A new commutativity property of exceptional orthogonal polynomials

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2024) 118:81 https://doi.org/10.1007/s13398-024-01570-7 ORIGINAL PAPER A new commutativity property of exceptional orthogonal polynomials M. M. Castro1 ·F. A. Grünbaum2 Received: 7 July 2023 / Accepted: 8 February 2024 / Published online: 29 March 2024 © The Author(s) 2024 Abstract We exhibit three examples showing that the “time-and-band limiting” commutative property found and exploited by D. Slepian, H. Landau and H. Pollak at Bell Labs in the 1960s, and independently by M. Mehta and later by C. Tracy and H. Widom in Random matrix theory, holds for exceptional orthogonal polynomials. The property in question is the existence of local operators with simple spectrum that commute with naturally appearing global ones. We illustrate numerically the advantage of having such a local operator. Keywords Time-band limiting ·Exceptional orthogonal polynomials ·Bispectral property Mathematics Subject Classification 33C45 ·33C47 ·33E30 ·42C05 1 A brief historical introduction The interest of one of us in what is now called the “bispectral problem” posed and solved in [16] arose from an effort to understand and extend a mathematical miracle, uncovered by D. Slepian, H. Landau and H. Pollak at Bell Labs back in the 1960s, [44,45,56–60]. It is of obvious importance in signal processing and it was motivated by work of C. Shannon, see [54]. This is mentioned in the introduction to [16] and is recalled in the next few lines. For an unknown signal f(t)supported in [–T,T] one observes its Fourier transform Ff(k) for frequencies kin the band [−W,W]. The numerically stable reconstruction of ffrom this data leads to the study of an integral operator in L2(−T,T)with kernel given by K(t,s)=sin W(t−s)/(t−s). One needs to compute numerically many of its eigenfunctions, and this gives a very ill-conditioned problem because the eigenvalues are (except for about 4TWof them) very close together. The miracle in question is that one can exhibit a selfadjoint BM. M. Castro [email protected] F. A. Grünbaum [email protected]y.edu 1Departamento de Matemática Aplicada II and IMUS, Escuela Politécnica Superior, Universidad de Sevilla, C. Virgen de Africa 7, 41011 Seville, Spain 2Department of Mathematics, University of California, Berkeley, CA 94705, USA 123 81 Page 2 of 20 M. M. Castro, F. A. Grünbaum second order differential operator that commutes with the integral operator, has a simple and well spread out spectrum resulting in a common set of eigenfunctions. One has replaced a very ill-posed problem by a very well posed one. The topic of exceptional orthogonal polynomials has developed into an active area of research in the last few years. The literature is large and we just mention a few papers, [17–22,41,50,53]. The potential applications of these polynomials are quite varied: they provide a vast extension of the classical orthogonal polynomials of Jacobi, Laguerre and Hermite which feature in countless areas of mathematics, both pure and applied. They each give a basis of polynomials that are joint eigenfunctions of a fixed differential operator Lof order two Lpn(x)=λnpn(x) The main difference with the classical ones is that the index n,thatindicatesthedegreeof pn, needs not run over the entire set 0,1,2,3,.... A way to increase the chances that this larger class of polynomials will be widely used is to study the extent to which they share certain properties with their classical counterparts. Such an effort will necessarily develop in an exploratory fashion. This is the spirit of this paper. The hope at the time of [16] was that situations exhibiting the highly unusual “bispectral property”, to be defined below, would give rise to extensions of the commutativity miracle mentioned above beyond the Fourier case exploited by the Bell Labs group. One says that one has a “bispectral situation” when a differential operator Lhas eigenfunctions f(x,k) that satisfy a differential equation in the spectral parameter k, i.e. we have Lf(x,k)=(k)f(x,k), (1) as well as Bf(x,k)=(x)f(x,k), (2) where Bis a differential or difference operator acting on k. The