Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 050, 14 pages
Automorphisms of Algebras and Bochner’s Property
for Vector Orthogonal Polynomials?
Emil HOROZOV †‡
† Department of Mathematics and Informatics, Sofia University,
5 J. Bourchier Blvd., Sofia 1126, Bulgaria
† Institute of Mathematics and Informatics, Bulg. Acad. of Sci.,
Acad. G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria
E-mail: horozov@fmi.uni-sofia.bg
Received January 26, 2016, in final form May 12, 2016; Published online May 19, 2016
http://dx.doi.org/10.3842/SIGMA.2016.050
Abstract. We construct new families of vector orthogonal polynomials that have the pro-
perty to be eigenfunctions of some differential operator. They are extensions of the Hermite
and Laguerre polynomial systems. A third family, whose first member has been found
by Y. Ben Cheikh and K. Douak is also constructed. The ideas behind our approach lie
in the studies of bispectral operators. We exploit automorphisms of associative algebras
which transform elementary vector orthogonal polynomial systems which are eigenfunctions
of a differential operator into other systems of this type.
Key words: vector orthogonal polynomials; finite recurrence relations; bispectral problem;
Bochner theorem
2010 Mathematics Subject Classification: 34L20; 30C15; 33E05
1 Introduction
Salomon Bochner [11] has classified all systems of orthogonal polynomials Pn(x), n = 0, 1, . . .
(with respect to some measure on the real line) that are also eigenfunctions of a second-order
differential operator
L(x, ∂x) = A(x)∂
2
x +B(x)∂x + C(x) (1.1)
with eigenvalues λ(n). Here the coefficients A, B, C of the differential equation do not depend
on the index (degree) n of the polynomial Pn(x). The orthogonality condition, due to a classical
theorem by Favard–Shohat is equivalent to the well known 3-terms recursion relation
xPn = Pn+1 + β(n)Pn + γ(n)Pn−1,
where β(n), γ(n) > 0 are constants, depending on n.1 Bochner’s theorem states that all polyno-
mial systems with such properties are the classical orthogonal polynomials of Hermite, Laguerre
and Jacobi. From now on we will replace the condition of orthogonality with respect to a measure
with orthogonality with respect to a non-degenerate functional on the space of the polynomials
of one variable C[x] with complex coefficients. In this setting we have to add to the classical
orthogonal polynomials also the Bessel polynomials.
Generalizations of Bochner’s result were suggested long ago by H.L. Krall [29]. He classified
all order 4 differential operators which have a family of orthogonal polynomials as eigenfunctions.
?This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applica-
tions. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html
1Here we use polynomials, normalized by the condition that the coefficient of their highest-order term is 1.
2 E. Horozov
Later many contributions in this direction were made by T. Koornwinder [28], L. Littlejohn [30],
J. Koekoek, R. Koekoek [27], etc.
In recent times there is much activity in generalizations and versions of the classical result
of Bochner, see, e.g., [17, 18, 20, 24, 25]. An important role in some of these generalizations
is played by ideas from the study of the bispectral problem, initiated in [16]. Even translating
the results of Bochner and Krall into this language already gives a good basis to continue
investigations [20]. It goes as follows. Let us introduce the function ψ(x, n) = Pn(x). If we
write the right-hand side of the 3-term recursion relation as a difference operator Λ(n) acting
on functions f(n) of the discrete variable n as
Λ(n)f(n) = f(n+ 1) + β(n)f(n) + γ(n)f(n− 1),
then the 3-term recursion relation can be written as
xψ(x, n) = Λ(n)ψ(x, n). (1.2)
This means that ψ(x, n) is an eigenfunction of the discrete operator Λ with eigenvalue x and of
a differential operator L(x, ∂x) with eigenvalue λn. Hence we can formulate the Bochner–Krall
problem as
Find all systems of polynomials Pn(x) that are eigenfunctions of a second-order difference ope-
rator Λ(n) (1.2), where γ(n) 6= 0, with eigenvalue x and of a differential operator L(x, ∂x) (1.1)
with eigenvalues λ(n).
