Open access peer-reviewed chapter

Pencils of Semi-Infinite Matrices and Orthogonal Polynomials

Written By

Sergey Zagorodnyuk

Reviewed: 03 January 2022 Published: 19 February 2022

DOI: 10.5772/intechopen.102422

From the Edited Volume

Matrix Theory - Classics and Advances

Edited by Mykhaylo 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 λ whose 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 pencil was 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=0 is a set of orthonormal OPRL and J is the corresponding Jacobi matrix, then the following relation holds:

JxEpx=0,E2

where px=p0xp1xT, is a vector of polynomials (here the superscript T means the transposition), and E is the identity matrix (having units on the main diagonal and zeros elsewhere). In other words, p is 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 J replaced by a block Jacobi matrix, and with p replaced 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 J3 is a Jacobi matrix, and J5 is 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,n we mean a set of all complex matrices of size m×n. By P we denote the set of all polynomials with complex coefficients. The superscript T means the transposition of a matrix.

By l2 we denote the usual Hilbert space of all complex sequences c=cnn=0=c0c1c2T with the finite norm cl2=n=0cn2. The scalar product of two sequences c=cnn=0,d=dnn=0l2 is given by cdl2=n=0cndn¯. We denote em=δn,mn=0l2, mZ+. By l2,fin we denote the set of all finite vectors from l2, that is, vectors with all, but finite number, components being zeros.

By BR we denote the set of all Borel subsets of R. If σ is a (non-negative) bounded measure on BR then by Lσ2 we denote a Hilbert space of all (classes of equivalences of) complex-valued functions f on R with a finite norm fLσ2=Rfx2. The scalar product of two functions f,gLσ2 is given by fgLσ2=Rfxgx¯. By f we denote the class of equivalence in Lσ2, which contains the representative f. By P we 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 B be an arbitrary linear operator in Lσ2 with the domain P. Let fλP be 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 H and H mean the scalar product and the norm in H, respectively. Indices may be omitted in obvious cases. For a linear operator A in H, we denote by DA its domain, by RA its range, by KerA its null subspace (kernel), and A means the adjoint operator if it exists. If A is invertible then A1 means its inverse. A¯ means the closure of the operator, if the operator is closable. If A is bounded then A denotes its norm.

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

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

J2N+1λnEpλ=0,E4

where pλ=p0λp1λT, is a vector of polynomials, and E is 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=0 of 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 Pn and 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 μ is a N×N matrix measure. In the case of real coefficients in (4), this property was obtained by Durán in ref. [17].

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

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

where Wλ is a non-decreasing matrix-valued function on LN\0; MRN,N, M0; LN=λR:λNR; ε is a primitive N-th root of unity. At λ=0 the 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 B was 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 C may 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 hλ instead of λN.

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3. Pencils J5xJ3 and orthogonal polynomials

Let J3 be a Jacobi matrix and J5 be 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 Θ one 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=0 are said to be associated with the Jacobi-type pencil of matrices Θ.

Observe that for each system of OPRL with p0=1 one can take J3 to be the corresponding Jacobi matrix, J5=J32, and α,β being 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. 1. Discretization of a 4-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,rC01 are uniformly positive, while the parameters c and α are 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].

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

pnx1c0g0j=0ncjgjx,nZ+,E13

are associated with a Jacobi-type pencil [25, Theorem 1]. Polynomials pn are 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 J3 and J5 have 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 ek with negative k we mean (vector) zero. The following operator:

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

with DA=l2,fin is called the associated operator for the Jacobi-type pencil Θ. In the sums in (18), only a finite number of ξn are 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 J3 and J5 define linear operators with the domain l2,fin, which we denote by the same letters.

For an arbitrary nonzero polynomial fλP of 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 σ on the Borel subsets of R (Favard’s theorem). Consider the following operator:

Un=0ξnen=n=0ξnrnx,ξnR,E23

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

A=Aσ=UAU1.E24

The operator A=Aσ is said to be the model representation in Lσ2 of the associated operator A.

Theorem 1.1 ([14]) Let Θ=J3J5αβ be a Jacobi-type pencil. Let rnλn=0, r0λ=1, be a system of polynomials satisfying (22) and σ be their (arbitrary) orthogonality measure on BR. The associated polynomials pnλn=0 satisfy the following relations:

RpnA1pmA1¯=δn,m,n,mZ+,E25

where A is the model representation in Lσ2 of the associated operator A.

