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Pencils of Semi-Infinite Matrices and Orthogonal Polynomials

Written By

Sergey Zagorodnyuk

Reviewed: January 3rd, 2022Published: February 19th, 2022

DOI: 10.5772/intechopen.102422

IntechOpen
Matrix Theory - Classics and AdvancesEdited by Mykhaylo Andriychuk

From the Edited Volume

Matrix Theory - Classics and Advances [Working Title]

Dr. Mykhaylo I. Andriychuk

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Abstract

Semi-infinite matrices, generalized eigenvalue problems, and orthogonal polynomials are closely related subjects. They connect different domains in mathematics—matrix theory, operator theory, analysis, differential equations, etc. The classical examples are Jacobi and Hessenberg matrices, which lead to orthogonal polynomials on the real line (OPRL) and orthogonal polynomials on the unit circle (OPUC). Recently there turned out that pencils (i.e., operator polynomials) of semi-infinite matrices are related to various orthogonal systems of functions. Our aim here is to survey this increasing subject. We are mostly interested in pencils of symmetric semi-infinite matrices. The corresponding polynomials are defined as generalized eigenvectors of the pencil. These polynomials possess special orthogonality relations. They have physical and mathematical applications that will be discussed. Examples show that there is an unclarified relation to Sobolev orthogonal polynomials. This intriguing connection is a challenge for further investigations.

Keywords

  • semi-infinite matrix
  • pencil
  • orthogonal polynomials
  • Sobolev orthogonality
  • difference equation

1. Introduction

In this section, we will introduce the main objects of this chapter along with some brief historical notes.

By operator pencils or operator polynomials one means polynomials of a complex variable λwhose coefficients are linear bounded operators acting in a Banach space X:

Lλ=j=0mλjAj,E1

where Aj:XX(j=0,,m), see, for example, [1, 2]. Parlett in ref. [3, p. 339] stated that the term pencilwas introduced by Gantmacher in ref. [4] for matrix expressions, and Parlett explained how this term came from optics and geometry. In this chapter, we shall be mainly interested in pencils of banded semi-infinite matrices that are related to different kinds of scalar orthogonal polynomials. The classical example of such a relation is the case of orthogonal polynomials on the real line (OPRL) and Jacobi matrices, see, for example, refs. [5, 6]. If pnxn=0is a set of orthonormal OPRL and Jis the corresponding Jacobi matrix, then the following relation holds:

JxEpx=0,E2

where px=p0xp1xT, is a vector of polynomials (here the superscript Tmeans the transposition), and Eis the identity matrix (having units on the main diagonal and zeros elsewhere). In other words, pis an eigenfunction of the pencil JxE. It is surprising that mathematicians rarely talked about the relation (2) in such a manner. The next classical example is the case of orthogonal polynomials on the unit circle (OPUC) and the corresponding three-term recurrence relation, see ref. [7, p. 159]. More recently there appeared CMV matrices, which are also related to OPUC, see, for example, ref. [8]. We should notice that besides orthogonal polynomials, there are other systems of functions that are closely related to semi-infinite matrices. Here we can mention biorthogonal polynomials and rational functions, see, for example, [9, 10] and references therein.

A natural generalization of OPRL is matrix orthogonal polynomials on the real line (MOPRL). MOPRL was introduced by Krein in 1949 [11]. They satisfy the relation of type (2), with Jreplaced by a block Jacobi matrix, and with preplaced by a vector of matrix polynomials. It turned out that MOPRL is closely related to orthogonal polynomials on the radial rays in the complex plane, see refs. [12, 13]. We shall discuss this case in Section 2.

Another possible generalization of relation (2) is the following one:

J5xJ3px=0,E3

where J3is a Jacobi matrix, and J5is a real symmetric semi-infinite five-diagonal matrix with positive numbers on the second subdiagonal, see ref. [14]. These polynomials contain OPRL as a proper subclass. In general, they possess some special orthogonality relations. These polynomials will be discussed in Section 3.

