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Some functions that generalize the Krall–Laguerre polynomials
F. Alberto Grunbauma;∗;1, Luc Haineb;2, Emil Horozovc;3 a
Department of Mathematics, University of California, Berkeley, CA 94720, USA
bDepartment of Mathematics, Universite Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium c
Department of Mathematics and Informatics, Soa University, Soa 1126, Bulgaria
Received 29 December 1998
Abstract
Let L() be the (semi-innite) tridiagonal matrix associated with the three-term recursion relation satised by the Laguerre polynomials, with weight function 1
(+1)z
e−z; ¿−1, on the interval [0;∞[. We show that, when is a positive integer, by performing at mostsuccessive Darboux transformations fromL(), we obtain orthogonal polynomials on [0;∞[ with ‘weight distribution’ 1
(−k+1)z
−ke−z+Pk j=1sj
(k−j)(z), with 1
6k6. We prove that, as a consequence of the rational character of the Darboux factorization, these polynomials are eigenfunctions of a (nite order) dierential operator. Our construction calls for a natural bi-innite extension of these results with polynomials replaced by functions, of which the semi-innite case is a limiting situation. c1999 Elsevier Science B.V. All rights reserved.
1. Introduction
The bispectral problem, as originally posed and solved by Duistermaat and Grunbaum [8], consists in nding all Schrodinger operatorsL= d2
dx2+V(x), for which some families of eigenfunctionsf(x; z), satisfying Lf=zf, are also eigenfunctions of a dierential operator of arbitrary (but xed) order in the spectral variable z; Bf=(x)f. The complete solution of this problem revealed its intimate connection with the theory of Darboux transformations and integrable systems. In a nutshell, all
solutions of the problem can be obtained by means of repeated application of the Darboux process to some of its basic solutions. By a ‘basic’ solution, we mean a solution for which the bispectral operator B is of lowest possible order, in this case of order 2.
∗Corresponding author.
1Supported in part by NSF Grant # DMS94-00097 and by AFOSR under Contract FDF49620-96-1-0127. 2
A Research Associate of the Belgian National Fund for Scientic Research.
3Supported in part by Grant MM-523 of Bulgarian Ministry of Education.
In [13] some of us formulated a discrete–continuous version of the original problem, where the Schrodinger operator is replaced by a doubly innite tridiagonal matrix
L=
with the convention that b1 is the (0;0)th entry of L. More precisely, the problem consists in
determining all bi-innite tridiagonal matrices L, such that at least one family of eigenfunctions
fn(z); n∈Z, given by
anfn−1(z) +bn+1fn(z) +fn+1(z) =zfn(z); (1.2)
is also a family of eigenfunctions of a dierential operator of order m (with coecients independent of n∈Z):
The ‘basic’ solutions correspond again to the case m= 2. They are obtained by shifting n to
n+ in the three-term recursion relation satised by either the Hermite, the Laguerre, the Jacobi or the (lesser known) Bessel polynomials, letting n run over all integers, see [13,16]. Observe that the ‘associated polynomials’, see [1,2], satisfy the corresponding three-term recursion relation. However, as soon as 6= 0, the family of common solutions to Eqs. (1.2) and (1.3) is not given by the associated polynomials, but rather by functions which can be specied in terms of an arbitrary solution of Gauss’ hypergeometric equation or some of its conuences, see [13]. It is only when=0, that one can put f−1(z) = 0 and f0(z) = 1 and replace the matrix L in Eq. (1.1) by the semi-innite
matrix obtained by chopping all the columns to the left of b1 and all the rows above b1. In this case
the fn’s become the Hermite, the Laguerre, the Jacobi and the Bessel polynomials. This gives back
the classical result of Bochner [6], characterizing the classical orthogonal polynomials as the only families of orthogonal polynomials, which are eigenfunctions of a second order dierential operator. The more general problem of determining all orthogonal polynomials which are eigenfunctions of a dierential operator of arbitrary order was formulated by Krall [24], back in 1938. He found that the operator had to be of even order and, in [25], he gave the complete solution for m= 4. The new orthogonal polynomials that he discovered are now coined under the name of ‘Krall polynomials’.
of polynomial eigenfunctions that satises a xed dierential equation of order 4. The same con-struction can be performed out of any instance of the Jacobi and the Laguerre polynomials [14]; the Krall polynomials are distinguished as the special cases when the above pentadiagonal matrix is the square of a tridiagonal matrix. This never happens in the case of the Hermite and the Bessel polynomials.
In this paper we show that repeated application of the Darboux process starting from the semi-innite matrix L(), associated with the three-term recursion relation satised by the Laguerre poly-nomials, for positive integer values of its parameter , leads to orthogonal polynomials which are eigenfunctions of a (nite order) dierential operator. Actually, we will see that it is natural to enlarge the problem to the two-parameters bi-innite extension L(; ) of the matrix L() mentioned above, which reduces to it when = 0. When 6= 0, we obtain in this way functions which provide higher order instances of solutions of the discrete-continuous version of the bispectral problem, and which have never appeared before, even in the case of order 4.
One of the key new ideas, that has been introduced in the area of the bispectral problem in the last few years, is due to Wilson [31,32], who raised the original question at the level of a commutative algebra of dierential operators, that is at the level of the ‘common eigenfunctions’ of the algebra. This led him to introduce the seminal idea of the bispectral involution, which amounts to interchanging the role of the ‘space’ and the ‘spectral’ variables in these eigenfunctions. In this way, he obtained a complete description of all (maximal) rank one commutative algebras of bispectral dierential operators. Recently, his ideas were further developed in the works of Bakalov et al. [3–5] and Kasman and Rothstein [17], aiming at obtaining further examples of higher order rank commutative algebras of bispectral dierential operators, the rst examples of which appeared already in the original work of Duistermaat and Grunbaum [8]. Although the methods of [8,31] appeared to be very dierent, these works allowed to see them as part of a general theory, by introducing the concept of a bispectral Darboux transformation. For the study of some new and intriguing examples of bispectral ordinary dierential operators, of a dierent nature than those in [8], we refer the reader to [4,5,11].
