# Generalized Pseudospectral Method and Zeros of Orthogonal Polynomials

Authors: Oksana Bihun; Clark Mourning
Journal: Advances in Mathematical Physics
 (2018)

---

## Abstract

Via a generalization of the pseudospectral method for numerical solution of differential equations, a family of nonlinear algebraic identities satisfied by the zeros of a wide class of orthogonal polynomials is derived. The generalization is based on a modification of pseudospectral matrix representations of linear differential operators proposed in the paper, which allows these representations to depend on two, rather than one, sets of interpolation nodes. The identities hold for every polynomial familypνxν=0∞ orthogonal with respect to a measure supported on the real line that satisfies some standard assumptions, as long as the polynomials in the family satisfy differential equations Apν(x)=qν(x)pν(x), where A is a linear differential operator and each qν(x) is a polynomial of degree at most n0∈N; n0 does not depend on ν. The proposed identities generalize known identities for classical and Krall orthogonal polynomials, to the case of the nonclassical orthogonal polynomials that belong to the class described above. The generalized pseudospectral representations of the differential operator A for the case of the Sonin-Markov orthogonal polynomials, also known as generalized Hermite polynomials, are presented. The general result is illustrated by new algebraic relations satisfied by the zeros of the Sonin-Markov polynomials.


---

## Body

## 1. Introduction and Main Results

### 1.1. Summary of Results

In this paper, we identify a class of algebraic relations satisfied by the zeros of a wide class of orthogonal polynomials. To prove the identities, we generalize the notion of pseudospectral matrix representations of linear differential operators, by allowing these representations to depend on two, rather than one, sets of interpolation nodes. The identities hold for all polynomialspνxν=0∞ orthogonal with respect to a measure satisfying some standard assumptions, as long as they satisfy the differential equations:(1)Apνx=qνxpνx,where A is a linear differential operator and each qν(x) is a polynomial of degree at most n0∈N; n0 does not depend on ν. This includes classical orthogonal polynomials [1, 2], polynomials in the Askey scheme [3] and Krall polynomials [4], and additional classes of nonclassical orthogonal polynomials. If applied to classical or Krall orthogonal polynomials, the proposed result reduces to known identities [5–7]; see Section 1.4.The motivation of this study stems from understanding that zeros of orthogonal polynomials play an important role in mathematical physics, numerical analysis, and related areas. For example, zeros of some orthogonal polynomials are equilibria of importantN-body problems [8–11]. They transpire as building blocks of remarkable isospectral matrices [12–15] and play an important role in construction of highly accurate approximation schemes for numerical integration [16–19].To prove certain algebraic identities satisfied by the zeros of the polynomialspνxν=0∞, we generalize and relate the notions of spectral and pseudospectral matrix representations of linear differential operators used in the corresponding numerical methods for solving differential equations. The standard pseudospectral matrix representations of linear differential operators are based on Lagrange collocation on the real line. These representations were proposed by Calogero in the context of numerical solving of eigenvalue and boundary value problems for linear ODEs [20, 21]; see also [8, 22–25]. Mitropolsky et al. set up a general algebraic-projection framework based on Calogero’s method and furthered its applications to solution of evolution equations in mathematical physics [26]. The convergence analysis of Calogero’s method was studied in [27].The standard pseudospectral method was utilized in [7] to prove new properties of the zeros of Krall polynomials. While Krall polynomials are eigenfunctions of linear differential operators, the polynomial families considered in this paper satisfy differential equations (1) with qn(x) beingpolynomials (as opposed to eigenvalues) of degree n0, where n0 does not depend on n. To prove new properties of the zeros of the latter polynomial families, we propose ageneralization of the standard pseudospectral method.Because, in general, the differential operatorA in (1) raises the degree of polynomials by a summand of n0, the standard pseudospectral method does not allowexact discretization of differential equations (1). The main idea of the proposed generalization is to construct Lagrange collocation type matrices thatexactly represent linear differential operators acting between spaces of polynomials of different degrees. In this paper, the last goal is achieved by allowing the matrix representations to depend on two rather than one set of interpolation nodes. We thus findexact discretizations of the differential equations (1) satisfied by the polynomials pνxν=0∞; the discretizations are constructed using the zeros of these polynomials as the nodes. By comparing the generalized pseudospectral and the generalized spectral matrix representations of the differential operator A, we derive a family of algebraic identities satisfied by the zeros of polynomials pνxν=0∞.The proposed generalization of the pseudospectral matrix representations of linear differential operators has applications beyond those outlined in this paper; one such application allows to simplify the process of incorporation of initial or boundary conditions into linear systems that discretize certain ODEs; see the discussion in Section4.We illustrate the general result of the main Theorem1 by applying it to the case of the Sonin-Markov polynomials, also known as generalized Hermite polynomials; see [16, 28, 29] and references therein. These zeros play an important role in the computation of integrals of singular or oscillatory functions [17, 18] as well as extended Lagrange interpolation on the real line [19].To prove the identities satisfied by the zeros of the Sonin-Markov polynomials stated in Theorem7, we compute the generalized pseudospectral matrix representations of the differential operators d/dx and d2/dx2 as well as the differential operator D associated with the Sonin-Markov family, see (26) and Section 2.2, assuming that the interpolation nodes are zeros of the Sonin-Markov polynomials. The formulas for these matrix representations as well as the identities of Theorem 7 have been verified using programming environment MATLAB, for several particular values of the relevant parameters (the degree N and the parameter β; see Theorem 7 and definition (21) and (22)).


### 1.2. The Orthogonal Polynomial Familypνxν=0∞

Letpνxν=0∞ be a sequence of polynomials orthogonal with respect to a measure ω and the corresponding inner product f,g=∫fgdω. We denote the norm associated with this inner product by ·; that is, f2=∫f2dω. Assume that ω is a Borel measure with support on the real line satisfying the following three conditions:(a)
ω is positive.(b)
all its moments∫xνdω exist and are finite.(c)
ω has infinitely many points in its support I=supp⁡ω.Under the above assumptions on the measureω, the zeros of each polynomial pν, ν≥1, are real and simple and belong to the convex hull of the support of ω; see, for example, [16].Notation 1.
Here and throughout the rest of the paperN denotes a fixed integer strictly larger than 1, while n0 is a fixed nonnegative integer. The small Greek letter ρ denotes an index that may take values N or N+n0. The small Greek letter ν denotes an integer index that usually takes values 0,1,2,…, unless otherwise indicated. The small Latin letters n, m, j, k, and so forth denote integer indices that usually run from 1 to N or from 1 to N+n0, see (2) and (1), and thus we indicate the range of the indices each time they are used. We reserve the letter l to denote polynomials in Lagrange interpolation bases.LetPν denote the space of all algebraic polynomials with real coefficients of degree at most ν. Assume that, for each ν, the polynomials pjxj=0ν form a basis of Pν. Let A be a linear differential operator acting on functions of one variable. Assume that A has the following property:(2)APν⊆Pν+n0,for all ν. Recall that n0 is a fixed nonnegative integer; it does not depend on ν. For example, the differential operator D=a0+a1xd/dx+⋯+aq(x)dq/dxq with q∈N and aj(x)∈Pj for all j=1,2,…,q has property (2) with n0=0, while the operator x2D has property (2) with n0=2.Suppose that the orthogonal polynomialspνxν=0∞ satisfy differential equations (1).