Another natural generalization of OPRL is Sobolev orthogonal polynomials, see a recent survey in ref. [15]. During last years there appeared several examples of Sobolev polynomials, which are eigenfunctions of pencils of differential or difference operators. This subject will be discussed in Section 4.

Notations.As usual, we denote by R,C,Z,Z+, the sets of real numbers, complex numbers, positive integers, integers, and nonnegative integers, respectively. By Cm,nwe mean a set of all complex matrices of size m×n. By Pwe denote the set of all polynomials with complex coefficients. The superscript Tmeans the transposition of a matrix.

By l2we denote the usual Hilbert space of all complex sequences c=cnn=0=c0c1c2Twith the finite norm cl2=n=0cn2. The scalar product of two sequences c=cnn=0,d=dnn=0l2is given by cdl2=n=0cndn¯. We denote em=δn,mn=0l2, mZ+. By l2,finwe denote the set of all finite vectors from l2, that is, vectors with all, but finite number, components being zeros.

By BRwe denote the set of all Borel subsets of R. If σis a (non-negative) bounded measure on BRthen by Lσ2we denote a Hilbert space of all (classes of equivalences of) complex-valued functions fon Rwith a finite norm fLσ2=Rfx2. The scalar product of two functions f,gLσ2is given by fgLσ2=Rfxgx¯. By fwe denote the class of equivalence in Lσ2, which contains the representative f. By Pwe denote a set of all (classes of equivalence which contain) polynomials in Lσ2. As usual, we sometimes use the representatives instead of their classes in formulas. Let Bbe an arbitrary linear operator in Lσ2with the domain P. Let fλPbe nonzero and of degree dZ+, fλ=k=0ddkλk, dkC. We set

fB=k=0ddkBkB0EP.

If f0, then fB0P.

If H is a Hilbert space then Hand Hmean the scalar product and the norm in H, respectively. Indices may be omitted in obvious cases. For a linear operator Ain H, we denote by DAits domain, by RAits range, by KerAits null subspace (kernel), and Ameans the adjoint operator if it exists. If Ais invertible then A1means its inverse. A¯means the closure of the operator, if the operator is closable. If Ais bounded then Adenotes its norm.

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2. Pencils J2N+1λNEand orthogonal polynomials on radial rays in the complex plane

Throughout this section Nwill denote a fixed natural number. Let J2N+1be a complex Hermitian semi-infinite 2N+1-diagonal matrix. Let pnλn=0, degpn=nbe a set of complex polynomials, which satisfy the following relation:

J2N+1λnEpλ=0,E4

where pλ=p0λp1λT, is a vector of polynomials, and Eis the identity matrix. Polynomials, which satisfy (4) with real coefficients, were first studied by Durán in ref. [16], following a suggestion of Marcellán. As it was already noticed in the Introduction, these polynomials are related to MOPRL. Namely, the following polynomials:

Pnx=RN,0pnNxRN,1pnNxRN,N1pnNxRN,0pnN+1xRN,1pnN+1xRN,N1pnN+1xRN,0pnN+N1xRN,1pnN+N1xRN,N1pnN+N1xE5

are orthonormal MOPRL [12, Theorem]. Here

RN,mpt=npnN+m0nN+m!tn,pP,0mN1.E6

Conversely, from a given set Pnx=Pn,m,jm,j=0N1n=0of orthonormal MOPRL (suitably normed) one can construct scalar polynomials:

pnN+mx=j=0N1xjPn,m,jxN,nN,0mN1,E7

which satisfy relation (4) [12]. Writing the corresponding matrix orthonormality conditions for Pnand equating the entries on both sides, one immediately gets orthogonality conditions for pn:

RRN,0pnxRN,1pnxRN,N1pnxRN,0pmxRN,1pmxRN,N1pmx¯==δn,m,n,mZ+,E8

where μis a N×Nmatrix measure. In the case of real coefficients in (4), this property was obtained by Durán in ref. [17].