An important message of this paper is to show that the bi-innite extension of Bochner’s original result [6], obtained in [13], is the natural context in which Wilson’s ideas can be adapted to the discrete-continuous version of the bispectral problem. Indeed, only bi-innite tridiagonal matrices possess a two-dimensionalkernel, from which bispectral Darboux transformations can be performed. In our context, Wilson’s bispectral involution becomes an anti-isomorphism from an algebra of matrices to an algebra of dierential operators. This anti-isomorphism is given explicitly by the three-term recursion relation, the dierentiation formula and the second-order equation which are satised by the two-dimensional space of ‘bispectral functions’, corresponding to the ‘basic’ solutions of the problem.
positive integer value of, these orthogonal polynomials are eigenfunctions of a dierential operator and, to the best of our knowledge, this result is new. When 6= 0, there is a family of functions
which solve simultaneously Eqs. (1.2) and (1.3) and become polynomials only when = 0. We propose to call these functions the Krall–Laguerre functions. Finally, in Section 5, we illustrate the theory by considering the simplest example which in the limit →0 leads to orthogonal polynomials with weight distribution involving not only the delta-function but also its rst derivative.
2. The bi-innite Laguerre matrix and its associated bispectral triple
The bi-innite Laguerre matrix L(; ) (in short, L) is a tridiagonal matrix as in Eq. (1.1) which is obtained by shifting n to n+ in the coecients of the standard recursion relation which denes the (monic) Laguerre polynomials, with the understanding that n runs over all integers:
an= (n+)(n++); bn= 2(n+) +−1: (2.1)
In the sequel, we shall denote by L() the semi-innite matrix which is obtained by chopping all the columns to the left of b1 and all the rows above b1 in L(;0), and denes the three-term recursion
relation satised by the usual (monic) Laguerre polynomials.
It is shown in [13] (see also [16]) that there is a two-dimensional space of functions {fn(z)}n∈Z satisfying the following three properties:
(i) {fn(z)} satisfy the three-term recursion relation
zf=L(; )f; (2.2)
with f≡(: : : ; f−1; f0; f1; : : :)T;
(ii) {fn(z)} satisfy a dierentiation formula
A
z; d
dz
f=Mf; (2.3)
with A a rst-order dierential operator and M a (bi-innite) tridiagonal matrix; (iii) {fn(z)} are eigenfunctions of a second-order dierential operator
B
z; d
dz
f=f; (2.4)
with the diagonal matrix of eigenvalues of B, = diag(: : : ; −1; 0; 1; : : :).
The data listed in (i) – (iii) can be described as follows. Pick w(z) an arbitrary solution of the equation
zw′′
(z) + (+ 1−z)w′
(z) +w(z) = 0; (2.5)
which is obtained from Gauss’ hypergeometric equation by conuence. Dene f0(z) and f1(z) by
f0(z) =w(z); (2.6)
and
f1(z) = (z−−1−)w(z)−zw
′
Then, the family of functions {fn(z)}n∈Z dened by the three-term recursion relation (2.2) with
f0(z) and f1(z) as in Eqs. (2.6) and (2.7), satises automatically Eqs. (2.3) and (2.4) with
A=z d
dz;
Mn; n+1= 0; Mn; n=n+; Mn; n−1= (n+)(n++);
(2.8)
B=−z d
2
dz2 + (z−−1)
d dz; n=n+:
(2.9)
One can always pick [9, 6.3(2)]
w(z) =1F1(−;+ 1;z); (2.10)
to be a solution of Eq. (2.5), with 1F1(a;c;z) denoting the hypergeometric series
1F1(a;c;z) =
∞
X
n=0
(a)n
n!(c)n
zn: (2.11)
Here, as well as in the rest of the paper, (x)n, n∈Z, is the shifted factorial (Pochammer notation):
(x)0= 1; (x)n=
(x+n)
(x) = (x+n−1)(x)n−1: (2.12) It follows from [9, 6.4(2),6.4(9)] that the solution of the three-term recursion relation (2.2) with initial conditions given by f0(z) and f1(z) as in Eqs. (2.6) and (2.7), and w(z) as in Eq. (2.10), is
given by
fn(z) = (−1)n(+ 1 +)n 1F1(−n−;+ 1;z): (2.13)
Notice that, when = 0, the functions fn(z), n¿0, reduce precisely to the Laguerre polynomials,
normalized to be monic, see [9, 10.12(14)]. For this reason, we shall call the functions fn(z)
dened in Eq. (2.13) theLaguerre functions. They are eigenfunctions of the second order dierential operator B dened in Eq. (2.9), which generalizes the standard second order dierential equation satised by the Laguerre polynomials, and they will play a basic role in what follows. As emphasized in the introduction, we notice that, although the three-term recursion relation satised by the functions
fn(z) is the same as the one satised by the associated Laguerre polynomials studied in [2], when
6= 0, the fn(z) are not polynomials. The ‘associated polynomials’ satisfy Eq. (2.2) but, for 6= 0,
do not satisfy Eq. (2.4). We shall denote by
B=hL; M; i; (2.14)
the subalgebra of the algebra of nite band bi-innite matrices generated by the matrices L, M,
appearing on the right-hand side of Eqs. (2.2) – (2.4). Similarly
B′
=hz; A; Bi; (2.15)
will denote the subalgebra of the algebra of dierential operators generated by z, A, B. Formulas (2.2) – (2.4) serve to dene an anti-isomorphism
b:B→B′
between these two algebras, i.e. it is given on the generators by
b(L) =z; b(M) =A and b() =B: (2.17)
More precisely, any monomial LiMjk in B, i; j; k¿0, acting on the original space of bispectral
functions {fn}n∈Z, gives
LiMjkf=BkAjzif;
i.e.