In [21, 22] and several other papers the authors make use of Darboux transformations to
construct families of systems of orthogonal polynomials, that are eigenfunctions of even-order
differential operators. Some properties of these polynomial systems are different from the pro-
perties of the classical orthogonal polynomials. For example P. Iliev [24, 25] has studied the
algebras of operators that have Bochner–Krall polynomials as eigenfunctions. These algebras
have two or more generators, whereas the algebras corresponding to the classical OP are iso-
morphic to the algebra C[X].
Here we also use ideas found in the studies of bispectral operators but of different nature.
Before explaining these as well as the main results let us introduce one more concept which
is central for the present paper. This is the notion of vector orthogonal polynomials (VOP),
introduced by J. van Iseghem [35]. Let {Pn(x)} be a family of monic polynomials such that
degPn = n. Assume that they satisfy a (d+ 2)-term recursion relation, d ≥ 1
xPn(x) = Pn+1 +
d∑
j=0
γj(n)Pn−j(x) (1.3)
with coefficients γj(n) independent of x and γd(n) 6= 0. Then by a theorem of P. Maroni [31]
there exist d functionals uj , j = 0, . . . , d− 1 on the space of all polynomials C[x] such that
uk(PnPm) = 0, m > nd+ k, n ≥ 0,
uk(PnPnd+k) 6= 0, n ≥ 0,
for each k ∈ Nd+1 := {0, . . . , d− 1}. When d = 1 this is the notion of orthogonal polynomials.
In the last 20–30 years there has been much activity in the study of vector orthogonal polyno-
mials and the broader class of multiple orthogonal polynomials (see the cited references below).
Applications of d-orthogonal polynomials include the simultaneous Pade´ approximation prob-
lem where the multiple orthogonal polynomials appear [2, 3, 4, 5, 14]. Multiple orthogonal
polynomials play an important role in random matrix theory [4, 10]. Applications to integrable
systems and quantum mechanics are found as well [12, 13].
Automorphisms of Algebras and Bochner’s Property for Vector Orthogonal Polynomials 3
One problem that deserves attention is to find analogs of classical orthogonal polynomials.
Several authors [26, 34] have found multiple orthogonal polynomials, that share a number of
properties with the classical orthogonal polynomials – they have raising and lowering operators,
Rodrigues type formulas, Pearson equations for the weights, etc. However one of the features
of the classical orthogonal polynomials – a differential operator for which the polynomials are
eigenfunction, is missing. Sometimes this property is relaxed to the property that the polyno-
mials satisfy linear differential equation, whose coefficients may depend on the degree of the
polynomial.
The ideas from bispectral theory that we use, allow us to to construct new VOP systems,
having the strict property to be eigenfunctions of one differential operator. This is even stronger
than one of the research problems suggested in [34]. We notice that Hahn’s characterization,
while valid for the polynomials of Hermite type (Appell polynomials) is not shared by the new
VOP, constructed in Sections 4 and 5 (this was guessed in [34]). In these cases the relevant
property is a generalization of Sheffer’s one: There exist an operator Q(x, ∂) such that
Q(x, ∂)Pn(x) = µ(n)Pn−1.
Recall that Sheffer’s property corresponds to µ(n) = n and Q = ∂. But in the constructions in
the present paper µ(n) is a polynomial of degree higher than one and Q can be a higher-order
differential operator.
We are looking for polynomials Pn(x), n = 0, 1, . . . that are eigenfunctions of a differential
operator L with eigenvalues depending on the discrete variable n (the index) as in (1.1) and
which at the same time are eigenfunctions of a difference operator in n, i.e., finite-term recursion
relation (1.3) with an eigenfunction, depending only on the continuous variable x. This is
a (d+ 2)-order recursion relation with some d ≥ 1. Systems with these properties will be called
polynomial systems with Bochner’s property.
One of our results includes an extension of Laguerre polynomials. We construct systems
of vector orthogonal polynomials {Pn(x)} which are eigenfunctions of a differential operator
starting from a very simple bispectral situation. Our approach uses ideas of the bispectral
theory from [6] but does not use Darboux transformations, which is usually the case, see, e.g.,
[19, 21, 22, 24, 25]. We use methods introduced in [6] which we describe below. Also a well
known extension of Hermite polynomials (see [1, 15, 32]) is presented. The reason to repeat
this well known result is that our construction is a new one. Also the case of Hermite-like
polynomials is the simplest and illustrates the idea without going to computations.