### 1.3. Generalized Pseudospectral and Spectral Matrix Representations of Linear Differential Operators

In this subsection, we introduce the notions of a generalized pseudospectral and a generalized spectral matrix representations of the linear differential operatorA introduced in the previous subsection. Note that, in general, the definitions of these generalized matrix representations hold for any linear differential operator D, which may or may not satisfy property (2), and for n0 being a (positive or negative) integer such that N+n0≥1.The definition of the standard pseudospectralN×N matrix representation A~c of A, as shown in [7, 25, 30, 31], is motivated by a search for an exact discretization of a differential equation(3)Au=f,under the assumption that it possesses a polynomial solution u∈PN-1. More precisely, choose a vector x→(N)=x1(N),…,xN(N) of distinct real nodes and define the isomorphism πN:PN-1→RN by(4)πNg=gx1N,…,gxNN.The inverse of πN is of course given in terms of the standard Lagrange interpolation basis lN-1,jxj=1N of PN-1 constructed using the nodes x→(N): for every vector g→(N)=g1(N),…,gN(N)∈RN, (5)πN-1g→N=∑j=1NgjNlN-1,jx. The standard pseudospectral N×N matrix representation A~c of the differential operator A is defined as the unique N×N matrix that satisfies the following condition: (6)πNAg=A~cπNg,for all g∈PN-1. Note that the superscript “c” in the notation of the matrix A~c stands for “collocation” in the spectral collocation method for numerical solving of differential equations, also known as the pseudospectral method [25]. It is not difficult to conclude that the matrix A~c is given componentwise by (7)A~kjc=AlN-1,jxx=xkN, where 1≤k, j≤N; see [7, 20, 21, 26, 27, 30, 31]. The last definition implies that a vector u→(N)∈RN solves the linear system(8)A~cu→N=πNf,if and only if the polynomial u(x)=(πN)-1u→(N)=∑j=1Nuj(N)lN-1,j(x) solves ODE (3). Of course, if ODE (3) does not possess a polynomial solution u∈PN-1, linear system (8)approximates ODE (3), in the sense that a solution u→(N) of system (8), if it exists, allows one to construct anapproximate solution of ODE (3) given by u(x)=(πN)-1u→N.Let us now generalize this notion of pseudospectral matrix representation of the differential operatorA to take advantage of its property (2).In addition to the interpolation nodesx→(N), consider another vector of distinct real nodes x→(N+n0)=x1(N+n0),…,xN+n0(N+n0). In short, we will work with two vectors of nodes x→(ρ), where ρ=N or ρ=N+n0, see Notation 1, and the respective Lagrange interpolation bases lρ-1,jxj=1ρ. Recall that, for each j∈1,…,ρ,(9)lρ-1,jx=ψρxψρ′xjρx-xjρ,where ψρ(x)=x-x1(ρ)x-x2(ρ)⋯x-xρ(ρ) is the node polynomial.We define the generalized pseudospectral(N+n0)×N matrix representation Acx→(N),x→(N+n0)≡Ac of the linear differential operator A componentwise by(10)Akjc=AlN-1,jxkN+n0,1≤k≤N+n0,1≤j≤N,where, as before, the superscript “c” stands for “collocation.” This definition is motivated by the following relation: (11)πN+n0Ag=AcπNg, for all g∈PN-1, where the isomorphisms πρ:Pρ-1→Rρ are defined by (4) with N being replaced with ρ,  ρ∈N,N+n0. In other words, if ODE (3) has a polynomial solution u∈PN-1, then the vector u→(N)∈RN solves the system of linear equations(12)Acu→N=πN+n0f if and only if the polynomial u(x)=(πN)-1u→(N)=∑j=1Nuj(N)lN-1,j(x) solves ODE (3).Using analogous motivation, we define the generalized spectral(N+n0)×N matrix representation Aτ of the linear differential operator A componentwise by(13)Akjτ=Apj-1,pk-1pk-12,1≤k≤N+n0,1≤j≤N.Here, the superscript “τ” indicates that the τ-variant of the spectral method is used [25].We prove that the(N+n0)×N matrices Aτ and Ac satisfy the following property:(14)LN+n0-1Ac=AτLN-1,where each of the two ρ×ρ matrices L(ρ-1) with ρ=N or ρ=N+n0 is the transition matrix from the orthogonal polynomial basis pjxj=0ρ-1 to the Lagrange interpolation basis lρ-1,jxj=1ρ; see Theorem 8 in Section 3.


### 1.4. Main Result: Algebraic Identities Satisfied by the Zeros of the Polynomialspνxν=0∞

Let us now assume thatx1(N),…,xN(N) are the zeros of the polynomial pN(x) from the orthogonal family pνxν=0∞, while x1(N+n0),…,xN+n0(N+n0) are the zeros of the polynomial pN+n0(x). We therefore use these two sets of zeros as the nodes in the definition of the pseudospectral matrix representation Ac of the differential operator A; see (10) and (9). In this case, each of the two matrices L(ρ-1) in relation (14), where ρ=N or ρ=N+n0, can be expressed in terms of the values pm-1(xj(ρ)) and the Christoffel numbers λj(ρ-1):(15)Lmjρ-1=pm-1xjρpm-12λjρ-1,1≤m,j≤ρ,where the Christoffel numbers are defined by(16)λjρ-1=∫lρ-1,jxdω,1≤j≤ρ,see Theorem 8.Recall that Christoffel numbers arise in the Gaussian quadrature numerical integration formulas; they are always positive [16]. Christoffel numbers play an important role in the proof of the main identity (18) presented in this paper, although they are eliminated from that identity in the process of inversion of the matrices L(ρ-1): the inverses of L(ρ-1) are given componentwise by(17)Lρ-1-1mj=pj-1xmρ,1≤m,j≤ρ,see (51) in the proof of Theorem 8.Using property (14) of the matrix representations Aτ and Ac, together with the neat formulas (17) for the matrices L(ρ-1)-1 in the case where x→(ρ) are the zeros of pρ(x),  ρ∈N,N+n0, we prove the following algebraic identities satisfied by the zeros of the polynomials in the family pνxν=0∞.Theorem 1.
The zerosx→(N)=x1(N),…,xN(N) of the polynomial pN(x) and the zeros x→(N+n0)=x1(N+n0),…,xN+n0(N+n0) of the polynomial pN+n0(x) in the orthogonal polynomial family pνxν=0∞ of generalized eigenfunctions of the linear differential operator A (see (1)) satisfy the following algebraic relations for all integer m,n such that 1≤m≤N+n0, 1≤n≤N:(18)∑k=1NAmkcx→N,x→N+n0pn-1xkN=∑k=max⁡n-n0,1min⁡n+n0,N+n0Aknτpk-1xmN+n0,where the (N+n0)×N pseudospectral and spectral matrix representations Acx→(N),x→(N+n0)≡Ac and Aτ, respectively, are defined by (10) and (13).Remark 2.
For every pair of integersn,m such that 0≤n≤N-1 and 1≤m≤N+n0, identity (18) relates the zeros of the polynomials pN(x), pN+n0(x) and pn(x) with the zeros of all the polynomials pk(x) such that the index k∈0,1,…,N+n0-1 satisfies n-n0≤k≤n+n0.Remark 3.
The main identity of Theorem1 may be recast as follows:(19)e→mN+n0TAcv→nN=w→mN+n0Aτe→nN,for all 1≤m≤N+n0 and 1≤n≤N, where each v→n(N)∈RN is a column-vector with the components v→n(N)k=pn-1xk(N), each w→m(N+n0)∈RN+n0 is a row-vector with the components w→m(N+n0)k=pk-1xm(N+n0) and the column-vectors e→kρk=1ρ form the standard basis of Rρ.Theorem1 is proved in Section 3.Identities (18) or, equivalently, (19), are remarkable in the sense that they reveal a deeper structure and relation between the linear operators Ac and Aτ, the generalized spectral and pseudospectral representations, respectively, of the differential operator A, for the case where these operators are constructed using two sets of zeros of orthogonal polynomials as the nodes.Let us compare the main result of Theorem1 with other results of this kind. By setting n0=0 in identities (18), (19), we obtain that(20)∑k=1NAmkcx→Npn-1xkN=qn-1pn-1xmN,for all m,n such that 1≤m, n≤N. In this case, of course, qν are constants, see (1), and pνxν=0∞ are either classical or Krall orthogonal polynomials. In this special case where n0=0, identities of Theorem 1 reduce to those reported in [7]. It was shown in [7] that, if applied to the classical Jacobi, Hermite, or Laguerre polynomials, identities (20) reduce to known identities for the zeros of these polynomials reported in [5, 6].Therefore, identities (18), (19) generalize similar results for classical orthogonal polynomials proved in [5, 6] and for Krall polynomials proved in [7]. These identities may be considered as analogues of the properties of the zeros of the Askey scheme and generalized hypergeometric polynomials proved in [12–15], for the case of the polynomial families considered in this paper. An application of the identities proved in this paper is related to the study of the asymptotic behavior of algebraic expressions involving the zeros of orthogonal polynomials of degree N as N→∞; see [32, 33].In Section2, we apply Theorem 1 to prove new identities satisfied by the zeros of the nonclassical Sonin-Markov orthogonal polynomials. In Section 3, “Proofs,” we elaborate on the proofs of most of the theorems of this paper, except for those that are straightforward consequences of another theorem. In Section 4, titled “Discussion and Outlook,” we summarize the results proposed in this paper and discuss their importance, possible applications, and further developments.