spectral parameter kruns over a discrete or continuous set. Several tools were used in [16] to classify all the situations when Lhas order two and B is a differential operator of arbitrary order. They include, among other things, the so called Darboux processes, the so called ad-conditions, as well as a careful study of the monodromy properties of L. The main surprise was the observation that the rational solutions of the Korteweg-deVries equation play a important role in half of the cases found in [16]. The role of its master symmetries was observed in a later paper [66]. The case when kis a discrete variable and B is a second order difference operator, as in the case of the classical orthogonal polynomials, was considered in a series of papers [27–29,31], where the Darboux process was applied to either the semi-infinite or doubly infinite banded matrix Bto obtain the Krall polynomials or functions that extend them. Here the role of the KdV equation is taken up by nonlinear evolutions such as the Toda flows. The appearance of these integrable isospectral systems when Bis a differential or a difference operator came about in these papers by using the full power of the Darboux process (see also [38,40]). Two earlier papers by M. Reach [51,52], based on his UC Berkeley thesis 1987, deal with applying the Darboux process to the second order operator L,asin[16], but allowing Bto be a difference recursion of arbitrary order. Just as in [16] this leads to increasing orders in the recursion relation given by B. See also [22,41] where one has a similar situation. This is not the place to review in detail the developments just mentioned, and we just recall the very important contributions by G. Wilson, [64,65], A. Kasman and M. Rothstein, 123 A new commutativity property of exceptional orthogonal… Page 3 of 20 81 [42], as well as those of B. Bakalov, E. Horozov and M. Yakimov, [2–4]. More references can be found in [25,30,39]. It is clear that a certain set of common tools were used both in the study of the bispectral problem as well as in the study of exceptional orthogonal polynomials. We just mention a few of these. Apparently the first explicit mention of the bispectral property in the second area is made in [53] (see Sect. 4 as well as the acknowledgments in this reference). It appears that the first paper to exploit the Darboux process in connection with exceptional orthogonal polynomials is [50]. The role of trivial monodromy has appeared in later papers such as [19, 20]. One can see points of contact between the considerations in [22,41] and those in the bispectral problem as mentioned by these authors (see also [20]). There are many differences between these two topics: all considerations in the bispectral problem are of a local nature, whereas in the case of exceptional orthogonal polynomials the issue of the completeness of these polynomials in some appropriate Hilbert space is of importance. Making heavy use of the bispectral property, the paper [24] establishes that the commutativity phenomenon alluded to above holds for the classical orthogonal polynomials. The paper [26] shows that the same property holds in connection with the “even family” in [16], the one connected with the master symmetries of the KdV equation. A general strategy to connect bispectrality and the commutativity property is given in [32]. The results in [24] were done by a “bare hands” approach. A more streamlined general method was developed, many years later, in [7–10]. The trivial cases of bispectrality when both physical and frequency space are the real line are given by Bessel and Airy operators (the Bessel case includes Fourier analysis). They have featured in important problems of mathematical physics, such as potential theory, electromagnetism and optics for a very long time. Both cases lead to integral operators admitting a commuting differential operator. For the Bessel case this was proved by D. Slepian, see [58]. The Airy case was observed by C. Tracy and H. Widom in the context of Random matrix theory, see [62]. In [9] one considers deformations of the Airy integral operator which preserve the commutativity property. For the Bessel case see [26]. For very recent numerical work on the eigenfunctions of the Airy integral operator which exploits the existence of the commuting one see [55]. This work has applications both in Random matrix theory as well as in optics. In view of these more recent papers, and once one notices that exceptional orthogonal polynomials give a bispectral situation and are connected to the classical ones by applications of the Darboux process, the results in the present paper are not totally unexpected. However, the nature of the explicit results given here makes it worthwhile presenting them separately. One should mention that there are matrix valued versions of the commutativity property too, see [11,12,33–35]. A last historical remark: many ideas appear again and again in the development of any area of mathematics and the issue of giving appropriate credit takes complicated turns. It is nice to be able to single out a paper by V. Bargmann, [1], where several of the concerns and issues in modern spectral theory originate. The very nice paper by C. Quesne [50]starts by referring to that pioneering paper. In fact, the introduction to her paper offers a rather instructive view of a rich area where many tools appear over and over again. A very complete discussion of the work of V.Bargmann, M.G. Krein, V.A. Marchenko, I.M. Gelfand and B.M. Levitan (among others) is given in the book [13]. In conclusion, we mention that the phenomenon mentioned above (in its original Fourier version) has found a rather unexpected use in a series of papers by A. Connes and collaborators in connection with the Riemann zeta function. For the most recent push in this direction, see [14]. For a commentary on this paper see [37]. For a recent use of this commutativity property 123 81 Page 4 of 20 M. M. Castro, F. A. Grünbaum (again in its original Fourier form) in connection with the Bethe ansatz and entanglement, see [5,6,15] and its references. It is interesting that many of the kernels that appear in connection with the bispectral problem play an important role in Random matrix theory, see [62,63]. For deformations of these kernels that preserve both phenomena see [7–10]. These observations argue for approaching the relatively new area of exceptional orthogonal polynomials by casting a wide net. The commutativity phenomenon alluded to above was extended to the case of classical orthogonal polynomials defined on a finite set, see [48,49]. It was later observed by R. Perline that the explicit form of the commuting operator could be given a simple and unified form, see [47]. It was subsequently seen, see [33,36] that this simple form of Perline applies unchanged to other situations. In [36] one observes that (as mentioned by R. Perline) the simple form of the commuting operator may result in one that does not have simple spectrum. This happens for the so called Bannai-Ito polynomials. In [36] one shows that a more complicated form of the commuting operator, still built with an appropriate extension of Perline’s construction, yields one with simple spectrum. Since the examples of exceptional orthogonal polynomials that we consider here, namely Jacobi, Laguerre and Hermite, involve recursion relations of orders five and seven respectively, the simple form put forward by R. Perline has to be modified. This point will be illustrated in some of the examples below. 2 The contents of the paper In Sect. 3we recall the definitions of the operator of time-band-time limiting as well as the operator of band-time-band limiting. In Sect. 4we consider the case of the exceptional Hermite polynomials which do not depend on free parameters. In Sect. 5we consider the simplest instance of the exceptional Jacobi polynomials. These ones depend on the usual parameters αand βand some of our results are given in terms of these parameters. Some results, which have been checked for multiple values of the parameters, are illustrated by a specific (but arbitrary) choice of them. In Sect. 6we consider an instance of exceptional Laguerre polynomials. Finally, in Sect. 7, in the spirit of [11, Section 6], we conclude by exploiting the numerical pay-off of the results of the previous sections. 