Here is a sketch of the method explained on the example of Hermite-like polynomials. We
start with the Weyl algebra W1, i.e., the algebra of differential operators in one variable x
with polynomial coefficients. Another algebra R is spanned by the shift operator T in n,
Tf(n) = f(n + 1), its inverse T−1 and n. There is a very simple example of the bispectral
problem with ψ(x, n) = xn. It is a joint eigenfunction of the operators H = x∂x and the shift
operator T . In more details
Hψ(x, n) = nψ(x, n), Tψ(x, n) = xψ(x, n).
This defines an anti-involution b : W1 → R, b(H) = n, b(T ) = x, b(∂) = nT−1. If we perform
an automorphism of the Weyl algebra σ : W1 → W1 we can obtain another anti-isomorphism
b′ = b ◦ σ−1 : W1 → R. This will define a new bispectral problem by taking H ′ = b ◦ σ(H)
and Λ = b ◦ σ−1(x). In the case described briefly here we obtain Hermite polynomials from the
simplest nontrivial automorphism σ = eadB with B = −∂2/2. Using other automorphisms we
obtain systems of vector orthogonal polynomials in general, which are known under the name
Appell polynomial systems [1, 15, 32].
Instead of W1 we can use other algebras. For example we can take a suitable enveloping
algebra of sl2. This will give us Laguerre polynomials and some of their vector orthogonal
4 E. Horozov
generalizations. Using a different algebra in Section 5, we come to new classes of VOP. In fact
its first member is also known and can be found in [8, 9].
In [36] the authors use automorphisms of the Heisenberg–Weyl algebra to construct vector
orthogonal polynomials of Hermite and Charlier type. A nice feature of their construction is
that a connection to representation theory is underlined and heavily used. Our construction uses
only automorphism of algebras as defined in [6] and seems to be simpler and easier to apply
elsewhere.
Here we keep the exposition as simple as possible. We do not study properties of the poly-
nomial systems and leave this for future work. The construction of the present paper can be
performed in much more general situations of VOP, including their discrete versions, see [23],
matrix orthogonal polynomials, multivariable orthogonal polynomials, etc. This will be done
elsewhere.
2 Elements of bispectral theory
The following introductory material is mainly borrowed from [6]. Here we present it in a form
suitable for the continuous-discrete version of the bispectral problem.
For i = 1, 2, let Ωi be two open subsets of C such that Ω2 is invariant under the translation
operator T : n 7→ n+ 1 and its inverse T−1.
A complex analytic difference operator on Ω2 is a finite sum of the form
∑
k∈Z
ck(n)T
k,
where ck : Ω1 → C are analytic functions.
By B1 we denote an algebra with unit, consisting of differential operators L(x, ∂) in one
variable x. By B2 we denote an algebra of difference operators Λ(n, T ) with unit. Denote byM
the space of complex analytic functions on Ω1 × Ω2. The space M is naturally equipped with
the structure of bimodule over the algebra of analytic differential operators L(x, ∂x) on Ω1 and
the difference operators on Ω2.
Assume that there exists an algebra isomorphism b : B1 → B2 and an element ψ ∈ M such
that Pψ = b(P )ψ, ∀P ∈ B1. Assume that Ai and Ki are subalgebras of Bi such that b(A1) = K2
and b(K1) = A2. In our case K1 will be an algebra of polynomials in x and K2 will be an algebra
of polynomials in n.
A continuous-discrete bispectral function is by definition an element of M (i.e., an analytic
function) ψ : Ω1 × Ω2 → C for which there exist analytic differential operator L(x, ∂x) on Ω1,
analytic difference operator Λ(n, T ) on Ω2, and analytic functions θ(x) and λ(n), such that
L(x, ∂x)ψ(x, n) = λ(n)ψ(x, n), Λ(n, Tn)ψ(x, n) = θ(x)ψ(x, n) (2.1)
on Ω1 × Ω2. In fact, as we would be interested in VOP, we will consider only the case when
θ(x) ≡ x. We will assume that ψ(x, n) is a nonsplit function of x and n in the sense that it
satisfies the condition
(∗∗) there are no nonzero analytic difference operators L(x, ∂x) and Λ(n, T ) that satisfy one of
the above conditions with f(n) ≡ 0 or θ(x) ≡ 0.