## 1.1. Summary of Results

In this paper, we identify a class of algebraic relations satisfied by the zeros of a wide class of orthogonal polynomials. To prove the identities, we generalize the notion of pseudospectral matrix representations of linear differential operators, by allowing these representations to depend on two, rather than one, sets of interpolation nodes. The identities hold for all polynomialspνxν=0∞ orthogonal with respect to a measure satisfying some standard assumptions, as long as they satisfy the differential equations:(1)Apνx=qνxpνx,where A is a linear differential operator and each qν(x) is a polynomial of degree at most n0∈N; n0 does not depend on ν. This includes classical orthogonal polynomials [1, 2], polynomials in the Askey scheme [3] and Krall polynomials [4], and additional classes of nonclassical orthogonal polynomials. If applied to classical or Krall orthogonal polynomials, the proposed result reduces to known identities [5–7]; see Section 1.4.The motivation of this study stems from understanding that zeros of orthogonal polynomials play an important role in mathematical physics, numerical analysis, and related areas. For example, zeros of some orthogonal polynomials are equilibria of importantN-body problems [8–11]. They transpire as building blocks of remarkable isospectral matrices [12–15] and play an important role in construction of highly accurate approximation schemes for numerical integration [16–19].To prove certain algebraic identities satisfied by the zeros of the polynomialspνxν=0∞, we generalize and relate the notions of spectral and pseudospectral matrix representations of linear differential operators used in the corresponding numerical methods for solving differential equations. The standard pseudospectral matrix representations of linear differential operators are based on Lagrange collocation on the real line. These representations were proposed by Calogero in the context of numerical solving of eigenvalue and boundary value problems for linear ODEs [20, 21]; see also [8, 22–25]. Mitropolsky et al. set up a general algebraic-projection framework based on Calogero’s method and furthered its applications to solution of evolution equations in mathematical physics [26]. The convergence analysis of Calogero’s method was studied in [27].The standard pseudospectral method was utilized in [7] to prove new properties of the zeros of Krall polynomials. While Krall polynomials are eigenfunctions of linear differential operators, the polynomial families considered in this paper satisfy differential equations (1) with qn(x) beingpolynomials (as opposed to eigenvalues) of degree n0, where n0 does not depend on n. To prove new properties of the zeros of the latter polynomial families, we propose ageneralization of the standard pseudospectral method.Because, in general, the differential operatorA in (1) raises the degree of polynomials by a summand of n0, the standard pseudospectral method does not allowexact discretization of differential equations (1). The main idea of the proposed generalization is to construct Lagrange collocation type matrices thatexactly represent linear differential operators acting between spaces of polynomials of different degrees. In this paper, the last goal is achieved by allowing the matrix representations to depend on two rather than one set of interpolation nodes. We thus findexact discretizations of the differential equations (1) satisfied by the polynomials pνxν=0∞; the discretizations are constructed using the zeros of these polynomials as the nodes. By comparing the generalized pseudospectral and the generalized spectral matrix representations of the differential operator A, we derive a family of algebraic identities satisfied by the zeros of polynomials pνxν=0∞.The proposed generalization of the pseudospectral matrix representations of linear differential operators has applications beyond those outlined in this paper; one such application allows to simplify the process of incorporation of initial or boundary conditions into linear systems that discretize certain ODEs; see the discussion in Section4.We illustrate the general result of the main Theorem1 by applying it to the case of the Sonin-Markov polynomials, also known as generalized Hermite polynomials; see [16, 28, 29] and references therein. These zeros play an important role in the computation of integrals of singular or oscillatory functions [17, 18] as well as extended Lagrange interpolation on the real line [19].To prove the identities satisfied by the zeros of the Sonin-Markov polynomials stated in Theorem7, we compute the generalized pseudospectral matrix representations of the differential operators d/dx and d2/dx2 as well as the differential operator D associated with the Sonin-Markov family, see (26) and Section 2.2, assuming that the interpolation nodes are zeros of the Sonin-Markov polynomials. The formulas for these matrix representations as well as the identities of Theorem 7 have been verified using programming environment MATLAB, for several particular values of the relevant parameters (the degree N and the parameter β; see Theorem 7 and definition (21) and (22)).


## 1.2. The Orthogonal Polynomial Familypνxν=0∞

Letpνxν=0∞ be a sequence of polynomials orthogonal with respect to a measure ω and the corresponding inner product f,g=∫fgdω. We denote the norm associated with this inner product by ·; that is, f2=∫f2dω. Assume that ω is a Borel measure with support on the real line satisfying the following three conditions:(a)
ω is positive.(b)
all its moments∫xνdω exist and are finite.(c)
ω has infinitely many points in its support I=supp⁡ω.Under the above assumptions on the measureω, the zeros of each polynomial pν, ν≥1, are real and simple and belong to the convex hull of the support of ω; see, for example, [16].Notation 1.
Here and throughout the rest of the paperN denotes a fixed integer strictly larger than 1, while n0 is a fixed nonnegative integer. The small Greek letter ρ denotes an index that may take values N or N+n0. The small Greek letter ν denotes an integer index that usually takes values 0,1,2,…, unless otherwise indicated. The small Latin letters n, m, j, k, and so forth denote integer indices that usually run from 1 to N or from 1 to N+n0, see (2) and (1), and thus we indicate the range of the indices each time they are used. We reserve the letter l to denote polynomials in Lagrange interpolation bases.LetPν denote the space of all algebraic polynomials with real coefficients of degree at most ν. Assume that, for each ν, the polynomials pjxj=0ν form a basis of Pν. Let A be a linear differential operator acting on functions of one variable. Assume that A has the following property:(2)APν⊆Pν+n0,for all ν. Recall that n0 is a fixed nonnegative integer; it does not depend on ν. For example, the differential operator D=a0+a1xd/dx+⋯+aq(x)dq/dxq with q∈N and aj(x)∈Pj for all j=1,2,…,q has property (2) with n0=0, while the operator x2D has property (2) with n0=2.Suppose that the orthogonal polynomialspνxν=0∞ satisfy differential equations (1).