Polynomials pnλn=0also satisfy the following orthogonality relations on radial rays in the complex plane [13]:

LNpnλpnλεpnλεN1dWλpmλpmλεpmλεN1¯++pn0pn0pnN10Mpm0pm0pmN10¯=δn,m,n,mZ+,E9

where Wλis a non-decreasing matrix-valued function on LN\0; MRN,N, M0; LN=λR:λNR; εis a primitive N-th root of unity. At λ=0the integral is understood as improper. Relation (9) can be derived from a Favard-type theorem in ref. [12, Theorem], but in ref. [13] we proceeded in another way. Relation (9) easily shows that the following classes of polynomials are included in the class of polynomials from (4):

  1. OPRL;

  2. orthogonal polynomials with respect to a scalar measure on radial rays LN;

  3. discrete Sobolev orthogonal polynomials on R, having one discrete Sobolev term.

A detailed investigation of polynomials in the case Bwas done by Milovanovic, see ref. [18] and references therein. In particular, interesting examples of orthogonal polynomials were constructed and zero distribution of polynomials was studied. Discrete Sobolev polynomials from the case Cmay possess higher-order differential equations. This subject has a long history, see historical remarks in recent papers [19, 20]. For polynomials (9) some simple general properties of zeros were studied in ref. [21], while a Christoffel type formula was constructed in ref. [22]. In ref. [12] there was studied a more general case of relation (4), with a polynomial hλinstead of λN.

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3. Pencils J5xJ3and orthogonal polynomials

Let J3be a Jacobi matrix and J5be a semi-infinite real symmetric five-diagonal matrix with positive numbers on the second subdiagonal. A set Θ=J3J5αβ, where α>0, βR, is said to be a Jacobi-type pencil (of matrices)[14]. With a Jacobi-type pencil of matrices Θone associates a system of polynomials pnλn=0, which satisfies the following relations:

p0λ=1,p1λ=αλ+β,E10

and

J5λJ3pλ=0,E11

where pλ=p0λp1λp2λT. Polynomials pnλn=0are said to be associated with the Jacobi-type pencil of matricesΘ.

Observe that for each system of OPRL with p0=1one can take J3to be the corresponding Jacobi matrix, J5=J32, and α,βbeing the coefficients of p1(p1λ=αλ+β). Then, this system is associated with Θ=J3J5αβ. Let us mention two other circumstances where Jacobi-type pencils arise in a natural way.

  1. Discretization of a4-th order differential operator.Ben Amara, Vladimirov, and Shkalikov investigated the following linear pencil of differential operators [23]:

    pyλy+cry=0.E12

The initial conditions are: y0=y0=y1=y1=0, or y0=y0=y1=py1+λαy1=0. Here p,rC01are uniformly positive, while the parameters cand αare real. Eq. (12) has several physical applications, which include a motion of a partially fixed bar with additional constraints in the elasticity theory [23]. The discretization of this equation leads to a Jacobi-type pencil, see ref. [24].

  • Partial sums of series of OPRL.Let gnxn=0(deggn=n) be orthonormal OPRL with positive leading coefficients. Let ckk=0be a set of arbitrary positive numbers. Then polynomials

    pnx1c0g0j=0ncjgjx,nZ+,E13

  • are associated with a Jacobi-type pencil [25, Theorem 1]. Polynomials pnare normed partial sums of the following formal power series:

    j=0cjgjx.

    We shall return to such sums below.