b(LiMjk) =b(k)b(Mj)b(Li) =BkAjzi: (2.17′)
The triple (B;B′; b) provides an instance of the notion of a bispectral triple which was introduced in [4]. To explain the terminology, we need to introduce the commutative subalgebras K⊂B and K′
⊂B′ generated respectively by and z. We shall refer to these subalgebras as the ‘algebras of functions’. Their images by b and b−1 will be denoted by A′
and A respectively and provide
obvious bispectral operators. In order to dene the notion of Darboux transformation in this abstract setting, we shall denote with a bar the elds of quotients of K, K′, A and A′. Obviously, b extends to isomorphisms K →A′ and A →K′. We now reproduce from [15] the main tool that
we need to produce out of L(; ) new non-trivial bispectral operators by means of the Darboux transformation, see also [4].
Theorem 1. Let L∈A be a constant coecients polynomial in L; which factorizes ‘rationally’
as
L=QP; (2.18)
in such a way that
Q=SV−1; P=−1R; (2.18′)
with R; S∈B and V ∈K. Then the Darboux transform of L given by
˜
L=PQ; (2.19)
is a bispectral operator. More precisely; dening ≡b(L)∈K′
and f˜ ≡Pf; with f satisfying
Eqs. (2:2)–(2:4) above; we have
˜
Lf˜=f;˜ (2.20)
˜
Bf˜=Vf;˜ (2.21)
with
˜
B=b(R)b(S)−1: (2.22)
Proof. Eq. (2.20) follows immediately from the denitions. Let
Pb ≡b(R) and Qb≡b(S)
−1: (2.23)
Clearly, using the anti-isomorphism introduced in Eqs. (2.16), (2.17), (2:17′ ), ˜
f=Pf=−1P
Since B has no zero divisors, Eqs. (2.18), (2.18′) imply
V =RL−1S; L−1∈A:
Applying the anti-isomorphism b, we obtain
b(V) =b(S)−1b(R): (2.25)
From Eqs. (2.23) – (2.25), we have
Vf=QbPbf=Qbf;˜
and thus, using Eq. (2.18) and this last relation, we deduce that
f=−1Qf˜=V−1Q
bf:˜ (2.26)
This equation combined with Eqs. (2.23) and (2.24) gives Eq. (2.21), which completes the proof of the theorem.
Remark. Observe thatB˜ in Eq.(2:22) is a Darboux transformation ofb(V)in Eq. (2:25).Since
V∈K; V is a polynomial in ; and thus b(V) is a polynomial in B. This shows that the
new bispectral operator B˜ is in fact obtained as a Darboux transform of a constant coecients
polynomial in the original (second order) bispectral operator B.
Notice that Theorem 1 leaves open the question of when a ‘rational’ factorization of the form (2.18), (2.18′
) can be performed. Our aim in the next two sections is to show that, for any positive integer, the successive powers L(; )k, k= 1;2; : : : ; , of the doubly innite Laguerre matrix admit
such a factorization with the additional property that the new operator ˜L in Eq. (2.19) is again the
kth-power of a tridiagonal matrix. The functions ˜fn= (Pf)n=−1Pbfn, obtained from the Laguerre
functions fn in Eq. (2.13), are thus solutions of the discrete-continuous version of the bispectral
problem (1.2), (1.3), with a bispectral operator B of order m ¿2. We shall refer to them as the
Krall–Laguerre functions. When = 0, these functions become polynomials which generalize the
Krall–Laguerre polynomials. For a precise denition, see Theorem 3 in Section 4.
3. One step of the Darboux process
In this section we rst perform a standard Darboux transformation on the doubly innite Laguerre matrix L(; ), when ¿0 and is arbitrary. In the limit = 0, we obtain in this way orthogonal polynomials on the interval [0;∞[ for the measure 1
()z
−1e−z+(z), where is the free parameter
of the Darboux transformation. These polynomials (as well as an extension to the Jacobi case) were found by Koornwinder [20]. In particular, he proved that, in general, they satisfy a second-order dierential equation with coecients depending on n. In the special case when is a positive integer, we show in Section 3.2 that the Darboux factorization of L(; ) can be recast in the form (2.18), (2.18’) called for in Theorem 1, and as a consequence, the Darboux transform of L(; ) is again bispectral, in the sense of Eqs. (1.2) and (1.3). When = 0 and = 1, we get back the classical Krall–Laguerre polynomials, discovered by Krall [25]. When ¿1, this result has been obtained (using other methods) by Littlejohn [29] when = 2, by Krall and Littlejohn [23] when
3.1. Factorizing the bi-innite Laguerre matrix
We remind the reader that the standard Darboux transformation (referred to in the sequel as elementary Darboux transformation) of a bi-innite tridiagonal matrix L [30] starts by factorizing L
as
L=QP; (3.1)
with the two factors Q and P denoting, respectively, upper and lower triangular matrices acting on a vector h= (: : : ; h−1; h0; h1; : : :)T as follows:
(Qh)n=xn+1hn+hn+1; (3.2)
(Ph)n=ynhn−1+hn: (3.3)
The most general factorization ofL, in the form (3.1), is obtained by picking anarbitraryelement
f∈kerL, so that the matrices P and Q are given by
(Ph)n=hn−
fn
fn−1
hn−1; (3.4)
(Qh)n=−an
fn−1
fn
hn+hn+1: (3.5)
Since the kernel of L is two-dimensional and only the ratios of the fn’s are involved, this
factoriza-tion depends (projectively) on one free parameter. The Darboux transformation ˜L of L is obtained by exchanging the order of the factors in Eq. (3.1)
˜
L=PQ: (3.6)
Explicitly, the entries ˜an and ˜bn of the new tridiagonal matrix ˜L are given by
˜
an=an−1
fn−2fn
f2 n−1
; (3.7)
˜
bn=bn+
fn
fn−1
−fn−1
fn−2
: (3.8)
As observed in [12], the above construction still makes sense in the case of semi-innite matrices, that is if one chops all the columns to the left of the (0;0)th entry and all the rows above the (0;0)th entry of the matrices L, P and Q. The crucial observation is that, in this limit, the upper–lower factorization (3.1) of L, still contains a free parameter, which can be picked to be x1. This would
no longer be the case, if instead we had chosen to perform a lower–upper factorization of L, in which case all the entries of P and Q are uniquely specied.