## 1.3. Generalized Pseudospectral and Spectral Matrix Representations of Linear Differential Operators

In this subsection, we introduce the notions of a generalized pseudospectral and a generalized spectral matrix representations of the linear differential operatorA introduced in the previous subsection. Note that, in general, the definitions of these generalized matrix representations hold for any linear differential operator D, which may or may not satisfy property (2), and for n0 being a (positive or negative) integer such that N+n0≥1.The definition of the standard pseudospectralN×N matrix representation A~c of A, as shown in [7, 25, 30, 31], is motivated by a search for an exact discretization of a differential equation(3)Au=f,under the assumption that it possesses a polynomial solution u∈PN-1. More precisely, choose a vector x→(N)=x1(N),…,xN(N) of distinct real nodes and define the isomorphism πN:PN-1→RN by(4)πNg=gx1N,…,gxNN.The inverse of πN is of course given in terms of the standard Lagrange interpolation basis lN-1,jxj=1N of PN-1 constructed using the nodes x→(N): for every vector g→(N)=g1(N),…,gN(N)∈RN, (5)πN-1g→N=∑j=1NgjNlN-1,jx. The standard pseudospectral N×N matrix representation A~c of the differential operator A is defined as the unique N×N matrix that satisfies the following condition: (6)πNAg=A~cπNg,for all g∈PN-1. Note that the superscript “c” in the notation of the matrix A~c stands for “collocation” in the spectral collocation method for numerical solving of differential equations, also known as the pseudospectral method [25]. It is not difficult to conclude that the matrix A~c is given componentwise by (7)A~kjc=AlN-1,jxx=xkN, where 1≤k, j≤N; see [7, 20, 21, 26, 27, 30, 31]. The last definition implies that a vector u→(N)∈RN solves the linear system(8)A~cu→N=πNf,if and only if the polynomial u(x)=(πN)-1u→(N)=∑j=1Nuj(N)lN-1,j(x) solves ODE (3). Of course, if ODE (3) does not possess a polynomial solution u∈PN-1, linear system (8)approximates ODE (3), in the sense that a solution u→(N) of system (8), if it exists, allows one to construct anapproximate solution of ODE (3) given by u(x)=(πN)-1u→N.Let us now generalize this notion of pseudospectral matrix representation of the differential operatorA to take advantage of its property (2).In addition to the interpolation nodesx→(N), consider another vector of distinct real nodes x→(N+n0)=x1(N+n0),…,xN+n0(N+n0). In short, we will work with two vectors of nodes x→(ρ), where ρ=N or ρ=N+n0, see Notation 1, and the respective Lagrange interpolation bases lρ-1,jxj=1ρ. Recall that, for each j∈1,…,ρ,(9)lρ-1,jx=ψρxψρ′xjρx-xjρ,where ψρ(x)=x-x1(ρ)x-x2(ρ)⋯x-xρ(ρ) is the node polynomial.We define the generalized pseudospectral(N+n0)×N matrix representation Acx→(N),x→(N+n0)≡Ac of the linear differential operator A componentwise by(10)Akjc=AlN-1,jxkN+n0,1≤k≤N+n0,1≤j≤N,where, as before, the superscript “c” stands for “collocation.” This definition is motivated by the following relation: (11)πN+n0Ag=AcπNg, for all g∈PN-1, where the isomorphisms πρ:Pρ-1→Rρ are defined by (4) with N being replaced with ρ,  ρ∈N,N+n0. In other words, if ODE (3) has a polynomial solution u∈PN-1, then the vector u→(N)∈RN solves the system of linear equations(12)Acu→N=πN+n0f if and only if the polynomial u(x)=(πN)-1u→(N)=∑j=1Nuj(N)lN-1,j(x) solves ODE (3).Using analogous motivation, we define the generalized spectral(N+n0)×N matrix representation Aτ of the linear differential operator A componentwise by(13)Akjτ=Apj-1,pk-1pk-12,1≤k≤N+n0,1≤j≤N.Here, the superscript “τ” indicates that the τ-variant of the spectral method is used [25].We prove that the(N+n0)×N matrices Aτ and Ac satisfy the following property:(14)LN+n0-1Ac=AτLN-1,where each of the two ρ×ρ matrices L(ρ-1) with ρ=N or ρ=N+n0 is the transition matrix from the orthogonal polynomial basis pjxj=0ρ-1 to the Lagrange interpolation basis lρ-1,jxj=1ρ; see Theorem 8 in Section 3.


## 1.4. Main Result: Algebraic Identities Satisfied by the Zeros of the Polynomialspνxν=0∞

Let us now assume thatx1(N),…,xN(N) are the zeros of the polynomial pN(x) from the orthogonal family pνxν=0∞, while x1(N+n0),…,xN+n0(N+n0) are the zeros of the polynomial pN+n0(x). We therefore use these two sets of zeros as the nodes in the definition of the pseudospectral matrix representation Ac of the differential operator A; see (10) and (9). In this case, each of the two matrices L(ρ-1) in relation (14), where ρ=N or ρ=N+n0, can be expressed in terms of the values pm-1(xj(ρ)) and the Christoffel numbers λj(ρ-1):(15)Lmjρ-1=pm-1xjρpm-12λjρ-1,1≤m,j≤ρ,where the Christoffel numbers are defined by(16)λjρ-1=∫lρ-1,jxdω,1≤j≤ρ,see Theorem 8.Recall that Christoffel numbers arise in the Gaussian quadrature numerical integration formulas; they are always positive [16]. Christoffel numbers play an important role in the proof of the main identity (18) presented in this paper, although they are eliminated from that identity in the process of inversion of the matrices L(ρ-1): the inverses of L(ρ-1) are given componentwise by(17)Lρ-1-1mj=pj-1xmρ,1≤m,j≤ρ,see (51) in the proof of Theorem 8.Using property (14) of the matrix representations Aτ and Ac, together with the neat formulas (17) for the matrices L(ρ-1)-1 in the case where x→(ρ) are the zeros of pρ(x),  ρ∈N,N+n0, we prove the following algebraic identities satisfied by the zeros of the polynomials in the family pνxν=0∞.Theorem 1.
The zerosx→(N)=x1(N),…,xN(N) of the polynomial pN(x) and the zeros x→(N+n0)=x1(N+n0),…,xN+n0(N+n0) of the polynomial pN+n0(x) in the orthogonal polynomial family pνxν=0∞ of generalized eigenfunctions of the linear differential operator A (see (1)) satisfy the following algebraic relations for all integer m,n such that 1≤m≤N+n0, 1≤n≤N:(18)∑k=1NAmkcx→N,x→N+n0pn-1xkN=∑k=max⁡n-n0,1min⁡n+n0,N+n0Aknτpk-1xmN+n0,where the (N+n0)×N pseudospectral and spectral matrix representations Acx→(N),x→(N+n0)≡Ac and Aτ, respectively, are defined by (10) and (13).Remark 2.
For every pair of integersn,m such that 0≤n≤N-1 and 1≤m≤N+n0, identity (18) relates the zeros of the polynomials pN(x), pN+n0(x) and pn(x) with the zeros of all the polynomials pk(x) such that the index k∈0,1,…,N+n0-1 satisfies n-n0≤k≤n+n0.Remark 3.
The main identity of Theorem1 may be recast as follows:(19)e→mN+n0TAcv→nN=w→mN+n0Aτe→nN,for all 1≤m≤N+n0 and 1≤n≤N, where each v→n(N)∈RN is a column-vector with the components v→n(N)k=pn-1xk(N), each w→m(N+n0)∈RN+n0 is a row-vector with the components w→m(N+n0)k=pk-1xm(N+n0) and the column-vectors e→kρk=1ρ form the standard basis of Rρ.Theorem1 is proved in Section 3.Identities (18) or, equivalently, (19), are remarkable in the sense that they reveal a deeper structure and relation between the linear operators Ac and Aτ, the generalized spectral and pseudospectral representations, respectively, of the differential operator A, for the case where these operators are constructed using two sets of zeros of orthogonal polynomials as the nodes.Let us compare the main result of Theorem1 with other results of this kind. By setting n0=0 in identities (18), (19), we obtain that(20)∑k=1NAmkcx→Npn-1xkN=qn-1pn-1xmN,for all m,n such that 1≤m, n≤N. In this case, of course, qν are constants, see (1), and pνxν=0∞ are either classical or Krall orthogonal polynomials. In this special case where n0=0, identities of Theorem 1 reduce to those reported in [7]. It was shown in [7] that, if applied to the classical Jacobi, Hermite, or Laguerre polynomials, identities (20) reduce to known identities for the zeros of these polynomials reported in [5, 6].Therefore, identities (18), (19) generalize similar results for classical orthogonal polynomials proved in [5, 6] and for Krall polynomials proved in [7]. These identities may be considered as analogues of the properties of the zeros of the Askey scheme and generalized hypergeometric polynomials proved in [12–15], for the case of the polynomial families considered in this paper. An application of the identities proved in this paper is related to the study of the asymptotic behavior of algebraic expressions involving the zeros of orthogonal polynomials of degree N as N→∞; see [32, 33].In Section2, we apply Theorem 1 to prove new identities satisfied by the zeros of the nonclassical Sonin-Markov orthogonal polynomials. In Section 3, “Proofs,” we elaborate on the proofs of most of the theorems of this paper, except for those that are straightforward consequences of another theorem. In Section 4, titled “Discussion and Outlook,” we summarize the results proposed in this paper and discuss their importance, possible applications, and further developments.


