2. Separated Operators of Basis Transformation
Let
be a n-dimensional vector space with basis
vectors
. The linear operator that transforms
these basis to another basis
,
normally is defined as a unique linear operator
in
the matrix form. In present theory we define a set of linear operators
;
each one acts separately on the corresponding basis as follows:
The result of the action of operator
and the set of
on the initial basis are the same, but
as separated basis transformations allow to choose
a wide range of
operators whose overall transformations are
equivalent to the operation of
. On the other hand, in many problems such as
Rodrigues type formulas the separated basis transformation
are more accessible than overall operator
. In the context of differential operators,
could be regarded as the operators that transform
the initial basis (monomials) in polynomial space to another basis as for
example we observe in Rodrigues formula for Laguerre polynomials as the
solutions (eigenfunctions) for Laguerre differential equation:
This equation can be interpreted as the
transformation of initial basis
to new basis
(Laguerre polynomials) by the action of the
operator
Respect to Rodrigues formula, for all related
differential equations such as Legendre, Chebyshev and Bessel Equations, there
are separate and independent operators for each basis. Therefor we can apply a
set of operators instead of a unique operator to transform the initial basis in polynomials
space. This method obviates the need to find the unique linear operator with the same action on all initial bases. We will
show the relation between and in proposition 2.1 after defining the projection
operators as follows.
Let introduce the projection operators
by the definition:
Where
is a vector expanded as:
From (2) we have: ; as the main condition for projection operators ( is identity operator).
Remark 1
projection operators defined in (2), are
linear operators.
We show that by basis transformations according to
(1), the projection operators
are transformed as:
Theorem 2.1.
the generalized form of
projection operator under separated basis transformations is:
Proof
.
respect to
basis transformation
and
(2) we have:
Again, with substitution
(5) reads as:
It is valid for all
. with identity
, equation (6) becomes:
As we expected.
Proposition
2.1.
The
transformation of projection operator defined in (4) is equivalent to a
similarity transformation
Proof
.
expansion of the
first two terms on left side considering
yields:
Therefor we have the right side of (8).
Equation (8) implies the similarity transformation
of under the basis transformation made by operator . Therefor the operator is the linear operator for transforming all basis. Operator also yields transformations of all linear operators in vector space under basis transformation by the similarity transformation
Remark
2
Equation (4) meets the projection operator
conditions:
The action of
and
on a basis
is the same.
This implies that the action of and on is equivalent.
b) Respect to (4) and (9) we conclude:
Middle terms in right side of (7) reduce to
identity operator and thus:
Recalling
and (4) we obtain:
This proves the idempotency of i.e.,
Where denoted as posterior probability analogy.
Equation (4) is the unique formula for
and other forms in spite of their validity for
satisfaction of projection operator conditions i.e., equation (3), are not the
right candidates. As an example, we may propose this formula for
:
It is straightforward to investigate that this
definition is compatible with conditions (3) but if we multiply both sides by
we obtain:
Respect to (1) and (2) we get:
One of the solutions results in a false outcome:
Proposition 2.2. The product of
projection operators is associative.
Proof.
Let projection operators and correspond toand as transformation groups of coordinates i.e.
And
Substitution of first relation into above equation
results in:
After vanishing of two central terms:
This is compatible with (4) by replacing with . Therefore, the corresponding projection operator
for two consecutive transformation and is equivalent the projection operator of transformation.
Remark 3
As is proved, the operators and are equivalent operators. Respect to equation (9),
the projection operator transforms as a similarity transformation under
the action of operator , therefor the initial basis should be transformed
by this operator and consequently and are equivalent.
We show the identity (4) is also valid in function
space, where the linear projection operators are defined.
Proposition
2.3.
Let
be
a n-dimensional function space over the real field
with
a set of orthogonal basis functions
and
inner product defined on a closed interval
.The
same definition in (2) can be applied on these basis
Where
(14)
Are the coefficients in expansion of square
integrable function in the basis calculated by inner product of and over the interval .
Proof
.
with the
identity (4) we conclude:
Then we have:
Respect to (13) and (14) we have:
Regarding (1) we can choose
as transformed basis that results in:
With the definition (2) of projection operator we
have:
Therefor the equation (16) respect to (14) reads
as:
So, the identity (9) is valid for operators in
function spaces.
