3 + e1 Rn−3 + f2 R1n−4 − f1 Rn−5 , Snn (x, y) = Rnn + d2 R1n−1 − d1 Rn−2

S nn (x, y) = 2

Rnn + 2

√

2e1 Rn−3 n −3 − 2

√

2e2 Rn−3 + |c|4 |a − c|2 Rn−6 n n −3 , 2

2

n even,

GERONIMUS TRANSFORMATIONS OF BIVARIATE LINEAR FUNCTIONALS

23

n where 2  k < n2 , n2 < m  n − 2, and Rkn ≡ Rkn (x, y), R−1 (x, y) = 0, Rnn−1 (x, y) = 0, and the coefficients are (a − c) + (¯ a − c¯) (a − c) − (¯ a − c¯) , d2 = , d1 = 2 2i

g1

=

c(¯ a − c¯) + c¯(a − c) , 2

g2

=

c(¯ a − c¯) − c¯(a − c) , 2i

e1

=

a − c¯) + c¯2 (a − c) c2 (¯ , 2

e2

=

a − c¯) − c¯2 (a − c) c2 (¯ , 2i

h1

=

c + c¯ 2 |c| |a − c|2 , 2

h2

=

c − c¯ 2 |c| |a − c|2 , 2i

c2 + c¯2 c2 − c¯2 |a − c|2 , |a − c|2 . f2 = 2 2i We remark that the explicit expression for the connecting relation between the orthogonal polynomials in real variables {Rkn (x, y) : n  0, 0  k  n} and {Skn (x, y) : n  0, 0  k  n} is not given in [35]. According to Theorem 4.4 there exists a polynomial λ(x, y) of degree 6 such that u = λ(x, y)v. Hence, v is a Geronimus transformation of u, defined by the weight function w1/2 with (x, y) dilated by a multiple of 3. The explicit expression for λ(x, y) can be deduced using (A.1), Theorem 4.3 in [35], and the relation between complex and real orthogonal polynomials studied in [34]. In fact, for c = 1 and a = 1/2, we obtain 1 4 4 4 4 8 λ(x, y) = + x2 + y 2 − x3 + xy 2 + 4x4 + 8x2 y 2 + 4y 4 − x5 9 3 3 9 3 3 16 4 100 4 2 x y − 20x2 y 4 + 4y 6 . + x3 y 2 + 8xy 4 + x6 + 3 9 3 Notice that the orthogonality with respect to w1/2 on the deltoid corresponds to the case c = a = 1 with (x, y) dilated by 3. In such a case, Gaussian cubature rules exist ([35]). However, nothing was known about the quasi-definite linear functional v. Here, we have established a relation between v and the well known weight w1/2 on the deltoid. f1

=

Appendix A. Proofs of Section 4 A.1. Proof of Theorem 4.4. Taking (4.3) for n = 0, we have u = λ(x, y) v with     −1   −1  −1 . (A.1) λ(x, y) = H0 Q N HN MN,N + · · · + Q1 H1 M1,1 + Q0 H0 From (4.1) and MN,N ≡ 0, we get PN = QN and deg λ(x, y) = N . This proves (i) ⇒ (ii). Now, suppose that u = λ(x, y) v, where λ(x, y) is a polynomial of total degree N and consider the expansion of Qn in terms of the polynomials Pn , n  Mn,k Pn−k , Qn = Pn + k=1

where

´ ´ ˜ F. MARCELLAN, M. E. MARRIAGA, T. E. PEREZ, AND M. A. PINAR

24

−1 −1  Mn,k = u, Qn P n−k Hn−k = λ(x, y) v, Qn Pn−k Hn−k −1 = v, Qn λ(x, y) P n−k Hn−k .

Then, for k  N + 1, Mn,k ≡ 0, and (4.1) follows. Namely, for n  N , Qn = Pn + Mn,1 Pn−1 + · · · + Mn,N Pn−N , and, for n  N − 1, we have Qn = Pn + Mn,1 Pn−1 + · · · + Mn,n P0 . Let λ(x, y) =

h N  

ah,j xh−j y j

with

h=0 j=0

N 

|aN,j | > 0,

j=0

be the explicit expression of the polynomial λ(x, y). Using (4.1), we get −1 −1 −1   MN,N = MN,N u, P0 P 0  H0 = u, QN P0  H0 = λ(x, y) v, QN P0  H0