The assumption (∗∗) implies that the map b : B1 → B2, given by b(P (x, ∂x)) := S(n, T ) is
a well defined algebra anti-isomorphism. The algebra A1 := b−1(K2) consists of the bispectral
operators corresponding to ψ(x, z) (i.e., differential operators in x having the properties (2.1))
and the algebra A2 := b(K1) consists of the bispectral operators corresponding to ψ(x, n) (i.e.,
difference operators in n having the properties (2.1)).
Automorphisms of Algebras and Bochner’s Property for Vector Orthogonal Polynomials 5
Below we present the differential-difference version of the general bispectral problem, suitable
in the set-up of vector orthogonal polynomial sequences, which are eigenfunctions of differential
operators. We are interested in the case when, for any fixed n, the function ψ(x, n) defining the
map b is a polynomial in x. We additionally assume that, for all n, polynomials ψ(x, n) are the
eigenfunctions of a fixed differential operator in the variable x and that, for any fixed x, the
function ψ(x, n) is an eigenfunction of a difference operator in n. We know that such a situation
occurs in case of the classical orthogonal polynomials.
Let B1 be the Weyl algebra W1 (spanned over C by x and ∂). To define another algebra, we
introduce three operators: the shift operator T acting on functions f(n) of the discrete variable n
by Tf(n) := f(n+1), its inverse T−1 shifting in the opposite direction, i.e., T−1f(n) := f(n−1),
and the operator n of multiplication by the variable n. We define the algebra R2 of difference
operators over C, spanned by T , T−1 and n with the obvious relations. LetM be a left module
over B1 andR2. In our case such a module is a linear space of bivariate analytic functions f(x, n),
where x is a continuous and n is a discrete variables.
Set L := x∂. Introducing ψ(x, n) := Sn(x) := xn, we get
LSn(x) = nSn(x), (2.2)
xSn(x) = Sn+1(x), (2.3)
∂Sn(x) = nSn−1(x). (2.4)
We see that (2.2) and (2.3) are providing an instance of a differential-difference bispectral pair
of operators. The equation (2.2) guarantees that ψ(x, n) is an eigenfunction of the differential
operator L, and (2.3) shows that the same function is an eigenfunction of the difference ope-
rator T . Equation (2.4) follows from the previous two; it is called the differentiation (or the
lowering) formula and is quite important.
We can define the map b : B1 → R2 as given by
b(∂) = nT−1, b(x) = T, b(L) = n.
(It is clear that the first two identities are sufficient to determine b completely. The last identity
is included because of its importance and convenience.) Then we put B2 = b(B1).
Next we will discuss how to construct new bispectral operators from already known ones.
First recall that for an operator L it is said that adL acts locally nilpotently when for any
element a ∈ B there exists k ∈ N, such that adkL(a) = 0.
The main tool for constructing bispectral operators in this paper will be the following simple
observation made in [6].
Proposition 2.1. Let B1, B2 be unital algebras with the properties described above and
b : B1 → B2. Let L ∈ B1 and adL : B1 → B1 be a locally nilpotent operator. Suppose that,
for any fixed n, eLψ(x, n) is a polynomial in x of degree n. Define a new map b′ : B1 → B2
via the new polynomial function ψ′(x, n) := eLψ(x, n). Then b′ = b(e− adL) and b′ : B1 → B2 is
a bispectral anti-involution.
3 Hermite-like polynomials
Observe that ad∂ acts locally nilpotently on the algebra B1. The same is true when we take
instead of ∂ any polynomial q(∂) without a free term. From this it follows that for any ele-
ment a ∈ B1, the series
∞∑
k=0
adk∂(a)
k!
6 E. Horozov
has only finitely many terms. For such a polynomial q(∂) we define the automorphism σ:=eadq(∂) .
Its action on L is explicitly given by
σ(L) = L+
∞∑
k=1
adkq(∂)(L)
k!
= L+ [q(∂), L] = L+ q′(∂)∂,
since the rest of the terms vanish.
Let us first compute the action of σ = eadq(∂) on the standard basis of B1. We have
σ(∂) = ∂, σ(x) = x+ q′(∂).