    From the definition of a Jacobi type pencil we see that matrices J3and J5have the following form:

    J3=b0a0000a0b1a1000a1b2a20,ak>0,bkR,kZ+;E14
    J5=α0β0γ0000β0α1β1γ100γ0β1α2β2γ200γ1β2α3β3γ3,αn,βnR,γn>0,nZ+.E15

    Set

    unJ3en=an1en1+bnen+anen+1,E16
    wnJ5en=γn2en2+βn1en1+αnen+βnen+1+γnen+2,nZ+.E17

    Here and in what follows by ekwith negative kwe mean (vector) zero. The following operator:

    Af=ζαe1βe0+n=0ξnwn,
    f=ζe0+n=0ξnunl2,fin,ζ,ξnC,E18

    with DA=l2,finis called the associated operator for the Jacobi-type pencilΘ. In the sums in (18), only a finite number of ξnare nonzero. In what follows we shall always assume this in the case of elements from the linear span. In particular, the following relation holds:

    AJ3en=J5en,nZ+.

    Then

    AJ3=J5.E19

    The matrices J3and J5define linear operators with the domain l2,fin, which we denote by the same letters.

    For an arbitrary nonzero polynomial fλPof degree dZ+, fλ=k=0ddkλk, dkC, we set fA=k=0ddkAk. Here A0El2,fin. For fλ0, we set fA=0l2,fin. The following relations hold [14]:

    en=pnAe0,nZ+;E20
    pnAe0pmAe0l2=δn,m,n,mZ+.E21

    Denote by rnλn=0, r0λ=1, the system of polynomials satisfying

    J3rλ=λrλ,rλ=r0λr1λr2λT.E22

    These polynomials are orthonormal on the real line with respect to a (possibly nonunique) nonnegative finite measure σon the Borel subsets of R(Favard’s theorem). Consider the following operator:

    Un=0ξnen=n=0ξnrnx,ξnR,E23

    which maps l2,finonto P. Here, by Pwe denote a set of all (classes of equivalence which contain) polynomials in Lσ2. Denote

    A=Aσ=UAU1.E24

    The operator A=Aσis said to be the model representation inLσ2of the associated operatorA.

    Theorem 1.1 ([14]) Let Θ=J3J5αβbe a Jacobi-type pencil. Let rnλn=0, r0λ=1, be a system of polynomials satisfying (22) and σbe their (arbitrary) orthogonality measure on BR. The associated polynomials pnλn=0satisfy the following relations:

    RpnA1pmA1¯=δn,m,n,mZ+,E25

    where Ais the model representation in Lσ2of the associated operator A.

    There appears a natural question: what are the characteristic properties of the operatorA?The answer is given by the following theorem.

    Theorem 1.2 ([24, Corollary 1]) Let σbe a nonnegative measure on BRwith all finite power moments, R=1, Rgx2>0, for any nonzero complex polynomial g. A linear operator Ain Lσ2is a model representation in Lσ2of the associated operator of a Jacobi-type pencil if and only if the following properties hold:

    1. DA=P;

    2. For each k+it holds:

      Axk=ξk,k+1xk+1+j=0kξk,jxj,E26

    where ξk,k+1>0, ξk,jR(0jk);

  • The operator AΛ0is symmetric. Here, by Λ0we denote the operator of the multiplication by an independent variable in Lσ2restricted to P.

  • There is a general subclass of Jacobi-type pencils, for which elements much more can be said about their associated operators and models [24]. Here we used some ideas from the general theory of operator pencils, see ref. [1, Chapter IV, p. 163].

    Let Θ=J3J5αβbe a Jacobi-type pencil and Abe a model representation in Lσ2of the associated operator of Θ. By Theorem 1.2 we see that AΛ0is symmetric:

    AΛ0uλvλLσ2=uλAΛ0vλLσ2,u,vP.E27

    Suppose that the measure σis supported inside a finite real segment ab, 0<a<b<+, that is, σR\ab=0. In this case, the operator Λof the multiplication by an independent variable has a bounded inverse on the whole Lσ2. Using (27) we may write:

    Λ1AλuλλvλLσ2=Λ1λuλAλvλLσ2,u,vP.E28

    Denote P0=ΛPand A0=AP0. Then

    Λ1A0fgLσ2=Λ1fA0gLσ2,f,gP0.E29

    Then A0is symmetric with respect to the form Λ1Lσ2. Thus, in this case, the operatorAis a perturbation of a symmetric operator.