Thus, in order to perform the (most general) factorization of the form (3.1) for the bi-innite Laguerre matrix L(; ), we need a description of its two-dimensional kernel. Introducing the lower shift matrix
(T−1h)
n=hn−1; (3.9)
and the matrix C dened by
one checks easily that
L(; ) =T−1C(C+I); (3.11)
withI the identity matrix. From this important observation, since the matricesC andC+I commute, we deduce immediately the next result.
Lemma 3.1. Let ¿0.The kernel of L(; ) is generated by the vectors= (: : : ; −1; 0; 1; : : :)T and = (: : : ; −1; 0; 1; : : :)T;where
n= (−1)n(1 +)n; (3.12)
n= (−1)n(1 ++)n; (3.13)
and(x)n denotes the shifted factorial.
Corollary 3.2. The most general factorization of the bi-innite Laguerre matrix L(; ) in the
form (3:1)–(3:3)is given by
Proof. From Lemma 3.1, we have that
f∈kerL(; )⇔f=+ ;
with an arbitrary free parameter (we allow =∞; i.e. f= ), which using Eqs. (3.4) and(3.5) gives Eq. (3.14) and establishes the corollary.
One checks easily that in the limit →0; y0 → 0 and x1 → =(1 +); and thus one obtains a
factorization of the semi-innite Laguerre matrix, where x1 is equivalent to the free parameter .
The Darboux transform ˜L (3.6) denes new orthogonal polynomials q−1;
n given by
n(z) denote the (monic) Laguerre polynomials with weight function 1 (+1)z
e−z on [0;∞[;
when ¿−1. Using the standard formulas (see, for instance, [9, 10.12(15),10.12(16)])
p
which precisely agrees with the formula given in [18,20] for Koornwinder’s generalized Laguerre
polynomials (normalized to be monic) with weight function 1
()z
−1e−z +(z) on the interval
3.2. Factorizing in the algebra B; when is a positive integer
In the special case when is a positive integer, it is possible to achieve the form of the factor-ization (2.18), (2:18′); called for in Theorem 1. For this, it will be convenient to use another basis of the algebra B=hL; M; i introduced in Eq. (2.14), namely
B=hN; T; i; (3.15)
where N denotes the strictly lower part of the bi-innite Laguerre matrix L=L(; ):
(Nh)n=anhn−1= (n+)(n++)hn−1; (3.16)
and T is the upper shift matrix
(Th)n=hn+1: (3.17)
One checks easily that the change of basis is given by
N=M−;
T=L−−M−(+ 1)I: (3.18)
The point of this change of basis is that now the three generators N; T; are lower, upper and diagonal matrices, respectively, with only one non-zero diagonal. They give the most convenient way to factorize a tridiagonal matrix. Using these new generators for B; we have
Lemma 3.3. Let be a positive integer. Then; the Darboux factorization (3:14) ofL(; ) can
be recast into the form called for in Theorem 1:
L(; ) =QP≡(SV−1)(−1R); (3.19)
with R; S∈B given by
R= (+I)() +N(+ 2I); (3.20)
S= (+I)() +T(+I); (3.21)
and
=()(+I); V =(+I); (3.22)
where() a polynomial of degree in
() =I +
−1
Y
j=0
(+jI); = (1 +)
: (3.23)
Proof. Using that is a positive integer and the denition of an in Eq. (2.1), the entries yn and xn
(3.14) of the factors Q and P in the Darboux factorization (3.1) become rational functions of n:
yn=
(n+)(n++ 1)
(n+) ; xn=
(n−1 ++)(n+−1)
(n+) ; (3.24)
with (n) a polynomial of degree in n:
Since
(Ph)n=hn+ynhn−1
= 1
(n+)((n+)hn+ (n+)(n++ 1)hn−1); (3.26)
remembering that is the diagonal matrix = diag(n); n=n+; with 0= at the (0;0)th entry,
we can rewrite P as follows
P=()−1[() +T−1(+I)(+ 2I)]: (3.27)
The (lower) shift matrixT−1 does not belong to the algebra B;but we observe that, by the denition
of the an’s (2.1), the matrix N∈B (3.16) can be written as
N=T−1(+I)(+ (1 +)I)
= (+I)T−1(+I): (3.28)
Therefore,
P=()−1[() + (+I)−1N(+ 2I)]
=()−1(+I)−1[(+I)() +N(+ 2I)]: (3.29)
Similarly,
(Qh)n=xn+1hn+hn+1
= (n++) (n+)
(n++ 1)hn+hn+1; (3.30)
and thus,
Q= (+I)()(+I)−1+T
= [(+I)() +T(+I)](+I)−1: (3.31)
Since the (upper) shift matrix T belongs to B; the factor between the brackets on the right-hand
side of the last equality belongs automatically toB. Combining Eqs. (3.25), (3.29) and (3.31) gives
the result announced in the lemma, which concludes the proof.