Next we define a new map b′ : B1 → B2 by b′ := b ◦ σ−1. We will introduce the new
function ψ1(x, n) defined via
ψ1(x, n) := P
q
n(x) := e
q(∂)xn = xn +
∞∑
m=1
(q(∂))mxn
m!
.
Due to the fact that q(∂) reduces the degrees of the polynomials it is clear that the latter series
contains only a finite number of terms. Hence it is a polynomial in x. (Here we use that q has
no constant term.) Define the operator L1 := σ(L). We already saw that
L1 = x∂ + q
′(∂)∂.
Lemma 3.1. The new bispectral anti-involution b′ satisfies:
b′(∂) = nT−1, b′(x) = T − q′
(
nT−1
)
, b′(L1) = n. (3.1)
Proof. We have b′(∂) = b ◦ σ−1(∂) = b(∂) = nT−1. In the same way we compute b′(x) =
b ◦ σ−1(x) = b(x − q′(∂)) = T − q′(nT−1). Finally for L1 we find b′(L1) = b ◦ σ−1(L1) =
b ◦ σ−1 ◦ σ(L) = b(L) = n. 
Theorem 3.2. The polynomials P qn have the following properties:
(i) they are eigenfunctions of the differential operator
L1 := q
′(∂)∂ + x∂; (3.2)
(ii) they satisfy the recurrence relation
xP qn(x) = P
q
n+1 − q
′(nT−1
)
P qn(x); (3.3)
(iii) the following differentiation formula (lowering operator)
∂P qn(x) = nP
q
n−1
holds.
Proof. Indeed, statement (i) that polynomials Pn(x) are eigenfunctions of the operator
L1 = q
′(∂)∂ + x∂
follows from the computation of the new bispectral involution b′ in Lemma 3.1. More exactly,
the third equation of (3.1) gives
(q′(∂)∂ + x∂)Pn(x) = nPn(x).
To prove recurrence (ii), observe the second equation in (3.1) provides
xPn(x) = Pn+1 − q
′(nT−1
)
Pn.
Alsothe first equation in (3.1) gives the differentiation formula ∂Pn = nPn−1, which set-
tles (iii). 
Automorphisms of Algebras and Bochner’s Property for Vector Orthogonal Polynomials 7
4 Laguerre-like VOP
We start with an algebra B1 spanned by the operators x, H = x∂ and B = x∂2 + β∂. The
commutation relations are
[H,x] = x, [B, x] = 2H + β, [H,B] = −B.
This shows that our algebra is an enveloping algebra of sl2 (not the universal one).
Next we define the algebraR2 of difference operators over C, as the one spanned by T , T−1, n.
LetM be a left module over B1 and R2. In our case such a module is a linear space of bivariate
functions f(x, n), where x is a continuous and n is a discrete variables. We define K1 to be the
algebra of polynomials in x and K2 to be the algebra of polynomials in n.
Let again ψ(x, n) := Sn(X) := xn. It is obviously nonsplit in the above defined sense (∗∗).
Define the map b by its action on the generators:
b(x) = T, b(H) = n, b(B) = n(n− 1 + β)T−1.
We define the algebra B2 to be the image b(B1) of B1 under the map b. Obviously the map
σ : B1 → B2 is an anti-automorphism.
Setting L := H we get
HSn(x) = nSn(x), xSn(x) = Sn+1(x), BSn(x) = n(n− 1 + β)Sn−1(x).
Using the above notation and following the scheme of the previous section, we now set up an
appropriate bispectral problem. We will use the algebras B1 and B2 introduced there.
Consider a polynomial q(X). For simplicity we assume that q(X) has no constant term. Let
us define an automorphism σ : B1 → B1
σ(A) = eadq(B)(A), A ∈ B1.
In the next lemma we compute the action of σ.
Lemma 4.1. σ acts on the generators as follows
σ(x) = x+ (2H + β)q′(B) + q′′(B)B + q′(B)2B,
σ(H) = H + q′(B)B, σ(B) = B.
Proof. In order to derive the first formula we need to compute the commutator
[Bm, x] = 2
m−1∑
j=0
BjHBm−1−j + βBm−1.
Notice that BjH = (H + j)Bj . From this we find
[Bm, x] = 2mHBm−1 + (m(m− 1) + βm)Bm−1,
which shows that
[q(B), x] = (2H + β)q′(B) + q′′(B)B.