3 The operators of time and band limiting We start with a very general setup of the time-band limiting problem for orthogonal polynomials (see for instance [24]) which will be applied later on to the different examples of sequences of exceptional orthogonal polynomials discussed in this paper. Let w=w(x)be a weight function in the open interval (a,b), for which all the moments b axnw(x)dx,n≥0, are finite. Let (pn(x))n≥0be a sequence of real valued orthonormal polynomials with respect to the weight w(x). Since we will be dealing with exceptional orthogonal polynomials, we do not assume that deg pn=n. Consider the following two Hilbert spaces: The space L2((a,b), w(x)dx), denoted here by L2(w), of all measurable functions f(x),x∈(a,b), satisfying b af2(x)w(x)dx <∞and the space 2(N0)of all real valued sequences (cn)n∈N0such that ∞ n=0c2 n<∞. 123 A new commutativity property of exceptional orthogonal… Page 5 of 20 81 The map F:2(N0)−→ L2(w) given by (cn)∞ n=0−→ ∞  n=0 cnpn(x) is an isometry. If the polynomials are dense in L2(w), this map is unitary with the inverse F−1:L2(w) −→ 2(N0)given by f(x)−→ cn=b a f(x)pn(x)w(x)dx. We denote our map by Fto remind the reader of the usual Fourier transform. Here N0 takes up the role of “frequency space” and the interval (a,b)the role of “physical space”. The band limiting operator,atlevelNacts on 2(N0)by simply setting equal to zero all the components with index larger than N.WedenoteitbyχN.Thetime limiting operator ,at level T, acts on L2(w) by multiplication by the characteristic function of the interval (a,T], T≤b. This operator will be denoted by χT. Consider the problem of determining a function ffrom the following data: fhas support on the finite set {0,...,N}and its Fourier transform Ffis known on the set (a,T].This can be formalized as follows χTFf=g=known,χ Nf=f. We can combine the two equations into Ef =χTFχNf=g. To analyze this problem we need to compute the singular vectors (and values) of the operator E:2(N0)−→ L2(w). These are given by the eigenvectors of the operators E∗E=χNF−1χTFχNand S2=EE∗=χTFχNF−1χT. The operator E∗E, acting in 2(N0)is just a finite dimensional matrix M, and each entry is given by (M)m,n=(E∗E)m,n=T a pm(x)pn(x)w(x)dx,0≤m,n≤N. The second operator S2=EE∗acts in L2((a,T), w(x)dx)by means of the integral kernel k(x,y)= N  j=0 pj(x)pj(y). Consider now the problem of finding the eigenfunctions of E∗Eand/or EE∗. For arbitrary Nand Tthere is no hope of doing this analytically, and one has to resort to numerical methods. This is a remarkably ill-conditioned problem since most of the eigenvalues are crowded together. Of all the strategies one can dream of for handling this problem, none sounds so appealing as that of finding a differential operator with simple-and spread out-spectrum which would have the same eigenfunctions as the original operators. This is exactly what Mehta as well as Slepian, Landau and Pollak did when dealing with the real line and the actual Fourier transform. They discovered (the analog of) the following properties: •For each N,Tthere exists a symmetric matrix Lwith a small number of diagonals, with simple spectrum, commuting with M. 123 81 Page 6 of 20 M. M. Castro, F. A. Grünbaum •For each N,Tthere exists a selfadjoint differential operator D, with simple spectrum, commuting with the integral operator S2=EE∗. In this paper we will see instances of exceptional orthogonal polynomials where this phenomenon holds. Once more we will see that the “bispectral property”, first considered in [16], guarantees the commutativity of these two operators, a global and a local one. For an up-to-date treatment of the important issue of computing numerically the eigenfunctions of D,see[46]. For the case of the Discrete Fourier Transform (DFT), see [23]. 4 The exceptional Hermite polynomials We consider the family of exceptional Hermite polynomials defined by ˆ H0=1, ˆ Hn=Hn+4nH n−2+4n(n−3)Hn−4,n≥3, where Hnare the classical Hermite polynomials given by the Rodrigues formula Hn=(−1)nex2Dn xe−x2,n=0,1,2.... These polynomials can also be defined by means of a determinant (see for instance [20]): ˆ Hn=1 8(n−1)(n−2)det ⎛ ⎝ HnH nH n H1H 1H 1 H2H 2H 2 ⎞ ⎠,n= 1,2. The exceptional Hermite polynomials satisfy the orthogonality relation: ∞ −∞ ˆ Hm(x)ˆ Hn(x)e−x2 (1+2x2)2dx =√π2nn! (n−1)(n−2)δn,m,n= 1,2. 