## 2. Application: Properties of the Zeros of the Sonin-Markov Orthogonal Polynomials

In this section, we illustrate Theorem1 by applying it to the case of the Sonin-Markov orthogonal polynomials, which are generalized eigenfunctions of a certain linear differential operator D; see (25) and (26). This application requires a computation of the generalized spectral and the generalized pseudospectral matrix representations of the differential operator D associated with the Sonin-Markov polynomials. We thus proceed as follows. In Section 2.1, we define the Sonin-Markov polynomials and state their basic properties. In Sections 2.2 and 2.3, respectively, we compute the generalized pseudospectral and the generalized spectral representations, respectively, of the differential operator D. Finally, in Section 2.4, we provide a family of new algebraic properties satisfied by the zeros of the Sonin-Markov polynomials.


### 2.1. Definition and Basic Properties of the Sonin-Markov Polynomials

The Sonin-Markov polynomialspνβxν=0∞, which are also known as generalized Hermite polynomials, are orthogonal on I=(-∞,∞) with respect to the weight w(x)=xβe-x2 (see [16, 28] and references therein). These polynomials can be expressed in terms of the generalized Laguerre polynomials as follows:(21)p2nβx=cnLnαx2,p2n+1βx=dnxLnα+1x2,where(22)α=β-12,and the coefficients,(23)cn=-1nn!Γn+1+α,dn=-1nn!Γn+2+α,are chosen to make the polynomials orthonormal and the leading coefficients positive. Note that the leading coefficients Kν of pνβ(x) are given by(24)K2n=n!Γn+1+α-1/2,K2n+1=n!Γn+2+α-1/2.The zeros of the Sonin-Markov polynomials are distinct, real, and symmetric with respect to the origin.The Sonin-Markov polynomials satisfy the following differential equations:(25)Dpνx=μνx2+ηνpνx,where D is the linear differential operator given by(26)D=x2d2dx2+x-2x2+2α+1ddx,(27a)μν=-2ν,(27b)ην=0ifν=2n,2α+1ifν=2n+1.These differential equations can be derived from the corresponding differential equations satisfied by the generalized Laguerre polynomials stated in [3]; see also [1] and formula (3.5) in [34]. Clearly, DPν⊆Pν+2, so we can apply Theorem 1 with n0=2 to these polynomials.


### 2.2. The Generalized Pseudospectral Matrix Representation of the Sonin-Markov Differential Operator (26)

Letx1(N),…,xN(N) be the zeros of the Sonin-Markov polynomial pN(x)≡pNβ(x) defined by (21) and let x1(N+2),…,xN+2(N+2) be the zeros of the Sonin-Markov polynomial pN+2(x)≡pN+2β(x). In this subsection, we find the generalized pseudospectral matrix representation Dcx→(N),x→(N+2)≡Dc of the Sonin-Markov differential operator (26) with respect to the two vectors of nodes x→(N)=x1(N),…,xN(N) and x→(N+2)=x1(N+2),…,xN+2(N+2). To compute this representation Dc, we use definition (10) with n0=2. In the following, we use the notation of Section 2.1; note the appropriate definitions of α, cn, dn and μν, ην.LetlN-1,jxj=1N be the Lagrange interpolation basis with respect to the nodes xjNj=1N defined by (9) with ρ=N and ψN(x) replaced with pN(x). Let Z(k)x→(N),x→(N+2)≡Z(k) be the (N+2)×N generalized pseudospectral matrix representation of the differential operator dk/dxk with respect to the two vectors of nodes x→(N) and x→(N+2). By definition (10), its components Zmn(k) are given by(28)Zmnk=dkdxklN-1,nxx=xmN+2.The generalized pseudospectral(N+2)×N matrix representation Dc of the differential operator D is given componentwise by(29)Dmnc=DlN-1,nxx=xmN+2=xmN+22Zmn2+xmN+2-2xmN+22+2α+1Zmn1;see definition (10).By using the fact that the Sonin-Markov polynomialpN(x) satisfies differential equation (25) and (26) and the differential equation obtained by differentiation of (25) and (26) with respect to x (with ν=N), we simplify the formulas for Zmn(k), where k=1,2, to read(30)Zmn1=BmnpN′xnNpN′xmN+2-BmnpNxmN+2ifxmN+2≠xnN,0ifxmN+2=xnN=0,(31)Zmn2=BmnpN′xnN2xmN+22-2α-1xmN+2-2BmnpN′xmN+2+μNxmN+22+ηNxmN+22+2Bmn2pNxmN+2ifxmN+2≠xnNandxmN+2≠0,-2pN′0pN′xnNBmn2ifxmN+2=0andxnN≠0,μN+22α+2ifxmN+2=xnN=0,where Bmn are defined by(32)Bmn=xmN+2-xnN-1.By using the last two expressions for the components of the matricesZ(1) and Z(2) in (29), we obtain the following formulas for the components of the generalized pseudospectral matrix representation Dc:(33a)Dmnc=BmnpN′xnN-2BmnxmN+22pN′xmN+2+μNxmN+22+ηN+2Bmn2xmN+22-BmnxmN+2-2xmN+22+2α+1pNxmN+2ifxmN+2≠xnNandxmN+2≠0,(33b)Dmnc=0ifxmN+2=0,where, again, Bmn is given by definition (32).Remark 4.
Ifρ is even, the roots {xj(ρ)}j=1ρ of the Sonin-Markov polynomial pρ(x)≡pρβ(x) are all distinct from zero because the roots of every generalized Laguerre polynomial are all distinct from zero; see definition (21). Therefore, case (33b) does not apply to even values of N.Remark 5.
It is known that two generalized Laguerre polynomialsLnα(x) and Ln+1α(x) do not have common roots [2, 16]. But then, by definition (21), two Sonin-Markov polynomials pNβ(x) and pN+2β(x) have a common root x^ if and only if N is odd and x^=0. This is why formulas (30) and (31) include the case where xm(N+2)=xn(N)=0 (which is possible for odd N but not for even N), but do not include the case where xm(N+2)=xn(N)≠0 (the latter case is not possible).