In order to compute ad2q(x) we also need [q(B), H] = q
′(B)B, which yields
[q(B), Hq′(B)] = q′2(B)B.
This gives that
ad2q(x) = 2q
′2(B)B, ad3q(x) = 0.
The two other formulas are obvious. 
8 E. Horozov
The lemma shows that adq(B) acts locally nilpotently on the algebra B1, i.e., when applied
to any element A ∈ B1, the series
∞∑
k=0
adkq(B)(A)
k!
has only finitely many terms.
Let us define a new map b′ : B1 → B2 given by b′ := b ◦ σ−1 and a new function ψ1(x, n)
defined via
ψ1(x, n) := P
q
n(x) := e
q(B)xn = xn +
∞∑
k=1
(q(B))kxn
k!
.
It is clear that the latter series contains only a finite number of terms as the operator q(B)
reduces the degrees of the polynomials. (Here we use that q has no constant term.) Hence it is
a polynomial in x.
Below and further in the paper we also use without mentioning that
σ−1 =
∞∑
j=0
(− ad)jq(B)
j!
.
Define the operator L1 := σ(L). One can easily see that L1 = x∂ + q′(B)B.
Lemma 4.2. The new bispectral involution b′ on B′ satisfies:
b′(B) = n(n− 1 + β)T−1,
b′(x) = T − q′(b(B))(2n+ β)− q′′(b(B))b(B) + q′2(b(B))b(B),
b′(L1) = n.
Proof. We have b′(B) = b ◦ σ−1(B) = b(B) = n(n− 1 + β)T−1. In the same way we compute
b′(x) = b ◦ σ−1(x) = b
(
x− (2H + β)q′(B)− q′′(B)B + q′2(B)B
)
= T − q′(b(B))(2n+ β)− q′′(b(B))b(B) + q′2(b(B))b(B).
Finally for L1 we find b′(L1) = b ◦ σ−1(L1) = b ◦ σ−1 ◦ σ(L) = b(L) = n. 
Theorem 4.3. The polynomials P qn have the following properties:
(i) they are eigenfunctions of the differential operator
L1 := q
′(B)B + x∂;
(ii) they satisfy the recurrence relation
xP qn(x) = P
q
n+1 − q
′(n(n− 1 + β)T−1
)
(2n+ β)P qn(x)
+
{
−2q′′
(
n(n− 1 + β)T−1
)
+ q′2
(
n(n− 1 + β)T−1
)}
n(n− 1 + β)P qn−1(x);
(iii) the following differentiation formula (lowering operator)
BP qn(x) = n(n− 1 + β)P
q
n−1
holds.
Proof. All the statements follow easily from the computation of the new bispectral involution b′
in Lemma 4.2. Essentially the proof is the same as the proof for Hermite-like polynomials. 
Automorphisms of Algebras and Bochner’s Property for Vector Orthogonal Polynomials 9
5 Automorphisms generated by third-order operators
We start with an algebra B1 spanned by the operators x, H = x∂ and B = x2∂3 + αx∂2 + β∂.
The commutation relations are
[H,x] = x, [B, x] = W := 3H2 + 2αH + β, [H,B] = −B.
We again use the algebra R2 of difference operators over C, spanned by T , T−1, n and the
bimoduleM of bivariate functions f(x, n), where x is a continuous and n is a discrete variables.
Let Sn(x) := ψ(x, n) := xn. It is obviously nonsplit in the above defined sense (∗∗). Define
the map b : B1 → R2 by its action on the generators:
b(x) = T, b(H) = n, b(B) = n[(n− 1)(n− 2) + α(n− 1) + β]T−1.
The algebra B2 is the image of B1 under the map b.
Set L := H. Then
HSn(x) = nSn(x), (5.1)
xSn(x) = Sn+1(x), (5.2)
BSn(x) =
[(
n
3
)
+ α
(
n
2
)
+ βn
]
Sn−1(x).
We see that (5.1) and (5.2) are providing an instance of a differential-difference bispectral
pair of operators. Namely, we get that (5.1) guarantees that ψ(x, n) is an eigenfunction of
the differential operator L, and (5.2) shows that the same function is an eigenfunction of the
difference operator T .