Proposition
2.4.
Differential
operators with certain eigenvalues and eigenfunctions can be linearly expanded
by their projection operators.
Proof.
Let the
differential operator
is
characterized by eigenfunctions relation:
The eigenfunctions are linearly independent and are the basis
vectors, i.e.: .
Where
is
the projection on
i-th subspace, then by the identity:
The validity of this equation for all
yields:
That proves the proposition.
Theorem
2.2.
Let
the initial basis
correspond
to some set of linearly independent non-homogenous polynomials such as the
regular bases
.
After transforming the bases by equation
to
new bases
which
correspond the new linearly independent polynomials
,
if
denoted
as the differential operator with
or
equivalently
(n-th
exponent of x) as its eigenfunctions (or eigenvector), then the corresponding
differential operator
with
eigenfunctions
can
be obtained by the relation:
Where
are
eigenvalues of
.
Proof.
Respect
to equation (19) the expansion of
in
terms of
reads
as:
Where
are
projection operators onto the i-th subspace ( i.e.,
).
Substitution of
in
equation (21) by equation (4) results in:
This proves the theorem.
Projection operators in terms of resolvents
Associated to any differential operator in Hilbert
space there are projection operators in terms of their resolvent
Using (21), (22) we obtain:
For
a unique transformation
for
all
we
get:
with
expansion of resolvents
as
a Neumann infinite series (polynomial), it is proved that the corresponding
differential operator after action of operator
on
the base functions
as
defined in proposition 2.1, can be presented by a similarity transformation:
Example
2.1
.
Eigenfunctions
of the differential operator
could
be found as
.
Transforming by
,
The resulting corresponding differential operator respect to proposition 2.1
after substituting
reads
as:
Action
of this operator on
gives:
Thus, the eigenfunctions of this operator are as expected. Because of the similarity relations
of operators and , their eigenvalues are identical.
Theorem
2.3.
Let
the linearly independent monomials
and
of
polynomials
and
of
degree
are
connected by the operators
as
defined in equation (1) i.e.,
Denote
as
projection operators that project functions of variable
on
basis
with
the definition of equation (2)
Then
the operator
acts
as
umbral
composition on
polynomial
.
Proof.
Let
expand
in terms of monomial basis
Then action of
on
gives
This implies that the action of
on
replaces the monomials
with
while the coefficients
in the expansion remains unchanged. This means
that the operation of
is equivalent with umbral composition by the
definition
This definition coincides the action of on. Application of this theorem for finding the
generating function of Hermite polynomials has been shown in section 3.1.
Forbenius covariant of operators
For other representation of projection operator in
terms of differential operator we apply the Forbenius covariants [7] as
projection operators (matrices) which are the coefficient of
Sylvester's formula.
For a differential operator in polynomial space, the projection operator
on the one-dimensional eigenfunction subspaces are given by
These operators act on the functions in function
space and yields their projections on basis which are the eigenfunctions of with corresponding eigenvalues
Similarity transformation
Respect to equation (9) and related proposition, if
we substitute
with
respect to the identity,
we have:
This equation is a similarity transformation of
under the operator
. This similarity transformation corresponds to the
basis transformation
. Actually,
as an operator
transforms all basis
to
and corresponds the coordinate transformation
matrix. From this equation we can deduce similarity transformation for other
operators provided that the operators in similarity transformation have common
eigenvalues. Therefor the differential operators with
identical eigenvalues
could be related by similarity transformation. As an example, differential
operator
with basis (eigenfunction)
transforms to another differential operator
with eigenfunction
after the basis transformation
. Therefor we have the similarity transformation:
If all
are the same namely
, (34) will be reduced to:
In these cases that the single operator transforms
all bases, the exact closed form of related differential operator could be
derived by this method. However, for cases with separate , the validity of the retrieved differential
operator relies on the action on the first two polynomials as
we
show in next sections. The following example clarifies the method.