### 2.3. The Generalized Spectral Matrix Representation of the Sonin-Markov Differential Operator (26)

The generalized spectral(N+2)×N matrix representation Dτ of the operator D is defined by formula (13). To find an explicit formula for the components of Dτ, we employ recurrence relations satisfied by the Sonin-Markov polynomials.Recall that the generalized Laguerre polynomials satisfy certain three-term recurrence relations [2, 16], which imply(34)x2pνx=αν+2pν+2x+βνpνx+γν-2pν-2x,with(35)αν=nn+α1/2ifν=2n,nn+1+α1/2ifν=2n+1,βν=ν+α+1,γν=n+1n+1+α1/2ifν=2n,n+1n+2+α1/2ifν=2n+1, where pk(x)≡pkβ(x) are assumed to equal zero if k<0. Using recurrence relations (34), we obtain(36)Dnjτ=ηn-1δnj+μj-1αn-1δn,j+2+βn-1δn,j+γn-1δn,j-2, where n=1,2,…,N+2, j=1,2,…,N, and the coefficients μν, ην and αν, βν, γν are given by (27a), (27b) and (35).Remark 6.
The coefficientsαν and βν defined in (35), must not be confused with the parameters α and β in the definition of the Sonin-Markov polynomials; see (21) and (22).For completeness, let us mention that, as a consequence of their orthogonality, Sonin-Markov polynomials satisfy the standard three-term recurrence relation:(37)xpνx=α~ν+1pν+1x+α~νpν-1x,where(38)α~ν=0ifν=0,cndnΓn+α+2n!ifν=2n+1,n≥0,-cndn-1Γn+α+1n-1!ifν=2n,n≥1, and pj(x)=0 if j<0. Of course, the three-term recurrence relation (34) is a consequence of the recurrence relation (37).


### 2.4. Algebraic Properties of the Zeros of the Sonin-Markov Polynomials

Having found the generalized pseudospectral and the generalized spectral matrix representationsDc and Dτ, respectively, of the differential operator D, we apply Theorem 1 to the case of the Sonin-Markov polynomials. We thus obtain the following algebraic identities satisfied by their zeros.Theorem 7.
For every pair of integersn,m such that 1≤n≤N and 1≤m≤N+2, the zeros x1(N+2),x2(N+2),…,xN+2(N+2) of the Sonin-Markov polynomial pN+2(x)≡pN+2β(x) and the zeros x1(N),…,xN(N) of the Sonin-Markov polynomial pN(x)≡pNβ(x) satisfy the following relations.
Ifxm(N+2)=0, then(39)ηn-1+μn-1βn-1pn-10+μn-1αn+1pn+10+μn-1γn-3pn-30=0,where pj(x)=0 if j<0.
Ifxm(N+2)≠0, then(40)∑k=1NBmkpn-1xkNpN′xkN-2BmkxmN+22pN′xmN+2+μNxmN+22+ηN+2Bmk2xmN+22-BmkxmN+2-2xmN+22+2α+1pNxmN+2=ηn-1+μn-1βn-1pn-1xmN+2+μn-1αn+1pn+1xmN+2+μn-1γn-3pn-3xmN+2,where Bmn are defined by (32) and, as before, pj(x)=0 if j<0.


## 2.1. Definition and Basic Properties of the Sonin-Markov Polynomials

The Sonin-Markov polynomialspνβxν=0∞, which are also known as generalized Hermite polynomials, are orthogonal on I=(-∞,∞) with respect to the weight w(x)=xβe-x2 (see [16, 28] and references therein). These polynomials can be expressed in terms of the generalized Laguerre polynomials as follows:(21)p2nβx=cnLnαx2,p2n+1βx=dnxLnα+1x2,where(22)α=β-12,and the coefficients,(23)cn=-1nn!Γn+1+α,dn=-1nn!Γn+2+α,are chosen to make the polynomials orthonormal and the leading coefficients positive. Note that the leading coefficients Kν of pνβ(x) are given by(24)K2n=n!Γn+1+α-1/2,K2n+1=n!Γn+2+α-1/2.The zeros of the Sonin-Markov polynomials are distinct, real, and symmetric with respect to the origin.The Sonin-Markov polynomials satisfy the following differential equations:(25)Dpνx=μνx2+ηνpνx,where D is the linear differential operator given by(26)D=x2d2dx2+x-2x2+2α+1ddx,(27a)μν=-2ν,(27b)ην=0ifν=2n,2α+1ifν=2n+1.These differential equations can be derived from the corresponding differential equations satisfied by the generalized Laguerre polynomials stated in [3]; see also [1] and formula (3.5) in [34]. Clearly, DPν⊆Pν+2, so we can apply Theorem 1 with n0=2 to these polynomials.


## 2.2. The Generalized Pseudospectral Matrix Representation of the Sonin-Markov Differential Operator (26)

Letx1(N),…,xN(N) be the zeros of the Sonin-Markov polynomial pN(x)≡pNβ(x) defined by (21) and let x1(N+2),…,xN+2(N+2) be the zeros of the Sonin-Markov polynomial pN+2(x)≡pN+2β(x). In this subsection, we find the generalized pseudospectral matrix representation Dcx→(N),x→(N+2)≡Dc of the Sonin-Markov differential operator (26) with respect to the two vectors of nodes x→(N)=x1(N),…,xN(N) and x→(N+2)=x1(N+2),…,xN+2(N+2). To compute this representation Dc, we use definition (10) with n0=2. In the following, we use the notation of Section 2.1; note the appropriate definitions of α, cn, dn and μν, ην.LetlN-1,jxj=1N be the Lagrange interpolation basis with respect to the nodes xjNj=1N defined by (9) with ρ=N and ψN(x) replaced with pN(x). Let Z(k)x→(N),x→(N+2)≡Z(k) be the (N+2)×N generalized pseudospectral matrix representation of the differential operator dk/dxk with respect to the two vectors of nodes x→(N) and x→(N+2). By definition (10), its components Zmn(k) are given by(28)Zmnk=dkdxklN-1,nxx=xmN+2.The generalized pseudospectral(N+2)×N matrix representation Dc of the differential operator D is given componentwise by(29)Dmnc=DlN-1,nxx=xmN+2=xmN+22Zmn2+xmN+2-2xmN+22+2α+1Zmn1;see definition (10).By using the fact that the Sonin-Markov polynomialpN(x) satisfies differential equation (25) and (26) and the differential equation obtained by differentiation of (25) and (26) with respect to x (with ν=N), we simplify the formulas for Zmn(k), where k=1,2, to read(30)Zmn1=BmnpN′xnNpN′xmN+2-BmnpNxmN+2ifxmN+2≠xnN,0ifxmN+2=xnN=0,(31)Zmn2=BmnpN′xnN2xmN+22-2α-1xmN+2-2BmnpN′xmN+2+μNxmN+22+ηNxmN+22+2Bmn2pNxmN+2ifxmN+2≠xnNandxmN+2≠0,-2pN′0pN′xnNBmn2ifxmN+2=0andxnN≠0,μN+22α+2ifxmN+2=xnN=0,where Bmn are defined by(32)Bmn=xmN+2-xnN-1.By using the last two expressions for the components of the matricesZ(1) and Z(2) in (29), we obtain the following formulas for the components of the generalized pseudospectral matrix representation Dc:(33a)Dmnc=BmnpN′xnN-2BmnxmN+22pN′xmN+2+μNxmN+22+ηN+2Bmn2xmN+22-BmnxmN+2-2xmN+22+2α+1pNxmN+2ifxmN+2≠xnNandxmN+2≠0,(33b)Dmnc=0ifxmN+2=0,where, again, Bmn is given by definition (32).Remark 4.
Ifρ is even, the roots {xj(ρ)}j=1ρ of the Sonin-Markov polynomial pρ(x)≡pρβ(x) are all distinct from zero because the roots of every generalized Laguerre polynomial are all distinct from zero; see definition (21). Therefore, case (33b) does not apply to even values of N.Remark 5.
It is known that two generalized Laguerre polynomialsLnα(x) and Ln+1α(x) do not have common roots [2, 16]. But then, by definition (21), two Sonin-Markov polynomials pNβ(x) and pN+2β(x) have a common root x^ if and only if N is odd and x^=0. This is why formulas (30) and (31) include the case where xm(N+2)=xn(N)=0 (which is possible for odd N but not for even N), but do not include the case where xm(N+2)=xn(N)≠0 (the latter case is not possible).