4.1 Bispectral operators Let us now consider the orthonormal sequence of exceptional Hermite orthogonal polynomials given by  Hn=√(n−1)(n−2) 4 √π√2nn!ˆ Hn. The exceptional polynomials  Hnsatisfy the differential equation  H n−2x+8x 1+2x2 H n+2n Hn=0 and the recurrence relation, written explicitly in [17] for a different normalization: αn−3 Hn−3+βn−1 Hn−1+βn Hn+1+αn Hn+3=(x) Hn,n≥0,(3) where the coefficients αnand βnand the function (x)are given by: αn=√2 3(n+3)(n−1)(n−2), βn=2(n+1)n(n−2), (x)=4x3 3+2x. (4) Here, we understand that  H1(x)= H2(x)=0aswellas  H−2(x)= H−1(x)=0. Notice that (x)=2(2x2+1)=1 2det H1H 1 H2H 2. This allows, in the spirit of [16], to relate (x)to the appropriate Sato’s τfunction. 123 A new commutativity property of exceptional orthogonal… Page 7 of 20 81 If we write (x)=( H0(x),  H3(x),  H4(x),...) t and B for the semi-infinite heptadiagonal symmetric matrix: B= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0α00... 0... α00β30α30... 0β30β40α40... 00β40β50α50... 0α30β50β60α60... 00α40β60β70α70... . . ............. ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,(5) one has B(x)=(x)(x)(see (2)), where the expression of (x)is given in (4). From the other side one has L(x)=(n)(x),see(1), where L=d2 dx2−2x+8x 1+2x2d dx (6) and (n)is the diagonal matrix whose entries are λn=−2n,n=0,3,4,.... 4.2 Time and band limiting Here,wedefinetheN×Nmatrix Mof truncated inner products depending on a real parameter T, whose entries are given by: Mm,n=T −∞  Hm(x) Hn(x)e−x2 (1+2x2)2dx,n,m=0,3,4,...,N+1. For fixed values of the parameters Nand Tone looks for a “narrow banded” commuting matrix L. One finds that there exists an heptadiagonal matrix Lcommuting with M. If we consider for instance the normalization LN,N=0andLN,N−3=1, this matrix is unique. We display here the symmetric time-band limiting matrix Mof size 7 that appears in the Hermite case. Here and in what follows, Iwill denote the identity matrix of appropriate size. 123 81 Page 8 of 20 M. M. Castro, F. A. Grünbaum M−1+Er f (T) 2I = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −T1 √3 T √2 2T2−1 √10 1 √3 T(2T2+3) 34T4+12 T2+3 2√6 √2T32T2+5 √15 T √24T4+12 T2+3 2√6 1 4T4T4+12T2+38T6+20 T4+10 T2+5 4√5 2T2−1 √10 √2T32T2+5 √15 8T6+20 T4+10 T2+5 4√5 T 5(5+5T2+8T4+4T6) T2T2−3 3√2 8T6+12 T4−18 T2−3 6√6 T8T6+12 T4−6T2+3 12 16 T8+16 T6+60 T2+15 12 √5 4T4−12 T2+3 2√42 T34T4−21 3√14 16 T8−56 T4−7 8√21 T38T6−4T4−14 T2+35 2√15√7 T4T4−20 T2+15 4√30 16 T8−32 T6−120 T4+72 T2+9 24√10 T16 T8−32 T6−88 T4+40 T2−15 16 √15 32 T10−80 T8−80 T6+200 T4−150 T2−25 80 √3 T2T2−3 3√2 4T4−12 T2+3 2√42 8T6+12 T4−18 T2−3 6√6 T34T4−21 3√14 T8T6+12 T4−6T2+3 12 16 T8−56 T4−7 8√21 16 T8+16 T6+60 T2+15 12 √5 T38T6−4T4−14 T2+35 2√15√7 T 36 16T8+120T2+2732 T10 −48 T8+336 T4+126 T2+63 24√21 32 T10−48 T8+336 T4+126 T2+63 24 √21 T16T10 −48T8+40T6+252T4+77T2+84 84 T32 T10−112 T8+48 T6+408 T4−150 T2+45 48√15 64 T12−320 T10 +464 T8+1120 T6−420 T4+1260 T2+315 96 √35 T4T4−20 T2+15 4√30 16 T8−32 T6−120 T4+72 T2+9 24√10 T16 T8−32 T6−88 T4+40 T2−15 16 √15 32 T10−80 T8−80 T6+200 T4−150 T2−25 80 √3 T32 T10−112 T8+48 T6+408 T4−150 T2+45 48√15 64 T12−320 T10 +464 T8+1120 T6−420 T4+1260 T2+315 96 √35 T64T12 −448T10 +1040T8+928T6−1284T4+3460T2+735 960 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ −e−T2 √π2T2+1. For N=7 we have the following expression for the corresponding commuting matrix L ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ − 2T14T2+27 √314 0 0 0 0 0 14 − 5T22T2+21 4√3 52T2+19 2√255 2T5 √200 0 52T2+19 2√2− T26T2+1 √325 33T2+146√3T√70 05 5 2T25 33T2+14− T46T2−27 2√3 3 2√15 2T2+53√35T 21 05 √26√3T3 2√15 2T2+5−√3T6T2−5√75T2+72√5T 00 √73√35T 2√75T2+7− T42T2−29 4√3 √356T2+5 2√3 00012 √5T √356T2+5 2√30 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . One may check that this matrix has simple spectrum. 