    Consider two examples of Jacobi-type pencils which show that Sobolev orthogonality is close to them.

    Example 3.1.([26]). Let σbe a nonnegative measure on BRwith all finite power moments, R=1, Rgx2>0, for any nonzero complex polynomial g. The following operator:

    Apλ=Λ0pλ+p0+d,pP,E30

    where c>1and dR, satisfies properties i-iiiof Theorem 1.2. Let J3be the Jacobi matrix, corresponding to the measure σ, and J5=J32. Define α,βin the following way:

    α=1ξ0,1Δ1,β=ξ0,1s1+ξ0,0ξ0,1Δ1.E31

    Here sjare the power moments of σ, while Δndetsk+lk,l=0n, nZ+, Δ11are the Hankel determinants. The coefficients ξk,jare taken from property (ii) of Theorem 1.2. Let Θ=J3J5αβ. Denote by pnλn=0the associated polynomials to the pencil Θ, and denote by rnλn=0the orthonormal polynomials (with positive leading coefficients) with respect to the measure σ. Then

    pnλ=1c+1rnλdc+1rnλrn0λ+cc+1rn0,nZ+;E32
    rnλ=c+1pnλ+c+1dpnλpndλdcpnd,nZ+.E33

    In (32), (33) we mean the limit values at λ=0and λ=d, respectively. The following recurrence relation, involving three subsequent associated polynomials, holds:

    λpnλ=pndc+1+d+an1pn1λ+bnpnλ+anpn+1λ,nZ+,λC.E34

    The following orthogonality relations hold:

    R\dpnλpndc+12λλd2c1λ+dλd2c1λ+dλd2+dλd2pmλpmd++pndpnd1c+1dc+1dc+12d2pmdpmdσd=δn,m,n,mZ+.E35

    Polynomials pnλcan have multiple or complex roots.

    Suppose additionally that σand J3correspond to orthonormal Jacobi polynomials rnλ=Pnλab(a,b>1) and c=0; d=1. In this case, the associated polynomial pn(nZ+):

    pnλ=rnλrnλrn0λ,E36

    is a unique, up to a constant multiple, real n-th degree polynomial solution of the following 4-th order differential equation:

    t+1tt12y4t+t1a+b+10t2+bat+4y3t++32a+2b+8t2+a+9b+22t+3a3byt+6a+b+2t+2a+6b+8yt++λntt1yt+22t1yt+2yt=0,E37

    where λn=nn+a+b+1.

    Moreover, there exists a unique λnR, such that differential Eq. (37) has a real n-th degree polynomial solution.

    Example 3.2.([25]). Recall that Jacobi polynomials Pnαβx:

    Pnαβx=n+αn2F1nn+α+β+1α+11x2,nZ+,

    are orthogonal on 11with respect to the weight wx=1xα1+xβ, α,β>1. Orthonormal polynomials have the following form:

    P̂0αβx=12α+β+1Bα+1β+1,P̂nαβx=2n+α+β+1n!Γn+α+β+12α+β+1Γn+α+1Γn+β+1Pnαβx,nN.

    Let c>0be an arbitrary positive number. Set

    Dα,β,cx21d2dx2+α+β+2x+αβddx+c,E38
    ln,cc+nn+α+β+1.E39

    Define the following polynomials:

    Pnαβct0xk=0n1lk,cP̂kαβt0P̂kαβx,nZ+,E40

    where t01is an arbitrary parameter. Notice that normed by eigenvalues polynomial kernels of some Sobolev orthogonal polynomials appeared earlier in literature, see ref. [27].