Notice that in this case, both and V in Eq. (3.22) belong to the algebra of ‘functions’ K
(i.e. the constant coecients polynomials in ), and thus the product V∈K. Since we already
observed that R and S in Eqs. (3.20) and(3.21) belong to the algebra B; using Theorem 1 with
=b(L(; )) =z; we deduce immediately
Corollary 3.4. Let be a positive integer. The Darboux transform L˜=PQ of L(; ); with P; Q
in Eq. (3:19) is again bispectral. Explicitly; dening
˜
fn= (Pf)n; (3.32)
with fn as in Eq. (2:13); we have
˜
Lf˜n=zf˜n;
˜
with
˜
B=b(R)b(S)z−1; (3.33)
whereb:B→B′ is the bispectral anti-isomorphism dened in Eqs. (2:16); (2:17);(2:17′).
To see that these formulas are actually constructive, we compute from Eqs. (2.17) and (3.18)
b() = B; b(N) =A−B;
b(T) = z−B−A−(+ 1); (3.34)
and then, using the fact that b in Eq. (2.16) is an anti-isomorphism, we get from Eqs. (3.20), (3.21) that
b(R) =(B)(B+) +(B+ 2)(A−B);
b(S) =(B)(B+) +(B+ 1)(z−B−A−(+ 1)):
(3.35)
Recalling the remark following Theorem 1, ˜B is a Darboux transform of b(V). Since (V)n =
(n++)(n+)(n++ 1) is a polynomial of degree 2+ 1 in n+; both b(V) and ˜B in Eq. (3.33) are operators of order 2(2+ 1). This order is larger than the order of the operator produced in [18]. The issue of getting the lowest possible order will be discussed in Section 5.
4. Iterating the Darboux process
In this section we rst describe the result of k iterations of the Darboux process starting from the bi-innite Laguerre matrix L(; ); ¿0; k ¡ + 1. We compute the moment functional for the resulting orthogonal polynomial sequence, when = 0. Then, in Section 4.2, when is a positive integer and the number of elementary Darboux transformations is less than or equal to ; we show that the process leads to functionswhich solve simultaneously Eqs. (1.2) and (1.3). In the limit=0;
these functions become orthogonal polynomials which are eigenfunctions of a dierential operator and generalize the Krall–Laguerre polynomials; for this reason, we call them the Krall–Laguerre functions.
4.1. Extending Koornwinder’s generalized Laguerre polynomials
In order to compute the eect of k successive elementary Darboux transformations starting from
L0 ≡L(; )
L0=Q0P0yL1=P0Q0=Q1P1y: : :
yLk−1=Pk−2Qk−2=Qk−1Pk−1 yLk=Pk−1Qk−1; (4.1)
Lemma 4.1. For j¿0; the vectors (j)=(: : : ; (j)
When j= 0, this formula is interpreted as kerP0=f(0), reproducing the result given in Lemma 3.1.
Thus, the product matrix P
P=Pk−1Pk−2: : : P0; (4.8)
Another way to put this is to say that the sequence of elementary Darboux transformations (4.1) can be performed in one shot by factorizing
L≡Lk
with P as in Eq. (4.9) and Q a uniquely determined upper triangular matrix with k+ 1 diagonals, with the entries on the top diagonal normalized to be one. Then, the Darboux transform is
˜
L=PQ= (Lk)k; (4.11)
with Lk the tridiagonal matrix obtained at the end of the chain of the elementary Darboux
transfor-mations (4.1). Indeed, since
and can be normalized as
(I; ) or (; I)≡(−1; I); (4.13)
depending on whether 0 6= 0 or 0 6= 0; with I the identity matrix. Thus; there are in fact only
k free parameters 0; : : : ; k−1 involved in the factorization (4.10); which can be thought of as the
new free parameters involved in the successive elementary Darboux transformations (4.1).
Our aim now is to compute explicitly the entries of the new tridiagonal matrix Lk in Eq. (4.11).
For this, we need to introduce a few notations that will be repeatedly used in this section. We can write f(j)
Now we come to some expressions which will play a crucial role. Section 5 illustrates their useful-ness in the simplest case =k= 2. We introduce the determinants
(k)(n) = det( ˜f(j)
i(n) =
The following lemma extends Eqs. (3.7) and (3.8) to an arbitrary number of (elementary) Darboux transformations:
Lemma 4.3. The entries a(k)
n and b(nk) of the tridiagonal matrix Lk;obtained from L0=L(; ) after
k elementary Darboux transformations; are given by
a(nk)= (n+)(n++−k)
The following sublemma (whose proof we omit) will be needed to establish Lemma 4.3, as well as Lemma 4.5 later in this section. Both of these lemmas, whose detailed proofs are given, are used to establish Theorem 2 and Theorem 3 in this section. They produce very concrete formulas which will be illustrated in Section 5.