## 2.3. The Generalized Spectral Matrix Representation of the Sonin-Markov Differential Operator (26)

The generalized spectral(N+2)×N matrix representation Dτ of the operator D is defined by formula (13). To find an explicit formula for the components of Dτ, we employ recurrence relations satisfied by the Sonin-Markov polynomials.Recall that the generalized Laguerre polynomials satisfy certain three-term recurrence relations [2, 16], which imply(34)x2pνx=αν+2pν+2x+βνpνx+γν-2pν-2x,with(35)αν=nn+α1/2ifν=2n,nn+1+α1/2ifν=2n+1,βν=ν+α+1,γν=n+1n+1+α1/2ifν=2n,n+1n+2+α1/2ifν=2n+1, where pk(x)≡pkβ(x) are assumed to equal zero if k<0. Using recurrence relations (34), we obtain(36)Dnjτ=ηn-1δnj+μj-1αn-1δn,j+2+βn-1δn,j+γn-1δn,j-2, where n=1,2,…,N+2, j=1,2,…,N, and the coefficients μν, ην and αν, βν, γν are given by (27a), (27b) and (35).Remark 6.
The coefficientsαν and βν defined in (35), must not be confused with the parameters α and β in the definition of the Sonin-Markov polynomials; see (21) and (22).For completeness, let us mention that, as a consequence of their orthogonality, Sonin-Markov polynomials satisfy the standard three-term recurrence relation:(37)xpνx=α~ν+1pν+1x+α~νpν-1x,where(38)α~ν=0ifν=0,cndnΓn+α+2n!ifν=2n+1,n≥0,-cndn-1Γn+α+1n-1!ifν=2n,n≥1, and pj(x)=0 if j<0. Of course, the three-term recurrence relation (34) is a consequence of the recurrence relation (37).


## 2.4. Algebraic Properties of the Zeros of the Sonin-Markov Polynomials

Having found the generalized pseudospectral and the generalized spectral matrix representationsDc and Dτ, respectively, of the differential operator D, we apply Theorem 1 to the case of the Sonin-Markov polynomials. We thus obtain the following algebraic identities satisfied by their zeros.Theorem 7.
For every pair of integersn,m such that 1≤n≤N and 1≤m≤N+2, the zeros x1(N+2),x2(N+2),…,xN+2(N+2) of the Sonin-Markov polynomial pN+2(x)≡pN+2β(x) and the zeros x1(N),…,xN(N) of the Sonin-Markov polynomial pN(x)≡pNβ(x) satisfy the following relations.
Ifxm(N+2)=0, then(39)ηn-1+μn-1βn-1pn-10+μn-1αn+1pn+10+μn-1γn-3pn-30=0,where pj(x)=0 if j<0.
Ifxm(N+2)≠0, then(40)∑k=1NBmkpn-1xkNpN′xkN-2BmkxmN+22pN′xmN+2+μNxmN+22+ηN+2Bmk2xmN+22-BmkxmN+2-2xmN+22+2α+1pNxmN+2=ηn-1+μn-1βn-1pn-1xmN+2+μn-1αn+1pn+1xmN+2+μn-1γn-3pn-3xmN+2,where Bmn are defined by (32) and, as before, pj(x)=0 if j<0.


## 3. Proofs

The proof of Theorem1 is based on the following result.Theorem 8.
LetA be a linear differential operator that satisfies condition (2). Let {lN-1,j(x)}j=1N be the Lagrange interpolation basis of PN-1 constructed using the N distinct real nodes x1(N),…,xN(N) and let {lN+n0-1,j(x)}j=1N+n0 be the Lagrange interpolation basis of PN+n0-1 constructed using the (N+n0) distinct real nodes x1(N+n0),…,xN+n0(N+n0); see (9). If the (N+n0)×N matrices Ac, Aτ are defined by (10) and (13), respectively, while the two ρ×ρ matrices L(ρ-1) with ρ=N or ρ=N+n0 are defined componentwise by(41)Lmjρ-1=lρ-1,j,pm-1pm-12,1≤m,j≤ρ,then(42)LN+n0-1Ac=AτLN-1.Moreover, if the interpolation nodes x1(ρ),…,xρ(ρ) are the distinct real zeros of the polynomial pρ(x) from the orthogonal family {pν(x)}ν=0∞ introduced in Section 1, then the transition matrix L(ρ-1) and its inverse L(ρ-1)-1 are given by (15) and (17), respectively, where ρ∈{N,N+n0}.Remark 9.
Let us note thatL(N-1) is the transition matrix from the polynomial basis {pm(x)}m=0N-1 to the basis {lN-1,m(x)}m=1N of PN-1, while L(N+n0-1) is the transition matrix from the polynomial basis {pm(x)}m=0N+n0-1 to the basis {lN+n0-1,m(x)}m=1N+n0 of PN+n0-1.The proof of this theorem is similar to the proof of Theorem1.1 in [7]. It is provided below for the convenience of the reader.Proof of Theorem8.
First, let us prove property (42). Let u be a polynomial of degree N-1. Then(43)ux=∑j=1NujclN-1,jx,and, on the other hand,(44)ux=∑j=1Nujτpj-1x,where the coefficients(45)ujc=uxjN,(46)ujτ=u,pj-1pj-12,respectively, are the components of the column-vectors u→c and u→τ, respectively, and 1≤j≤N. To prove relation (42), we will show that u→τ=L(N-1)u→c and L(N+n0-1)Acu→c=Aτu→τ.
Let us expand(47)lN-1,jx=∑m=1NLmjN-1pm-1x,where the coefficients Lmj(N-1) are given by (41). Upon a substitution of (47) into (43), we obtain(48)u→τ=LN-1u→c.
To obtain the equalityL(N+n0-1)Acu→c=Aτu→τ, we first notice that because u∈PN-1 and the operator A satisfies APN-1⊆PN+n0-1, we have(49)Aux=∑j=1N+n0Acu→cjlN+n0-1,jx=∑j=1N+n0Acu→cj∑m=1N+n0LmjN+n0-1pm-1x=∑m=1N+n0LN+n0-1Acu→cmpm-1x.On the other hand,(50)Aux=A∑j=1Nujτpj-1x=∑j=1NujτApj-1x=∑j=1Nujτ∑m=1N+n0Amjτpm-1x=∑m=1N+n0Aτu→τmpm-1x.By comparing expansions (49) and (50), we obtain L(N+n0-1)Acu→c=Aτu→τ. Because u→τ=L(N-1)u→c, we conclude that L(N+n0-1)Ac=AτL(N-1) and so finish the proof of relation (42).
Second, let us assume thatx1(ρ),…,xρ(ρ) are the zeros of the polynomial pρ(x), where ρ=N or ρ=N+n0. Let us prove that the transition matrix L(ρ-1) is given componentwise by (15). The Gaussian rule for approximate integration with respect to the measure ω based on these nodes x1(ρ),…,xρ(ρ) has degree of exactness 2ρ-1; see, for example, Theorem 5.1.2 of [16]. Therefore, for the polynomial lρ-1,j(x)pm-1(x) of degree ρ-1+m-1≤2ρ-2 we have(51)pm-12Lmjρ-1=∫lN-1,jxpm-1xdω=∑n=1ρlN-1,jxnρpm-1xnρλnρ-1=pm-1xjρλjρ-1,which implies (15). By applying the Gaussian rule to the polynomial pm-1(x)pn-1(x), where 0≤m, n≤N, we obtain(52)δmn=∫pm-1xpn-1xpm-12dω=∑j=1ρpn-1xjρLmjρ-1, which implies (17).Proof of Theorem1.
Theorem1 is a straightforward consequence of identity (42) rewritten as AcL(N-1)-1mn=L(N+n0-1)-1Aτmn, where 1≤m≤N+n0 and 1≤n≤N, and formula (17) both proved in Theorem 8.
We note that the indexk∈{1,…,N+n0} in the sum from the right hand side of identity (18) ranges from n-n0 to n+n0 rather than from 1 to N+n0. This is due to the following consideration. Because the polynomial family {pν(x)}ν=0∞ is orthogonal and has the property that {pj(x)}j=0ν is a basis of Pν for each ν, this family must satisfy a three-term recurrence relation(53)xpνx=aν,ν+1pν+1x+aν,νpνx+aν,ν-1pν-1x, where aν,k are constants that equal zero if k<0 or k is outside of the range {ν-1,ν,ν+1} [16]. From the last recurrence relation we derive(54)qνxpνx=∑k=ν-n0ν+n0bν,kpkx, where bν,k are constants that equal zero if k<0. Thus, for all integer n,k such that 1≤k≤N+n0 and 1≤n≤N, (55)Aknτ=qn-1pn-1,pk-1pk-12=0, if the index k is outside of the range {n-n0,…,n+n0}.