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

Theorem 1.2 ([24, Corollary 1]) Let σ be a nonnegative measure on BR with all finite power moments, R=1, Rgx2>0, for any nonzero complex polynomial g. A linear operator A in Lσ2 is a model representation in Lσ2 of the associated operator of a Jacobi-type pencil if and only if the following properties hold:

  1. i. DA=P;

  2. ii. For each k+ it holds:

Axk=ξk,k+1xk+1+j=0kξk,jxj,E26

where ξk,k+1>0, ξk,jR (0jk);

  1. iii. The operator AΛ0 is symmetric. Here, by Λ0 we denote the operator of the multiplication by an independent variable in Lσ2 restricted 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=J3J5αβ be a Jacobi-type pencil and A be a model representation in Lσ2 of the associated operator of Θ. By Theorem 1.2 we see that AΛ0 is symmetric:

AΛ0uλvλLσ2=uλAΛ0vλLσ2,u,vP.E27

Suppose that the measure σ is supported inside a finite real segment ab, 0<a<b<+, that is, σR\ab=0. In this case, the operator Λ of 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=ΛP and A0=AP0. Then

Λ1A0fgLσ2=Λ1fA0gLσ2,f,gP0.E29

Then A0 is symmetric with respect to the form Λ1Lσ2. Thus, in this case, the operator A is 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 σ be a nonnegative measure on BR with 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>1 and dR, satisfies properties i-iii of Theorem 1.2. Let J3 be the Jacobi matrix, corresponding to the measure σ, and J5=J32. Define α,β in the following way:

α=1ξ0,1Δ1,β=ξ0,1s1+ξ0,0ξ0,1Δ1.E31

Here sj are the power moments of σ, while Δndetsk+lk,l=0n, nZ+, Δ11 are the Hankel determinants. The coefficients ξk,j are taken from property (ii) of Theorem 1.2. Let Θ=J3J5αβ. Denote by pnλn=0 the associated polynomials to the pencil Θ, and denote by rnλn=0 the 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=0 and 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 pnλ can have multiple or complex roots.

Suppose additionally that σ and J3 correspond 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 11 with 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>0 be 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 t01 is 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 An are some positive numbers and

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

For Pnαβc1x the following differential equation holds:

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

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

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4. Pencils of banded matrices and Sobolev orthogonality

Let K denote 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 polynomials ynzn=0 on K, satisfying the following two properties:

  1. a. Polynomials ynz satisfy the following differential equation:

Rynz=λnSynz,n=0,1,2,,E44

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

  1. b. Polynomials ynz obey the following difference equation:

Lyz=zMyz,yz=y0zy1zT,E45

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

Relation (44) shows that yn is an eigenfunction of the operator pencil RλS, while relation (45) means that vectors of ynz are 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 [20, 31, 32] 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,b of 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 (pn has degree n and real coefficients) be orthogonal polynomials on abR with respect to a weight function wx:

abpnxpmxwxdx=Anδn,m,An>0,n,mZ+.E46

The weight w is supposed to be continuous on ab. Denote

Dξyx=k=0ξdkxykx,E47

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

Dξyx=pnx,E48

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

Condition 1. Suppose that for each nZ+, the differential Eq. (48) has a real n-th degree polynomial solution yx=ynx.

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

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

where

Mxd0xd1xdξxd0xd1xdξx,xab.E50

Moreover, if pn satisfy a differential equation, then yn satisfy 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 D be 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=ynz of degree n;

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

  2. (C) The following conditions hold:

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

where dj,l means the coefficient by zl of the polynomial dj.

If one of the statements A,B,C holds 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 property A. 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 pFq is a usual notation for the generalized hypergeometric function, and δ is 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 yn appeared in numerators of some biorthogonal rational functions, see [38].

Let gntn=0 be 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,u are 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 z0 is a root of p with the smallest modulus. This ensures that Ftw is 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 F is analytic in CT1,R1. Expand the function Ftw in Taylor’s series by w with a fixed t:

Ftw=n=0φntwnn!,twCT1,R1,E59

where φnt are some complex-valued functions. Then the function φnt is 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 pz be as in (60). Let gntn=0 be a system of OPRL or OPUC, having a generating function Gtw from (57) and Ftw be given by (58). Fix some positive T1,R1, such that Ftw is analytic in the polydisk CT1,R1. Polynomials

φnz=j=0nnjbjgnjt,nZ+,E61

where bj=1puwj0, have the following properties:

  1. Polynomials φn are Sobolev orthogonal polynomials:

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

where

M˜=c0c1cdTc0¯c1¯cd¯.

Here dμg is the measure of orthogonality of gn.

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

  2. Polynomials φn have the following integral representation:

φnt=n!2πiw=R21puwfwetuwwn1dw,nZ+,E62

where R2 is 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=0 satisfy the following recurrence relation:

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

where φr0, r!1, for rZ:r<0.

Polynomials φntn=0 obey 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=0 satisfy 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=0 obey the following differential equation:

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

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

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5. 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: 03 January 2022 Published: 19 February 2022