Lemma 4.4. Let V andW be vector spaces of respective codimension 2 and1 in a vector space E
of dimension k+ 2. The (incidence) relationV⊂W amounts to the system of quadratic equations
satised by the Plucker coordinates p1:::i:::ˆ j ::: kˆ +2; 16i ¡ j6k + 2; (resp. 1:::i::: kˆ +2; 16i6k+ 2) of
V (resp. W):
1:::i::: kˆ +2p1:::j :::ˆ l ::: kˆ +2−1:::j ::: kˆ +2p1:::i :::ˆ l ::: kˆ +2
+1:::l ::: kˆ +2p1:::i :::ˆ j ::: kˆ +2= 0;
for all 16i ¡ j ¡ l6k + 2: (4.21)
Proof. See [10, p. 134, Lemma 2], where the more general case of the incidence relations satised
Proof of Lemma 4.3. The proof is done by induction on k. For k= 1, we have that
Assume now that formula (4.19) holds for k, we establish it for k+ 1. Indeed, since by Eq. (4.7), the kernel of Lk is spanned by
f=Pk−1Pk−2: : : P0f(k); (4.22)
using Eq. (3.7), we get that
a(nk+1)=a(nk−)1
and, therefore, by induction hypothesis,
a(nk+1)=a(nk−)1
Similarly, to establish Eq. (4.20), we rst observe that, from Eqs. (3.8) and (4.14), we have that
b(1)
Assume now that formula Eq. (4.20) holds for k, we establish it for k + 1. Indeed, by Eq. (3.8), we have that
with fn as in Eq. (4.23). By induction hypothesis, this gives that
Consider now the (k+ 2)×(k+ 1) matrix
( ˜fn(−j−k−1)2+i)16i6k+2 16j6k+1;
and the subspaces V (resp. W) of Rk+2 generated by the rst k columns (resp. all the columns)
of this matrix. Applying Lemma 4.4 to this situation, one checks easily that Eq. (4.21) with i= 1,
j=k+ 1 and l=k+ 2, amounts to
(k)(n−1)(k+1)(n+ 1) +1(n)(k+1)(n) =(k)(n) (k+1) 1 (n);
with (1k+1)(n) the (k + 1)×(k+ 1) determinant obtained by replacing k by k+ 1 into Eq. (4.17). Thus, Eq. (4.24) coincides with Eq. (4.20), with k replaced byk+ 1. This establishes Lemma 4.3. Remember that, as explained in Section 3, in the case of semi-innite matrices, the (0;0)th entry
x1 of Q (3.2) is equivalent to the free parameter of the elementary Darboux transformation. Thus
our construction still makes sense in the limit = 0 and the k free parameters can be thought of to be the entries x(1j), 06j6k−1, of the matrices Qj in Eq. (4.1). The matrix Lk which is obtained at
Lk are non-zero, there exists a unique (up to a multiplicative constant) moment functional M(k) for
which the sequence {q(k) n (z)}
∞
n=0 is an orthogonal polynomial sequence, that is
M(k)[q(k)
We remind the reader that the moment functional M corresponding to a sequence of complex
numbers {n}∞n=0 is a complex valued function dened on the vector space of all polynomials by
M(zn) =
n; n= 0;1;2; : : : ;
M(c11(z) +c22(z)) =c1M(1(z)) +c2M(2(z));
for all complex numbers ci and all polynomials i(z) (i= 1;2).
The next theorem shows that, iterating the Darboux process starting from the Laguerre poly-nomials, leads to a natural extension of Koornwinder’s generalized Laguerre polynomials (which correspond to k= 1).
Theorem 2. The sequence of polynomials {q(k)
n (z)} ∞
n=0; obtained after k successive elementary
Darboux transformations starting from the Laguerre matrixL();withk ¡ + 1;is an orthogonal
sequence of polynomials with moment functional M(k) given by the weight distribution
1
where x1(0); : : : ; x(1k−1) denote the successive free parameters in the elementary Darboux transforma-tions ((0;0)th entries of Q0; : : : ; Qk−1); (j)(z) denotes the jth derivative of the delta function and
Proof. Start with any orthogonal polynomial sequence{pn(z)}∞n=0 for a moment functional M, with M[zn] =
n and 0= 1. One just needs to observe that, after one elementary Darboux transformation,
if the new matrix ˜L in Eq. (3.6) satises the hypothesis of Favard’s theorem (i.e. ˜an 6= 0, n¿1),
the resulting sequence of polynomials {qn(z)}∞n=0 dened by ˜L will be orthogonal for the moment
functionalM˜ given by
˜
M[1] = 1; M˜[zn] =x
1n−1; n¿1; (4.27)
with x1 the free parameter ((0;0)th entry of Q) of the Darboux transformation. Indeed, sinceq=Pp,
one also has
zp=Qq ⇔ zpn=xn+1qn+qn+1:
Applying M˜ to both sides, we obtain for n= 0 that M˜[z] =x1, and inductively, using the classical
formula expressing the orthogonal polynomials in terms of the moments
pn(z) =
From (4.27) it follows that, after k elementary Darboux transformations (again assuming that at each step we can apply Favard’s theorem), the resulting polynomials {q(k)
n (z)} ∞
n=0 will be orthogonal
for the moment functional M(k) dened by
M(k)[1] = 1;
and thus the argument above applies. Since the moments {sn}∞n=0 of the weight distribution (4.25)
are given by
we see that this distribution denes the same moment functional as M(k), provided that we pick the
rj’s as in Eq. (4.26), which establishes Theorem 2.
4.2. The bispectral property
of the argument is to exploit in this case the rational character of the Darboux factorization (4.10). The denominators from the left-hand side and the right-hand side factors are taken out to produce part of the ‘function’ V∈K. Since the upper-shift matrix T is in the algebra B, what remains
to the left is automatically in B. We expand what remains on the right-hand side in terms of the lower-shift matrix T−1, then express the various powers of T−1 in terms of N via Eq. (3.28) and
absorb the denominator in . We now proceed with the details of our program. Put
Eqs. (4.16) – (4.18), become now polynomials in the variable n+. Thus, we shall write
(k)(n)≡
expanding the numerator of Eq. (4.9) along the last column, we get
(Ph)n=
From the previous formula, it is clear that, in terms of matrices, the operator P can be written as
P=0()
From Eq. (3.28) we deduce easily that
and so, we can rewrite P as
P=−1R; (4.36)
with
=0() k−1
Y
j=0
(+ (−j)I); (4.37)
R=
k
X
i=0
(−1)i k−1
Y
j=i
(+ (−j)I)Ni
i(+iI): (4.38)
It remains to express
Q=Q0Q1: : : Qk−1; (4.39)
as Q=SV−1, with S∈B and V a polynomial in . For this we need the following
Lemma 4.5. The operator Q in Eq. (4:39) acting on a vector h= (: : : ; h−1; h0; h1; : : :)T is given by
(Qh)n= k
X
i=0
(−1)i+k n
Y
j=n−k+i+1
(++j) i(n++i)
0(n++i+ 1)
hn+i; (4.40)
with i(n) as in Eq. (4:32).