    Theorem 1.3. Let α,β>1; c>0, and t01, be arbitrary parameters. Polynomials Pnx=Pnαβct0x, from (40), are Sobolev orthogonal polynomials on R:

    11PnxPnxPnxMα,β,cxPmxPmxPmxt0x1xα1+xβdx==Anδn.m,n,mZ+,E41

    where Anare some positive numbers and

    Mα,β,c==cα+β+2x+αβx21cα+β+2x+αβx21.E42

    For Pnαβc1xthe following differential equation holds:

    Dα+1,β,0Dα,β,cPnαβc1x=ln,0Dα,β,cPnαβc1x,nZ+,E43

    where Dα,β,c, ln,care defined by (38), (39).

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    4. Pencils of banded matrices and Sobolev orthogonality

    Let Kdenote the real line or the unit circle. The following problem was stated in ref. [28], see also ref. [29]:

    Problem 1.To describe all Sobolev orthogonal polynomialsynzn=0onK, satisfying the following two properties:

    1. Polynomialsynzsatisfy the following differential equation:

      Rynz=λnSynz,n=0,1,2,,E44

    where R,Sare linear differential operators of finite orders, having complex polynomial coefficients not depending on n; λnC;

  • Polynomialsynzobey the following difference equation:

    Lyz=zMyz,yz=y0zy1zT,E45

  • where L,Mare semi-infinite complex banded (i.e., having a finite number of non-zero diagonals) matrices.

    Relation (44) shows that ynis an eigenfunction of the operator pencil RλS, while relation (45) means that vectors of ynzare eigenfunctions of the operator pencil LzM. We emphasize that in Problem 1 we do not exclude OPRL or OPUC. They are formally considered as Sobolev orthogonal polynomials with the derivatives of order 0. In this way, we may view systems from Problem 1 as generalizations of systems of classical orthogonal polynomials (see, e.g., the book [30], and papers [31, 32, 20] for more recent developments on this subject, as well as references therein). Related topics are also studied for systems of biorthogonal rational functions, see, for example, ref. [33]. Conditions a,bof Problem 1 are close to bispectral problems, and in particular, to the Bochner-Krall problem (see refs. [31, 34, 35, 36] and papers cited therein).

    One example of Sobolev orthogonal polynomials, which satisfy conditions of Problem 1, we have already met in Example 3.2. In ref. [37] there was proposed a way to construct such systems of polynomials. Let pnxn=0(pnhas degree nand real coefficients) be orthogonal polynomials on abRwith respect to a weight function wx:

    abpnxpmxwxdx=Anδn,m,An>0,n,mZ+.E46

    The weight wis supposed to be continuous on ab. Denote

    Dξyx=k=0ξdkxykx,E47

    where dkand yare real polynomials of x: dξ0. Let us fix a positive integer ξ, and consider the following differential equation:

    Dξyx=pnx,E48

    where Dξis defined as in Eq. (47), and nZ+. The following assumption plays a key role here.

    Condition 1.Suppose that for eachnZ+, the differentialEq. (48)has a realn-th degree polynomial solutionyx=ynx.

    If Condition 1 is satisfied, by relations (46),(48) we immediately obtain that ynxn=0are Sobolev orthogonal polynomials:

    abynxynxynξxMxymxymxymξxwxdx=Anδn,m,n,mZ+,E49

    where

    Mxd0xd1xdξxd0xd1xdξx,xab.E50

    Moreover, if pnsatisfy a differential equation, then ynsatisfy a differential equation as well. Question: when Condition 1 is satisfied? An answer is given by the following proposition.

    Proposition 1 ([28, Proposition 2.1]) Let Dbe a linear differential operator of order rN, with complex polynomial coefficients:

    D=k=0rdkzdkdzk,dkzP.

    Let unzn=0, degun=n, be an arbitrary set of complex polynomials. The following statements are equivalent:

    1. (A) The following equation:

    Dyz=unz,E51

    for each n+, has a complex polynomial solution yz=ynzof degree n;

    1. (B) Dznis a complex polynomial of degree n, n+;

    2. (C) The following conditions hold:

    degdkk,0kr;E52
    j=0rnjdj,j0,nZ+,E53

    where dj,lmeans the coefficient by zlof the polynomial dj.