=

N  j=0

−1 aN,j v, QN xN −j y j X 0  H0 =

⎛

N ⎝ =H

N 

N 

−1  aN,j v, QN X N Γj H0

j=0

⎞ ⎠ H0−1 , aN,j Γ 0,j

j=0

where Γk,j =

j−1 

Lk+i,2

i=0

N −1 

Lk+i,1 ,

i=j

and Ln,k , k = 1, 2, were defined in Proposition 2.3. Notice that Γ0,j = (0, . . . , 1, . . . 0) is the 1 × (N + 1) row vector with a 1 in the (j + 1)-th entry. Thus, ⎛ ⎞ aN,0 ⎜ aN,1 ⎟ ⎟ N ⎜ MN,N = H ⎜ .. ⎟ H0−1 ≡ 0. ⎝ . ⎠ aN,N This proves (ii) ⇒ (i). Suppose that there exists a polynomial λ(x, y) of degree exactly N such that v = λ(x, y) u. The expression of λ(x, y) Pn in terms of the polynomials Qn yields λ(x, y) Pn =

n+N 

Kn,n+N −k Qk ,

k=0

where   −1   −1  −1 Kn,n+N −k = v, λ(x, y) Pn Q k Hk = λ(x, y) v, Pn Qk Hk = u, Pn Qk Hk .

By (ii) ⇒ (i), {Pn }n0 and {Qn }n0 satisfy a (4.1), and, therefore, we get ⎧ ⎨ 0,  −1 , Hn H Kn,n+N −k = n ⎩   −1 , Hn Mk,k−n H k

linear relation of order N such as for 0  k  n − 1, for k = n, for k  n + 1.

GERONIMUS TRANSFORMATIONS OF BIVARIATE LINEAR FUNCTIONALS

25

 n are non-singular, Kn,N is non-singular as well, and Moreover, since Hn and H (4.4) follows. This proves (ii) ⇒ (iii). Finally, assume that {Pn }n0 and {Qn }n0 satisfy (4.4), and let {Un }n0 be the dual basis associated with {Pn }n0 . Since {Un }n0 is a basis of Π , then λ(x, y) v =

+∞ 

U k Ek ,

k=0

where Ek = λ(x, y) v, Pk  = v, λ(x, y) Pk . Using (4.4), Ek = v, Kk,0 Qk+N + · · · + Kk,N −1 Qk+1 + Kk,N Qk  =



0, K0,N H 0,

for k = 0, for k  1.

 −1 and U0 = H −1 u, we get u = λ(x, y) v. This proves (iii) ⇒ Since K0,N = H0 H 0 0 (ii). A.2. Proof of Proposition 4.5. (1) If rank Mh,h = 0 for every 1  h  N , that  −1 )v. Thus is, Mh,h ≡ 0 for 1  h  N , then from (4.3) for n = 0 we get u = (H0 H 0 Pn = Qn , for all n  0, and from (4.1) and the fact that {P0 , . . . , Pn−1 } is a basis for Πn−1 , Mn,h ≡ 0 for 1  h  N holds. (2) If rank MN,N = 1, that is, MN,N ≡ 0, then again when n = 0 in (4.3), u = λ(x, y) v, where λ(x, y) is a polynomial of total degree N . Namely N  h 

λ(x, y) =

ah,j xh−j y j

with

h=0 j=0

N 

|aN,j | > 0.

j=0

Using (4.1), we get   Mn,N Hn−N = Mn,N u, Pn−N P n−N  = u, Qn Pn−N  = λ(x, y) v, Qn Pn−N  N 

=

aN,j v, Qn xN −j y j X n−N  =

j=0

n =H

N 

 aN,j v, Qn X n Γn−N,j

j=0 N 

aN,j Γ n−N,j .

j=0

Observe that Γn−N,j = ( 0 | In−N +1 | 0 ) , is a matrix of size (n − N + 1) × (n + 1) such that its first j and last N − j columns have zeros as entries. Therefore, ⎛ ⎞ N  ⎠ rank ⎝ aN,j Γ n−N,j j=0

⎛

aN,0

⎜ ⎜ = rank ⎜ ⎝

aN,1 aN,0

··· aN,1 .. .

and rank Mn,N = rank Mn,N Hn−N

⎞

aN,N ··· .. .

aN,N .. .

aN,0

aN,1

⎟ ⎟ ⎟ = n − N + 1, ⎠

..