Example
2.2
:
Let the vector space
spanned by the linearly independent basis
which are the eigenfunctions of operator
.If these basis transforms to the new set of basis
by multiplying with
i.e.,
then the corresponding operator with these new
basis as its eigenfunctions could be obtained by (34). In this case
. Thus, the equation (34) reduces to:
The term
is not just the derivative of
, but an operator that is equal to:
Then we have:
The eigenfunctions of this operator are
with eigenvalues
n as expected. It is
noteworthy to note that the expression for probabilist’s Hermite polynomial
with the definition:
Differs from , because in this definition the term is not an operator but merely the derivative of .
It is easy to prove that any function of
can be expanded in terms of
as follows:
Separated basis transformation method based on
Forbenius covariants
Another approach to find
in terms of
and
is to apply the Forbenius covariant operators as
projection operators as mentioned in (32).
These operators are projection operators onto the
k-th one-dimensional sub-space (basis) [8].
substituting
these projectors in equation (34) results in
denoted as the dimension of function or polynomial
space.
This method in comparison with previous methods are
more applicable because the calculation of inverse of a product of differential
operators is easier than other methods.
In the following sections we introduce an
applicable method to find
in terms of
. Taking into account the equation (22) we have:
If all
are the same i.e.,
, then (37) reduces to
The condition of identical eigenvalues for and is not necessary in equation (37) and the case of
identical eigenvalues are special case of this equation. we apply this equation
restricted to the first two polynomials i.e., two-dimensional polynomial space.
Substitution of in (37) by Forbenius covariants (35) yields an
applicable method as we will show in examples. It is noteworthy to recall that
the term stands for a linear operator (equivalent to a
matrix) that transforms the basis of polynomials space to another basis. For
example, it transforms basis to Hermite polynomial as new linearly independent basis by the
techniques that is presented in next section.
Rodrigues’ formula as a special case of separated basis transformation
In this section we prove the compatibility of
Rodrigues’ formula with our presented techniques and show that substitution of in equation (43) by Rodrigues’ formula
transformation, yields the corresponding differential operators and equations.
Proof
Due to the presented theory, we showed that if the
bases
of a vector space
which are the eigenfunctions of differential
operator
, are transformed separately by operators
, the transformed differential operator obeys the
equation (43) i.e.,
We check the basis transformation by Rodrigues
formula [3]:
Where defined by the relation with as a polynomial of first degree.
If we choose monomial
as the original basis of vector space:
The Rodrigues formula could be chosen as the action
of operator:
On these basis. Therefor it is a special case of
separated basis transformation.
The suitable operator with eigenfunctions
can be presented as
Where
denoted
as the derivative of
The
should
replace
in
(35):
Respect
to (43) by replacing
by
Rodrigues formula and
by
equation (46) we get:
Let
then
we obtain:
Taking into account the 2-dimensional space, and
using (44) and (47) we have:
If
we assume
(
as a crucial assumption in Rodrigues formula) this equation reduces to:
Acting
both side on
as
the second eigenfunction of
,
we have:
The
term
equals
,
thus:
The
term
will
be a constant
,
thus:
This implies that Rodrigues formula gives the
solutions (or eigenfunctions) of the differential operator
and related differential equation up to a constant
coefficient
. i.e.,
The following examples clarify this technique for
some polynomials.
Example 2.3: Laguerre differential equation
Let we intend to find the differential equation
which corresponds to a set of linearly independent polynomials in variable . For example, we are given a few first Laguerre
polynomials i.e., and we know the operator that maps the standard
basis to Laguerre basis i.e., operator presented in
(42).
We can recover the corresponding Laguerre
differential equation (operator) via the formula:
Proof in 2 dimension (first 2 polynomials)
We restrict calculation in 2-dimensional polynomial
space with basis .These polynomials are transformed by to Laguerre polynomials in the same dimension
i.e., . Thus, the corresponding operator will be transformed to Laguerre differential
operator by the equation (43).