    If one of the statements A,B,Cholds true, then for each nZ+, the solution of (51) is unique.

    Observe that condition (53) holds true, if the following simple condition holds:

    d0,0>0,dj,j0,jZ1,r.E54

    Thus, there exists a big variety of linear differential operators with polynomial coefficients that have propertyA. This leads to various Sobolev orthogonal polynomials.

    In ref. [37] there were constructed families of Sobolev orthogonal polynomials on the real line, depending on an arbitrary finite number of complex parameters. Namely, we considered the following hypergeometric polynomials:

    Lnx=Lnxακ1κδ==δ+1Fδ+1n11α+1κ1+1κδ+1x,E55
    Pnx=Pnxαβκ1κδ==δ+2Fδ+1nn+α+β+111α+1κ1+1κδ+1x,α,β,κ1,,κδ>1,nZ+.E56

    Here pFqis a usual notation for the generalized hypergeometric function, and δis a positive integer. These families obey differential equations. As for recurrence relations, they were only constructed for the case δ=1.

    In ref. [29] a family of hypergeometric Sobolev orthogonal polynomials on the unit circle was considered:

    ynx=1ρn!xn2F0nρ1x,

    depending on a parameter ρN. Observe that the reversed polynomials to ynappeared in numerators of some biorthogonal rational functions, see [38].

    Let gntn=0be a system of OPRL or OPUC, having a generating function of the following form:

    Gtw=fwetuw=n=0gntwnn!,tC,w<R0,R0>0,E57

    where f,uare analytic functions in the circle w<R0, u0=0. Such generating functions for OPRL were studied by Meixner, see, for instance, ref. [39, p. 273]. In the case of OPUC, we do not know any such a system, besides znn=0. Consider the following function:

    Ftw=1puwGtw=1puwfwetuw,tR,w<R1<R0,R1>0,E58

    where pP: p00. In the case uz=z, one should take R1z0, where z0is a root of pwith the smallest modulus. This ensures that Ftwis an analytic function of two variables in any polydisk CT1,R1=twC2:t<T1w<R1, T1>0. In the general case, since pu0=p00, there also exists a suitable R1, which guarantees that Fis analytic in CT1,R1. Expand the function Ftwin Taylor’s series by wwith a fixed t:

    Ftw=n=0φntwnn!,twCT1,R1,E59

    where φntare some complex-valued functions. Then the function φntis a complex polynomial of degree n, nZ+, see [28, Lemma 3.5]. Suppose that degp1, and

    pz=k=0dckzk,ckC,cd0;c00;dN.E60

    Theorem 1.4 ([28, Theorem 3.7]) Let dN, and pzbe as in (60). Let gntn=0be a system of OPRL or OPUC, having a generating function Gtwfrom (57) and Ftwbe given by (58). Fix some positive T1,R1, such that Ftwis analytic in the polydisk CT1,R1. Polynomials

    φnz=j=0nnjbjgnjt,nZ+,E61

    where bj=1puwj0, have the following properties:

    1. Polynomials φnare Sobolev orthogonal polynomials:

    φntφntφndtM˜φmtφmtφmdt¯dμg=τnδn,m,τn>0,n,mZ+,

    where

    M˜=c0c1cdTc0¯c1¯cd¯.

    Here dμgis the measure of orthogonality of gn.

    1. Polynomials φnhave the generating function Ftw, and relation (59) holds.

    2. Polynomials φnhave the following integral representation:

    φnt=n!2πiw=R21puwfwetuwwn1dw,nZ+,E62

    where R2is an arbitrary number, satisfying 0<R2<R1.

    There are two cases of gn, which lead to additional properties of φn, namely, to differential equations and recurrence relations. The next two corollaries are devoted to them.