Using the above notation and following the scheme of Section 3, we now set up an appropriate
bispectral problem.
Consider a polynomial q(X) and for simplicity we assume that q(X) has no constant term.
Define an automorphism of the algebra B1:
σ(A) = eadq(B)(A), A ∈ B1.
Let us compute the action of σ. Put R = 3(2H + 1) + 2α.
Lemma 5.1. σ acts on the generators as follows
σ(x) = x+Wq′(B) +Rq′′(B)B/2 + q′′′(B)B2
+
{
3q′′(B)B + (R+ 6)q′(B)/2 + q′2(B)B
}
Bq′(B),
σ(H) = H + q′(B)B, σ(B) = B.
Proof. To prove the first identity we will need to compute the commutator [Bm, x]. First we
have
[Bm, x] =
m−1∑
j=0
BjWBm−1−j .
Notice that BW = WB +RB. We also need BjR = (R+ 6j)Bj , from which we find
BjW = WBj +
j−1∑
k=0
BkRBj−k =
(
W +
j−1∑
k=0
(R+ 6k)
)
Bj = [W + jR+ 3j(j − 1)]Bj .
10 E. Horozov
Hence
[Bm, x] =
m−1∑
j=0
[W + jR+ 3j(j − 1)]Bm−1.
Then using the formulas for sums of powers we transform this expression into
[Bm, x] = [mW +m(m− 1)R/2 +m(m− 1)(m− 2)]Bm−1,
which shows that
[q(B), x] = Wq′(B) +Rq′′(B)B/2 + q′′′(B)B2.
Using the commutation relation [B,H] = B we compute
[q(B), H] = q′(B)B
and
[q(B),W ] = 3/2q′′(B)B2 +Rq′(B)B.
This gives that
ad2q(x) = {6q
′′(B)B + (R+ 6)q′(B)}Bq′(B).
Finally we find
ad3q(x) = 6q
′3(B)B2, ad4q(x) = 0.
This gives the first identity. The other two are obvious. 
The lemma shows that adq(B) acts locally nilpotently on the algebra B, i.e., when applied to
any element A ∈ B, the series
∞∑
k=0
adkq(B)(A)
k!
has only finite number of terms.
Let us define a new map b′ : B1 → B2 given by b′ := b ◦ σ−1 and the new function ψ1(x, n)
defined via
ψ1(x, n) := P
q
n(x) := e
q(B)xn = xn +
∞∑
k=1
qk(B)xn
k!
.
It is clear that the latter series contains only a finite number of terms as the operator q(B)
reduces the degrees of the polynomials. Hence ψ1(x, n) is a polynomial in x. (Here we use
that q has no constant term.) Define the operator L1 := σ(L). One can easily see that
L1 = x∂ + q
′(B)B.
In order to simplify the notations let us put
Q0(B) = −q
′′′(B)B2 +
{
2q′′(B)B + 3q′(B)− q′2(B)B
}
Bq′(B),
Q1(B) =
(
q′2(B)− q′′(B)
)
B
2
.
Automorphisms of Algebras and Bochner’s Property for Vector Orthogonal Polynomials 11
Lemma 5.2. The new bispectral anti-involution b′ satisfies:
b′(B) = n[(n− 1)(n− 2) + α(n− 1) + β]T−1,
b′(x) = T − q′(b(B))
(
3n2 + 2αn+ β
)
+Q1(b(B))(6n+ 3 + α) +Q0(b(B)),
b′(L1) = n.
Proof. Repeats the proof of Lemma 4.2 and will be omitted. 
We formulate the main results about the polynomial system P qn(x) in the next theorem.
Theorem 5.3. The polynomials P qn(x) have the following properties:
(i) they are eigenfunctions of the differential operator
L1 := q
′(B)B + x∂;
(ii) they satisfy the recursion relation
xP qn(x) = P
q
n+1 +
{
−3q′(b(B))
(
3n2 + 2αn+ β
)
+Q1(b(B))(6n+ 3 + α)
+Q0(b(B))
}
P qn(x); (5.3)
(iii) the following differentiation formula (lowering operator)
BP qn(x) = n[(n− 1)(n− 2) + α(n− 1) + β]P
q
n−1
holds.
Proof repeats that of Theorem 3.2.