## 4. Discussion and Outlook

The identities of the main Theorem1 are derived using a very specialexact discretization of the differential equation:(56)ApNx=qNxpNx;compare with (1). This discretization is constructed using the fact that ODE (56) has apolynomial solution pN(x), where pN(x) is a member of the nonclassical orthogonal polynomial family {pν(x)}ν=0∞. Because the differential operator A maps PN to PN+n0 (see property (2)) we choose to represent the operator by a (N+n0)×N matrix Acx→(N),x→(N+n0)≡Ac defined by (10) in terms oftwo vectors of interpolation nodes, x→(N) and x→(N+n0). We thusgeneralize the notion of pseudospectral matrix representations of a linear differential operator.This generalization can be utilized to resolve the issues related to the incorporation of the initial or boundary conditions into pseudospectral methods for solving a linear ODELu=f, where L is a linear differential operator; see [30, 31]. For example, if the differential operator L has the property LPν⊆Pν-k0, where k0 is a positive integer, it is beneficial to use the generalized pseudospectral N×(N-k0) matrix representation Lc of the differential operator L with respect to two vectors of nodes x→(N)∈RN and x→(N-k0)∈RN-k0. Of course, Lc is defined componentwise by Lkjc=(LlN-1,j)xkN-k0, 1≤k≤N-k0, 1≤j≤N; compare with (10), where {lN-1,j(x)}j=1N is the standard Lagrange interpolation basis with respect to the nodes x→(N).We may then approximate the ODELu=f by the N×(N-k0) system of linear equations Lcu→(N)=f→(N-k0), where f→(N-k0)=fx1(N-k0),…,fxN-k0(N-k0), compare with (4). The linear system Lcu→(N)=f→(N-k0) for the N components of u→(N) consists of N-k0 linear equations, which allows incorporating k0 boundary or initial conditions into the system without making the system overdetermined. Once the solution u→(N) of this augmented linear system is found, an approximate solution of the ODE Lu=f is given by u(x)≈(πN)-1u→(N)=∑j=1Nuj(N)lN-1,j(x), where {lN-1,j(x)}j=1N is the standard Lagrange interpolation basis with respect to the nodes x→(N).The ODE (56) may also be discretized using only one vector of nodes x→(N+n0) and the standard pseudospectral (N+n0)×(N+n0) matrix representation A^c of the differential operator A defined componentwise by(57)A^kjc=AlN+n0-1,jxx=xkN+n0,1≤k,j≤N+n0; see [7] and compare with (10). Indeed, in the proof of Theorem 8 we may choose to represent the polynomial u∈PN-1 in terms of the Lagrange interpolation basis lN+n0-1,j(x)j=1N+n0 or the basis {pj-1(x)}j=1N+n0; see (43) and (44). This modification allows proving the following relation:(58)LN+n0-1A^c=A^τLN+n0-1,where the standard spectral (N+n0)×(N+n0) matrix representation A^τ is defined by(59)A^kjτ=Apj-1,pk-1pk-12,1≤k,j≤N+n0.Let us assume again that x→(N+n0) are the zeros of the polynomial pN+n0(x). Then relation (58) rewritten as A^cL(N+n0-1)-1mn=L(N+n0-1)-1A^τmn together with expression (17) for the inverse of the transition matrices L(N+n0-1) yields the following identities:(60)∑k=1N+n0A^mkcx→N+n0pn-1xkN+n0=∑j=max1,n-n0minN+n0,n+n0A^jnτpj-1xmN+n0,which hold for all integer m,n such that 1≤m, n≤N+n0. Given a fixed integer n such that 0≤n≤N+n0-1, the last identity relates the zeros of the polynomials pN+n0(x) and pn(x) with the zeros of all the polynomials pj(x) such that the integer index j∈{0,1,…,N+n0-1} satisfies n-n0≤j≤n+n0.The set of identities (60) is a straightforward generalization of Theorem 1.1 in [7]; thus it is not the main focus of this paper. We invite the reader to apply identities (60) to the Sonin-Markov polynomials, replacing A^c and A^τ with D^c and D^τ, respectively, the latter being matrix representations of the differential operator D associated with the Sonin-Markov family; see (26). The pseudospectral matrix representation D^c can be found using definition (57) and the techniques outlined in Appendix A of [7]. The spectral matrix representation D^τ can be found using definition (59) and recurrence relations (34).In summary, the method of generalized pseudospectral matrix representations proposed in this paper can be utilized to prove new identities satisfied by nonclassical orthogonal polynomials that are generalized, rather than standard, eigenfunctions of linear differential operators. The proposed generalized matrix representations of linear differential operators are quite useful in the process of incorporation of initial or boundary conditions into a discretization of a linear ODE via a pseudospectral method. The last application deserves further exploration involving numerical tests, which the authors may pursue in the near future. Other interesting developments of the method proposed in this paper may involve its generalization to exceptional orthogonal polynomials [35–37].



---

*Source: 4710754-2018-02-07.xml*