123 A new commutativity property of exceptional orthogonal… Page 9 of 20 81 Let us write BNfor the truncated matrix of size N×Nmade up to the first Nrows and columns of Band Nthe diagonal N×Nmatrix N=diag(0,−6,−8,...,−2(N+1)). We write ad X(Y)=XY −YX (7) for the usual commutator and (ad X)n(Y)=ad X(ad X)n−1(Y),n≥2. Remark 4.1 The following relation holds true for every value of N,N≥1: −1 40 (ad N)4(BN)+(ad N)2(BN)−18 5BN=0. where 0 is the zero matrix of size N×N. A non obvious consequence of this is that if one tries to write the commuting matrix Lin terms of BNand Nfor arbitrary values of Nand T, in the spirit of Perline [47], one would need to use monomials of degree higher than five. We will face a much better situation in the next section. Remark 4.2 Interestingly, one can see that using the expression of (x)in (4) and the one of Lin (6), one obtains −1 40 (ad L)4() +(ad L)2() −18 5=0. in the spirit of M. Reach in [51,52], who was the first to consider mixed bispectral situations involving differential and difference operators. Remark 4.3 We point out that we have made no use of the Darboux process which has always played an important role in terms of the bispectral property. For the case of exceptional Hermite polynomials discussed here we notice that the differential operator Lin (6) is Darboux connected, see [19, Theorem 1.2 and Definition 3.7], to the one for classical Hermite polynomials, namely H=d2 dx2−2xd dx, by the relation LS=SH, with S=(2x2+1)d2 dx2−4xd dx +4. 5 The exceptional Jacobi polynomials 5.1 Bispectrality Let αand βbe the classical real Jacobi parameters, with α, β > −1andα= β. We write a=β−α 2,b=β+α (β −α),c=b+1 a.(8) 123 81 Page 16 of 20 M. M. Castro, F. A. Grünbaum 7 The benefit of having a commuting local matrix The previous sections have exhibited, for a collection of examples of exceptional orthogonal polynomials, a pair of matrices. The first one, a full matrix M=MT,N, is obtained by forming the inner products of the normalized OP over a restricted range in physical space, thus implementing “time limiting” with parameter T. In the resulting N×Nmatrix the parameter Nimplements “band limiting”. In each case the second matrix, denoted by L=LT,N,is a narrow banded one that commutes with the first matrix, and has simple spectrum. From a numerical point of view the entire purpose of the search for this second matrix is that it reduces the problem of computing the eigenvectors of the first one, a seriously ill-conditioned one, into a very well conditioned one. The eigenvectors of MT,Nare of paramount importance since they give the singular vectors of the signal processing problem at hand, as described in Sect.1. We display below the results of some small size numerical computations that illustrate the problem of computing the eigenvectors of a full matrix, such as MT,N, some of whose eigenvalues are very close together. In each case, we give the eigenvalues of both the full matrix MT,Nand those of the narrow banded matrix LT,N. It should be clear that in the case when Nis large the problems indicated below get to be much worse. Our point is that they already appear for small values of N. We use the QR algorithm as implemented in LAPACK. In each of the three situations, Hermite, Jacobi and Laguerre, we will denote by XMthe matrix of eigenvectors of MT,N (normalized and given as columns of XM. We will denote by YLthe matrix of eigenvectors of LT,N(normalized and given as columns of YL). In theory the eigenvectors of MT,Nshould agree (up to order and signs) with those of LT,N. If we compute the matrix of inner products given by YT LXM we expect to have the identity matrix up to some permutation and possibly some signs due to the normalization of the eigenvectors which are the columns of XMand YL. 7.1 Hermite The choice of parameters is N=7andT=5. The eigenvalues of MT,Nare 1., 1., 1., 1., 1., 1., 0.999989, and those of LT,Nare −2186.9,−2127.47,−1985.6,−1690.49,−1227.68,−611.685,100.033. The matrix YT LXMis ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0.0663098 0.384538 0.914298 −0.108586 0.00078648 0.0000796039 −0.0000275025 0.250947 0.784559 −0.396329 −0.405483 −0.000552959 −0.0000929769 −0.0000926722 −0.568584 0.481897 −0.0828753 0.661528 0.000385822 0.000140271 0.000264937 0.780602 0.0661256 −0.0106201 0.621429 −0.000970967 −0.000161057 −0.000249407 0.00106371 9.69168 ×10−6−0.000916543 0.000209371 0.999999 −0.00028544 −0.000399834 0.000223739 −0.000014601 −0.0000999801 −0.0000216885 0.000285012 1.