    Corollary 1 ([28]) In conditions of Theorem 1.4 suppose that gnt=tn, nZ+; fw=1, uw=w. Polynomials φntn=0satisfy the following recurrence relation:

    n+1k=0dφn+1ktckn+1k!==tk=0dφnktcknk!,nZ+,E63

    where φr0, r!1, for rZ:r<0.

    Polynomials φntn=0obey the following differential equation:

    tk=0dckφnk+1t=nk=0dckφnkt,nZ+.

    Corollary 2 ([28]) In conditions of Theorem 1.4 suppose that gnt=Hnt, nZ+, are Hermite polynomials; fw=ew2, uw=2w. Polynomials φntn=0satisfy the following recurrence relation:

    n+1k=0dφn+1ktck2kn+1k!+2k=0dφn1ktck2kn+1k!==2tk=0dφnktck2knk!,nN,E64

    where φr0, r!1, for rZ:r<0; and

    c0φ1t+2c1φ0t=2c0tφ0t.E65

    Polynomials φntn=0obey the following differential equation:

    k=0dckφnkt2tk=0dckφnk+1t=2nk=0dckφnkt,nZ+.E66

    Observe that polynomials φnfrom the last two corollaries fit into the scheme of Problem 1.

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    Conclusion

    The theory of orthogonal polynomials is closely related to semi-infinite matrices, as well as to their finite truncations. This interplay has shown its productivity in classical results. Nowadays there appeared new kinds of orthogonality, such as Sobolev orthogonality. It is not yet clear what kind of matrices can be attributed to them. One of candidates is a pencil of matrices, since it appeared in examples. In Section 3 there appeared a pencil of semi-infinite symmetric matrices, while in Section 4 it was a pencil of some banded matrices. In Section 2 we also met a pencil, but it was more close to classical eigenvalue problems of single operators.

    The above-mentioned examples of Sobolev orthogonal polynomials also showed that pencils of differential equations appeared here in a natural way. Moreover, there is a large number of differential operators, which have polynomial solutions with Sobolev orthogonality. This fact promises that Sobolev orthogonal polynomials can find their applications in mathematical physics.

    We think that Problem 1 is an appropriate framework for a search and a construction of new Sobolev orthogonal polynomials having nice properties. Notice that one can produce such systems using classical OPRL or OPUC. The differential equation, if it existed, is inherited by new systems of polynomials. The more complicated question is the existence of a recurrence relation.

    Besides new families of Sobolev orthogonal polynomials, it is of a big interest finding classes of systems of Sobolev orthogonal polynomials, having recurrence relations. One such a class (orthogonal polynomials on radial rays) was described in Section 2. Thus, it looks reasonable to start not only from Sobolev orthogonality, but from the other side, i.e., from recurrent relations. One such an example of derivation was given by orthogonal polynomials on radial rays from Section 2.

    Another possible way was given in Section 3, where we described Jacobi-type pencils. The associated polynomials of a Jacobi type pencil have special orthogonality relations. The associated operator yet has not a suitable functional calculus. As we have seen, under some conditions this operator is a perturbation of a symmetric operator. However, it is not clear how to calculate effectively a polynomial of this operator.

    In general, it is a classical situation that the operator theory stands behind special classes of semi-infinite matrices and related objects. The operator theory of single operators is well promoted and it is well recognized by any mathematician. It seems that the theory of operator pencils is less known to the mathematical community. This fact can explain the situation that pencils of semi-infinite matrices and related polynomials appeared on a mathematical scene just recently. We hope that, as in the classical case, these new orthogonal polynomial systems will shed some new light on the theory of operator pencils.

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    Acknowledgments

    The author is grateful to Professors Zolotarev and Yantsevich for their permanent support.

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    Written By

    Sergey Zagorodnyuk

    Reviewed: January 3rd, 2022Published: February 19th, 2022