Proof. The proof is by induction on k. For k= 1, we have that
(Q0h)n=−an
fn(0)−1
fn(0)
hn+hn+1; using Eq:(3:5);
=− anf˜ (0) n−1
(n+) ˜fn(0)
hn+hn+1; by Eq:(4:14);
=−(n++) 0(n+)
0(n++ 1)
hn+hn+1;
using Eqs:(2:1);(4:16);(4:32) for k= 1:
Suppose now that the result is true for k elementary Darboux transformations, we deduce it for
k+ 1 elementary Darboux transformations. Indeed:
(Qkh)n=−a(nk)
fn−1
fn
hn+hn+1; using Eq:(3:5) withf as in Eq:(4:22);
=−(n++−k)
(k)(n+ 1)(k+1)(n)
(k)(n)(k+1)(n+ 1)hn+hn+1;
Assuming Eq. (4.40) with Q as in Eq. (4.39), from the formula (4.41) for Qk, we compute that
the (n; n+i)th entry of QQk, for 06i6k, is given by
(QQk)n; n+i=Qn; n+i−1
−Qn; n+i(++n−k+i)
(k)(n+i+ 1)(k+1)(n+i)
(k)(n+i)(k+1)(n+i+ 1)
=(−1)
i+k+1Qn
j=n−k+i(++j)
(k)(n+i)(k+1)(n+i+ 1)
×[i−1(n++i−1)(k+1)(n+i+ 1) +i(n++i)(k+1)(n+i)]; (4.42)
where, in the last equality, we have used Eq. (4.32) and, by convention, we put −1(n) = 0. Thus, to
establish that Eq. (4.40) holds for QQk, one just needs to show that the term between the brackets
[: : :] in the last equality of Eq. (4.42) is
[: : :] =(k)(n+i)(k+1)
i (n++i); (4.43)
where (ik+1)(n) denotes the (k+ 1)×(k+ 1) determinant obtained by replacing k by k+ 1 in the denition of i(n), see Eqs. (4.32), (4.16) – (4.18).
We now establish Eq. (4.43) for 16i6k. The case i= 0 is trivial remembering that −1(n+−
1) = 0. Consider the (k+ 2)×(k+ 1) matrix
( ˜fn(+l−i−1)k−2+j)16j6k+2 16l6k+1
;
with j and l denoting respectively the line and the column indices. We take V (resp. W) to be the subspace of Rk+2 generated by the rst k columns (resp. all the columns) of this matrix. Applying
Lemma 4.4 to this choice of V and W, if we pick in (4.21) (i; j; l)≡(1; k+ 2−i; k+ 2), we get that
p2::: k+11:::k+2[−i::: k+2=p1:::k+2[−i::: k+12::: k+2+p2:::k+2[−i:::k+21:::k+1: (4.44)
Using the denitions (4.16) – (4.18) and (4.32) of the various determinants involved in Eq. (4.43), one checks easily that this relation amounts precisely to Eq. (4.44), i.e. it is identically satised. This concludes the proof of Lemma 4.5.
It follows immediately from Lemma 4.5, that, in terms of matrices, the operator Q takes the form
Q=SV−1; (4.45)
with
S=
k
X
i=0
(−1)i+kTi k−1
Y
j=i
(+ (−j)I)i(); (4.46)
By combining the results of this section we obtain
Theorem 3. Let be a positive integer and be arbitrary. Then; the (bi-innite) tridiagonal
matrix which is obtained after k elementary Darboux transformations starting from L(; ) as in
Eq. (4:1); with 16k6; is bispectral.
Proof. The result follows immediately by applying Theorem 1 to the factorization (4.10), with
; R; S and V as in Eqs. (4.37), (4.38), (4.46) and (4.47), using that =b(L(; )k) =zk. Given
any family of functions {fn(z)}n∈Z belonging to the two-dimensional space of bispectral functions of L(; ), which satisfy Eqs. (2.2) and (2.4), we get that the new functions
˜
fn= (Pf)n; (4.48)
with P as in Eq. (4.33), satisfy
B(k)f˜ n=
(k)
n f˜n; (4.49)
with
B(k)=b(R)b(S)z−k; (k)
n =0(n++ 1)0(n+) k−1
Y
j=0
(n++−j); (4.50)
0(n) as in Eq. (4.32) and b the bispectral anti-isomorphism dened in Eqs. (2.16), (2.17), (2:17′).
This establishes Theorem 3.
The functions ˜fn(z) that we obtain in Eq. (4.48) by taking f={fn(z)}n∈Z to be the Laguerre
functions introduced in Eq. (2.13), will be called the Krall–Laguerre functions. We remind the
reader (see Remark 4:2) that the matrix P in Eq. (4.48), involved in the Darboux factorization (4.10), depends on k-free parameters 0; : : : ; k−1 (equivalent to 0; : : : ; k−1 in Eq. (4.31)), when
we normalize the functions f(j) in Eq. (4.5) as in Eq. (4.13). Thus, the Krall–Laguerre functions
˜
fn(z) depend on the parameters (; ; 0; : : : ; k−1). When = 0, the Laguerre functions fn(z), n¿0,
reduce to the (monic) Laguerre polynomials. In this case, the ˜fn’s dened via Eq. (4.48) are also orthogonal polynomials (see Theorem 2); they are eigenfunctions of the dierential operator B(k) in
Eq. (4.50) and thus provide higher order generalizations of the Krall–Laguerre polynomials.