. ···

aN,N

26

´ ´ ˜ F. MARCELLAN, M. E. MARRIAGA, T. E. PEREZ, AND M. A. PINAR

⎛ n ⎝ = rank H

N 

⎞

⎛

⎠ = rank ⎝ aN,j Γ n−N,j

j=0

N 

⎞ ⎠. aN,j Γ n−N,j

j=0

 n and Hn−N are non-singular matrices and Hence, rank Mn,N = n − N + 1, since H the rank condition is invariant through multiplication by non-singular matrices. A.3. Proof of Theorem 4.6. First, (2.3). Let ⎛ ⎞ ⎛ B0,k Q0 ⎜Q1 ⎟ ⎜C  ⎜ ⎟ ⎜ 1,k Jk = ⎜ Q = ⎜Q2 ⎟ , ⎝ ⎠ ⎝ .. . ⎛

⎞

P0 ⎜P1 ⎟ ⎜ ⎟ P = ⎜P2 ⎟ , ⎝ ⎠ .. . and

⎛

B0,k ⎜C1,k ⎜ Jk = ⎜ ⎝

assume that {Pn }n0 is an OPS satisfying L0,k 1,k B 2,k C

L0,k B1,k C2,k

⎞ L1,k 2,k B .. .

L2,k .. .

L1,k B2,k .. .

L2,k .. .

..

..

⎟ ⎟ ⎟, ⎠

k = 1, 2,

. ⎞ ⎟ ⎟ ⎟, ⎠

k = 1, 2,

.

⎛

1 ⎜ M1,1 ⎜ ⎜ M2,2 ⎜ ⎜ .. ⎜ . ⎜ L = ⎜MN,N ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎞ I2 M2,1 .. .

I3 .. .

··· .. .

Mn,2 .. .

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

..

Mn,N

. Mn,1 .. .

IN +1 .. .

··· .. .

Mn,2 .. .

..

. Mn,1 .. .

In+1 .. .

..

.

From (2.3), (4.1), and (4.5), we get xk Q = Jk Q,

xk P = Jk P,

k = 1, 2,

where x1 := x and x2 := y, and Q = L P. Combining these equations, we deduce L Jk = Jk L,

k = 1, 2.

This matrix equation makes sense since each block matrix of LJk and Jk L is the combination of a finite number of block matrices in Jk , Jk and L. We obtain (4.6)-(4.10) from block matrix multiplication of LJk and Jk L and comparing the corresponding block entries. Conversely, write (4.6)-(4.10) in matrix form to obtain LJk = Jk L, k = 1, 2. Observe that the matrix L is a lower triangular matrix with 1 as diagonal entries. Then L has an inverse L−1 and, hence, it is the matrix of change of bases between {Qn }n0 and {Pn }n0 . Therefore, xk P = xk L−1 Q = L−1 Jk Q = L−1 Jk L P = Jk P, It follows that {Pn }n0 satisfies (2.3).

k = 1, 2.

GERONIMUS TRANSFORMATIONS OF BIVARIATE LINEAR FUNCTIONALS

27

To conclude, consider the linear functional u0 := U0 . We have just proved that {Pn }n0 satisfies three term relations such as (2.3). Then, using the same argument as in Theorem 3.3.7 of [17], we obtain the orthogonality condition u0 , Pi P j  = 0,

i = j.

Next, we show that, for every n  0, the symmetric matrix Hn = u0 , Pn P n  is non-singular, that is, it has full rank. Taking into account (2.2) for {Qn }n0 , (4.2) for n = 0, we have u0 = λ(x, y)v, where v is the quasi-definite linear functional associated with the OPS {Qn }n0 , and  −1 MN,N + · · · + Q  −1 M1,1 + Q  −1 . λ(x, y) = Q NH 1H 0H 1

N

0

As in the proof of Proposition 4.5, we deduce that ⎛ ⎞ N  n ⎝ ⎠, aN,j Γ Mn,N Hn−N = H n−N,j

n  N.

j=0

  N   a Γ We know that rank N,j n−N,j = n − N + 1. Since the matrix Hn is nonj=0 singular and the rank condition is invariant under multiplication by non-singular matrices, we get ⎛ ⎞ ⎛ ⎞ N N   n ⎝ ⎠ = rank ⎝ ⎠ = n − N + 1. rank H aN,j Γ aN,j Γ n−N,j n−N,j j=0

j=0

By using the Sylvester inequality, we deduce n − N + 1 = rank Mn,N Hn−N  min{rank Mn,N , rank Hn−N }  rank Hn−N  n − N + 1. Therefore, rank Hn−N = n − N + 1 for n  N + 1. Moreover, for n = N , Hn−n = H0 = u0 , P0 P 0  = u0 , 1 = 1, is a non-singular matrix, and so for every n  0, Hn is non-singular. Thus, {Pn }n0 is an OPS associated with u0 and the proof is completed. A.4. Proof of Theorem 4.7. The necessary condition has already been proved in Theorem 4.6. Conversely, write (4.6)-(4.10) in matrix form to obtain M Jk = Jk M , k = 1, 2. Then, xk Q = xk M P = M Jk P = Jk M P = Jk Q,