−0.000257904 0.000370889 −0.0000278978 7.33262 ×10−6−0.0000607604 0.000399533 0.000257706 1. ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 123 A new commutativity property of exceptional orthogonal… Page 17 of 20 81 7.2 Jacobi The choice of parameters is given by α=4, β=3, N=7andT=1/3. The eigenvalues of MT,Nare 0.00271069,0.0397751,0.752568,0.977212,1.0016,1.0,1.0. Notice that the value 1.0016 above is due to numerical instability. Those of LT,Nare −225.02,−131.013,−47.8296,180.78,22.1365,119.864,74.8259. The matrix YT LXMis ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0.00747678 −0.00879137 −0.00787297 0.997507 0.00605771 0.0686547 −0.00573508 0.0299278 −0.0351898 −0.0315137 0.239956 0.0242519 0.968588 −0.0232993 0.188064 −0.221129 −0.198037 0.0204167 0.154553 −0.0312631 0.922623 0.470392 −0.553098 −0.49561 0.00362164 0.476008 −0.00544977 0.0235557 −0.533333 0.6271 0.556811 2.82016 ×10−4−0.110716 −4.23774 ×10−40.00171454 0.643064 −0.753535 −0.136571 −7.37868 ×10−73.89179 ×10−41.10886 ×10−6−4.50479 ×10−6 −0.994134 0.108155 9.28542 ×10−54.00526 ×10−10 −2.1141 ×10−7−6.01908 ×10−10 2.44529 ×10−9 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 7.3 Laguerre The choice of parameters is given by α=7, N=7andT=1/2. The eigenvalues of MT,Nare 1.0856 ×10−4,6.4646 ×10−9,2.4992 ×10−13,−7.9725 ×10−17,−1.8132 ×10−18, 9.8169 ×10−17,6.469 ×10−17, and those of LT,Nare 169.43,119.107,78.6334,46.531,21.5219,2.5574,−11.0609 The matrix YT LXMis ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 4.53595 ×10−14 7.10889 ×10−15 −3.33066 ×10−15 4.84889 ×10−14 4.16888 ×10−14 −4.76174 ×10−13 −1.0 −2.24211 ×10−92.66194 ×10−10 1.39646 ×10−9−1.14752 ×10−9−1.08174 ×10−80.999999 −4.76896 ×10−13 6.62053 ×10−5−1.75441 ×10−5−6.23886 ×10−5−2.52534 ×10−40.999999 1.08165 ×10−84.19664 ×10−14 0.218337 0.447717 0.653743 0.569647 1.78046 ×10−41.13123 ×10−10 3.85524 ×10−14 0.423228 0.637769 −0.0176169 −0.643286 −1.80397 ×10−46.32598 ×10−11 −7.10542 ×10−15 −0.104047 0.513037 −0.717294 0.459842 8.72632 ×10−51.1602 ×10−92.35263 ×10−14 −0.873152 0.359929 0.240438 −0.224153 2.24938 ×10−5−2.64713 ×10−9−4.8397 ×10−14 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Observe that some of the entries of these matrices YT LXMare indeed very close to the theoretically correct values, while others are terribly off. The reason is that a few eigenvalues of the full matrix of inner products MT,Nare just too close together. This produces numerical instability in the computation of the corresponding eigenvectors. On the other hand, all the eigenvalues of the commuting matrix LT,Nare nicely separated and the corresponding eigenvectors can be trusted. In summary, a good way to obtain reliable numerical values for the eigenvectors of the global matrix Mis to forget about Maltogether and to compute numerically the eigenvectors of L. Not only we will then be dealing with a very sparse matrix for which the QR algorithm works very fast (most of the work is avoided) but the problem is numerically very well conditioned. 123 81 Page 18 of 20 M. M. Castro, F. A. Grünbaum 8 Conclusions The classical work of Slepian, Landau and Pollak has found, over the years, innumerable applications in various forms of signal processing reaching all the way to geophysics. This was extended in [24] when one consideres not Fourier expansions but expansions in terms of classical orthogonal polynomials. In this paper we show that expansions in terms of some exceptional orthogonal polynomials exhibit the same basic commutativity property, and we illustrate the numerical importance of this algebraic fact. Future applications involving exceptional orthogonal polynomials could benefit from the results in this paper. Funding Funding for open access publishing: Universidad de Sevilla/CBUA The work of the first author was partially supported by PID2021-124332NB-C21 (FEDER(EU)/Ministerio de Ciencia e Innovación-Agencia Estatal de Investigación) and FQM-262 (Junta de Andalucía). Data availability Not applicable. Declarations Conflict of interest The authors declare that they have no conflict of interest. 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