6 Examples
In the examples below we construct the differential operator L1 that has the vector orthogonal
sets P qn(x) = eq(B)xn as eigenfunctions. In some cases we also write explicitly the finite-term
recursion relation.
Example 6.1. Set q(∂) = −1/k∂k. Then we obtain
L1 = x∂ − ∂
k.
Observe that, for k = 2, the operator −L1 is the standard Hermite operator. For k > 2,
−L1 appeared earlier in [33]. The recurrence relation for Pn(x) is given by (3.3). In the present
case it reads
xPn(x) = Pn+1(x) +
n!
(k − 1)!
Pn−k+1(x),
which agrees with the results in [33]. In general all the examples of Section 3 have been studied
thoroughly starting with the classical paper [1]. In the context of vector orthogonal polynomials
the results have been found in [15].
Example 6.2. Here we take B = x∂2 + β∂. Let us take the polynomial q(B) = B. In this case
we have
L1 = H + q
′(B)B = x∂2 + β∂ + x∂,
which up to change of the variables is the Laguerre operator.
12 E. Horozov
Example 6.3. The simplest new example is given by the polynomial q(B) = B2/2, B =
x∂2 + β∂. In this case we have
L1 = H + q
′(B)B =
(
x∂2 + β∂
)2
+ x∂ = x2∂4 + 2x(1 + β)∂3 +
(
β + β2
)
∂2 + x∂.
The 5-terms recurrence relation reads
xPn(x) = Pn+1 − n(n− 1 + β)(2n− 1 + β)Pn−1 +
n!
3!
(
n− 1 + β
3
)
Pn−3.
Here are the first few polynomials
P0(x) = 1, P1(x) = x, P2(x) = x
2 + β(1 + β),
P3(x) = x
3 + 3(1 + β)(2 + β)x, . . . .
Example 6.4. Let us take the polynomial q(B) = B, B = x2∂3 + αx∂2 + β∂. Then
L1 = H + q
′(B)B = x2∂3 + αx∂2 + β∂ + x∂.
The corresponding polynomials satisfy the 4-term recurrence relation
xP qn(x) = P
q
n+1 + (3− 2α)P
q
n(x)− 3n[(n− 1)(n− 2 + α) + β]P
q
n−1
− n(n− 1)[(n− 1)(n− 2 + α) + β][(n− 2)(n− 3 + α) + β]P qn−2.
This example has been produced in [7] within the context of vector orthogonal polynomials
which are generalized hypergeometric functions.
Example 6.5. Let us take the polynomial q(B) = B2/2, B = x2∂3. Then
L1 = H + q
′(B)B =
(
x2∂3
)2
+ x∂ = x4∂6 + 6x3∂5 + 6x2∂4 + x∂.
This example seems to be new as most of the examples in Sections 4 and 5.
The polynomials satisfy a 7-term recurrence relation which we skip as it is not much simpler
than the corresponding general formula (5.3). Here are the first few polynomials:
P0(x) = 1, P1(x) = x, P2(x) = x
2,
P3(x) = x
3 + 3 · 23x, P4(x) = x
4 + 4 · 33 · 22x2, . . . .
Acknowledgements
The author is deeply grateful to Boris Shapiro for showing and discussing some examples of
systems of polynomials studied here and in particular the examples from [33]. Without this
probably the project would have never be even started. Also Plamen Iliev made valuable com-
ments on some results generalizing Laguerre polynomials, which helped me to place my work
among the rest of the research. Milen Yakimov pointed out several errors. Many thanks go to
the referees and the editor who pointed out a number of incorrect formulas and misprints and
thus helped me to improve considerably the initial text. The author is grateful to the Mathe-
matics Department of Stockholm University for the hospitality in April 2015. Last but not least
I am extremely grateful to Professors T. Tanev and K. Kostadinov, and Mrs. Z. Karova from
the Bulgarian Ministry of Education and Science and Professor P. Dolashka, who helped me in
the difficult situation when I was sacked by Sofia university in violations of the Bulgarian laws2.
2See EMS Newsletter, p. 71: http://www.ems-ph.org/journals/newsletter/pdf/2015-12-98.pdf.
Automorphisms of Algebras and Bochner’s Property for Vector Orthogonal Polynomials 13
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