k = 1, 2,

and the result follows. A.5. Proof of Theorem 4.8. From Theorem 4.4, {Pn }n0 and {Qn }n0 are related by (4.1) and (4.4). Then, we can write Q = LP

and

λ(x, y) P = U Q,

´ ´ ˜ F. MARCELLAN, M. E. MARRIAGA, T. E. PEREZ, AND M. A. PINAR

28

where

⎛

1 ⎜ M1,1 ⎜ ⎜ M2,2 ⎜ ⎜ .. ⎜ . ⎜ L = ⎜MN,N ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ and

⎛ ⎜ ⎜ ⎜ U =⎜ ⎜ ⎜ ⎝

K0,N

⎞ I2 M2,1 .. .

I3 .. .

··· .. .

Mn,2 .. .

K0,N −1 K1,N

Mn,N

··· K1,N −1 .. .

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

..

. Mn,1 .. .

IN +1 .. .

··· .. .

Mn,2 .. .

K0,0 ··· .. . Kn,N

..

. Mn,1 .. .

In+1 .. .

..

. ⎞

K1,N .. . Kn,N −1 .. .

⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

..

. ··· .. .

Kn,0 .. .

..

.

First, we have λ(J1 , J2 ) Q = λ(x, y) Q = λ(x, y) (L P) = L (λ(x, y) P) = L U Q. Since {Qn }n0 is a PS, we deduce λ(J1 , J2 ) = L U . In a similar way, λ(J1 , J2 ) P = λ(x, y) P = U Q = U L P, and, since {Pn }n0 is a PS, we get λ(J1 , J2 ) = U L. References [1] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, 9th printing. Dover, New York (1972). [2] M. Alfaro, A. Pe˜ na, T. E. P´erez, M. L. Rezola, On linearly related orthogonal polynomials in several variables, Numer. Algorithms 66 (2014), 525-553. [3] M. Alfaro, F. Marcell´ an, A. Pe˜ na, M. L. Rezola, When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?, J. Comput. Appl. Math. 233 (2010), 1446-1452. [4] M. Alfaro, F. Marcell´ an, A. Pe˜ na, M. L. Rezola, Orthogonal polynomials associated with an inverse quadratic spectral transform, Comput. Math. Appl. 61 (2011), 888-900. [5] G. Ariznabarreta, M. Ma˜ nas, Christoffel transformations for multivariate orthogonal polynomials, J. Approx. Theory 225 (2018), 242-283. [6] D. Beghdadi, P. Maroni, On the inverse problem of the product of a semi-classical form by a polynomial, J. Comput. Appl. Math. 88 (1998), 377-399. [7] F. Bracciali, F. Marcell´ an, S. Varma, Orthogonality of quasi-orthogonal polynomials, Filomat 32(20) (2018), 6953-6977. [8] A. Branquinho, F. Marcell´ an, Generating new classes of orthogonal polynomials, Int. J. Math. Sci. 19 (1996), 643-656. [9] M. I. Bueno, F. Marcell´ an, Darboux transformation and perturbation of linear functionals, Linear Algebra Appl. 384 (2004), 215-242. [10] M. I. Bueno, F. Marcell´ an, Polynomial perturbations of bilinear functionals and Hessenberg matrices, Linear Algebra Appl. 41 (2006), 64-83. [11] T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, vol. 13, Gordon and Breach, New York (1978). [12] D. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms, 4th edition, Springer-Verlag, New York, 2015.

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´ ticas, Universidad Carlos III de Madrid (Spain) (F. Marcell´ an) Departamento de Matema Email address: [email protected]

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´ ´ ˜ F. MARCELLAN, M. E. MARRIAGA, T. E. PEREZ, AND M. A. PINAR

´ tica Aplicada, Ciencia e Ingenier´ıa de Ma(M. E. Marriaga) Departamento de Matema ´ nica, Universidad Rey Juan Carlos (Spain) teriales y Tecnolog´ıa Electro Email address: [email protected] ´ ticas IEMath - GR & Departamento de (T. E. P´ erez, M. A. Pi˜ nar) Instituto de Matema ´ tica Aplicada, Facultad de Ciencias. Universidad de Granada (Spain) Matema Email address: [email protected], [email protected]