5. An illustrative example
In this last section, we spell out in detail the case =k= 2, that is the case of two elementary Darboux transformations performed from L(2; ). We hope that it will help the reader to have a better grasp of some of the key determinant formulas (4.16) – (4.18) and (4.32) in the previous section, that may have appeared a bit intimidating at rst sight. From Theorem 2, formula (4.25), it follows that this case leads, when = 0, to orthogonal polynomials with weight distribution
e−z+r
2(z)−r1
′
(z); (5.1)
Remembering Remark 4.2, formula (4.13), we can always pick (when k= 2) 0= 1, 1= 0, so
that 0 and 1 become the independent free parameters of the two successive elementary Darboux
transformations, equivalent to the parameters 0 and 1 in Eq. (4.31). It is easy to relate 0 and 1
to the free parameters r1 and r2 in (5.1). Indeed, when = 0, from Eq. (4.26) we get that
which, using Eq. (5.2) gives
r1= 20 and r2= 2(1−0): (5.4)
From Eqs. (4.32) and (4.16), we compute
0(n) =
and, from Eqs. (4.32) and (4.17):
1(n) =
Then, Eqs. (4.32) and (4.18) tell us that
2(n) =0(n+ 1): (5.7)
From Lemma 4.3, the tridiagonal matrix L2, obtained from L(2; ) after two elementary Daboux
transformations, has entries a(2)
n and b(2)n given by
From Eqs. (4.38) and (4.46), we nd that
R= (+ 2I)(+I)0()−(+I)N1(+I) +N22(+ 2I);
S= (+ 2I)(+I)0()−T(+I)1() +T22():
Applying the anti-isomorphism b dened in Eqs. (2.16), (2.17),(2.17′), remembering Eq. (3.34), this gives that
Pb ≡b(R) =0(B)(B+ 1)(B+ 2)−1(B+ 1)(A−B)(B+ 1) +2(B+ 2)(A−B)2; (5.9)
Qb≡b(S)z−2=0(B)(B+ 1)(B+ 2)z−2 −1(B)(B+ 1)(z−A−B−3)z
−2+
2(B)(z−A−B−3)2z
−2; (5.10)
with A and B as in (2.8), (2.9) with = 2.
Let now fn(z), n∈Z, be the Laguerre functions dened in Eq. (2.13) with = 2. Following Eq.
(2.24), we dene
˜
fn≡(Pf)n=
1
n
Pbfn; (5.11)
with P as in Eq. (4.33)
(Pf)n=fn−
(n+)1(n+)
0(n+)
fn−1+
(n+)(n+−1)2(n+)
0(n+)
fn−2;
and n the (n; n)th entry of in Eq. (4.37)
n= (n++ 1)(n++ 2)0(n+):
Theorem 3 guarantees that the new functions ˜fn(z) (the Krall–Laguerre functions with =k= 2), satisfy the pair of equations:
L2f˜=zf;˜
PbQbf˜n= (n++ 1)(n++ 2)0(n+)0(n++ 1) ˜fn;
(5.12)
with L2 the tridiagonal matrix with entries as in Eq. (5.8) andPb, Qb the dierential operators found
in Eqs. (5.9) and (5.10), i.e. L2 is again bispectral! Observe that in Eq. (5.12) the dependence in
is only involved in the entries of L2 and the eigenvalues of the bispectral operator PbQb via the
shift n7→ n+; the operator PbQb itself is independent of . Explicit evaluation of Eqs. (5.9) and
(5.10) gives, with D= d=dz,
Pb=
r2 1z6
12 D
10−r21(z−8)z5
3 D
9+· · · −2(3r
2z+ 2z−3r1)D−2;
Qb=
r2 1z4
12 D
10−r12(3z−14)z3
6 D
9+· · ·+ (: : :)D−(r
2+ 1); (5.13)
showing that the bispectral operator PbQb in Eq. (5.12) is of order 20. The question arises naturally
to determine the bispectral operator of lowest possible order.
We observe that any bispectral operator ˜B with eigenfunctions as in Eq. (5.11) and polynomial (in n+) eigenvalues (n+), satises necessarily
˜
with B as in Eq. (2.9) (with = 2). Indeed, if
for alln, which implies Eq. (5.14). We observe that in this example and several other ones one has
ring of bispectral operators ˜B =
ring of polynomials(n) (in n) such that(n)−(n−1) is divisible by 0(n):
We notice that, because of the factor 0(n++ 1)0(n+), the bispectral operator given in Eq.
(5.12) satises this condition. Also a similar rule applies (generically) in the case of the continuous– continuous version of the bispectral problem, with the discrete derivative replaced by the usual one, see [8].
Since in our example, 0(n) in Eq. (5.5) is of degree 4, this suggests that the lowest possible
order for (n) is 5, and the corresponding bispectral operator ˜B in Eq. (5.14) should be of order 10. One computes easily that (up to a constant)
(n) =r
(5.15), can be solved for an operator ˜B of order 10. Explicitly,
˜
When r1 = 0, the polynomials orthogonal with respect to the measure (5.1) become the
clas-sical Krall–Laguerre polynomials and the operator ˜B reduces to the original fourth order operator discovered by Krall [25].
6. For further reading
[19,21,22,26 –28]
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