Substitution of
by equation (42) and taking
as the eigenvalues of Laguerre differential
equation in 2-dimensional space of polynomials and replacing projection
operators
for basis
by equation (35) into equation (37) results in:
We have
and
. Then equation (44) reduces to:
By equations (35) and (42) we obtain:
And:
(
)(
(57)
If
we denote the
as
,
we can reduce the equation (46) as follows:
Is
the linear operator which transforms the basis
to
Laguerre basis
and
vice versa If we restrict the action of operators to 2-dimensional polynomial
space. Therefor we have:
If
we act both sides of (47) on another basis
we
get as well:
Or
briefly:
(61)
(Note
that
The
equation (61) implies that the action of both operators
and
on
basis
are
identical and therefor the simplest form of operator
which
its eigenfunctions are Laguerre polynomials and its related transformation
operators are
,
reads as:
This is the exactly the Laguerre differential
equation with positive eigenvalues, i.e.:
Action of this operator on the first basis i.e.,
“1” gives 0 as the first eigenvalue and therefor the required conditions for
validity of this differential operator are met.
Proof in 3 dimension (first 3 polynomials)
In 3-dimension with basis
of Laguerre polynomial and
of original basis, considering eigenvalues
the
reads as:
Here
denotes
the last term in (64). Acting both side on basis
results
in:
Respect
to (60) and the identity
we
have:
In
this dimension
can
be find as:
As the Laguerre differential operator.
Example 2.4: Hermite differential equation
The same technique could be applied to derive
Hermite differential equation by the formula (37). Because all
that transforms basis
to Hermite polynomials are equal to
as is shown in (41), after getting
by (35) and substitute them in (37) we have:
Respect
to
we
get:
Expanding
the sum for eigenvalues
and
substitution of
by
(41)we have:
From
(35) we calculate
as:
We
know
are
the eigenfunctions of
,
thus by
we
have:
This
equation can be interpreted as a similarity transformation that maps
into
after
basis changes. This will be hold just for the cases that eigenvalues are common
between
and
as
we see in Hermite and Laguerre differential equations.
Expansion
of
and
results
in:
In
the 2-dimensional space of polynomials the orders higher than 2 for
will
be omitted, as it could be verified by action of both side on basis
.
By omitting the higher orders, we obtain:
Omitting
results
in:
This is the well-known Hermit probabilist's Hermite
differential operator with Hermite polynomial as its eigenfunctions and
positive eigenvalues 0 , 1 , 2 , …. as its eigenvalues.
Example 2.5: Legendre differential equation
For Legendre polynomials we have:
That
transforms the basis set
to
Legendre polynomials. We can choose the appropriate operator
whose
eigenfunctions are these basis. Simply we write:
Eigenfunctions
of this operator are members of the set
.
The
transforming operator is
In
this case the eigenvalues of
and
(Legendre
differential operator) are not identical and therefor the similarity
transformation is not valid. However, we can apply the equation (37) after
determining the
from
(35).
For
calculating
in
2-dimension, we have:
From equation (37) and (63) we get:
Where
.
By
action of both sides of (75) on basis
as
the second basis of Legendre polynomials in two dimension, we have:
Respect
to
,
(65) reads as:
Expansion of (77) reads as:
Which
is the Legendre differential operator with positive eigenvalues
.
3. Hermit, Laguerre, and Legendre differential operator as Cartan subalgebra of
and
Let
denote
the linear transformation that maps vector space
onto
itself. In this section we present isomorphic Lie algebras to
defined
by
module
on vector space
which
is a linear map
defined
by
that
preserves the commutator relations of
algebra
[4,8].
This
representation is
module
on vector space
.
First,
we review the structure of irreducible vector field representation of
.The
generators of this algebra in matrix representation are as follows:
The
commutation relations for this representation of
are:
Let
define a representation of
as
its module on
that
preserves commutation relations by differential operators as its generators:
With the similar commutation relations
The
Cartan sub-algebra
produces
a decomposition of representation space:
are
the eigenspace (eigenfunction) of generator
as
Cartan sub-algebra of
and
provide the solutions to the related differential equation.
In
present paper the eigenspaces
are
one dimensional and coincide the basis of polynomial space. These basis are
called
weight vectors. For a finite dimensional representation there is
a
highest weight
that
determines the dimension of representation space by
.
As an example, the Cartan subalgebra of
can
be represented by
with
as
its
weight vectors (eigenfunctions) and integer
as
eigenvalues. Due to the properties of
,
the operator
acts as
lowering
operator
and
as
raising
operator
.
The action of these operator on representation basis (eigenfunction) of
lowers
or raise the power of
.
In
the following sections we will construct a set of isomorphic Lie algebras to
based
on differential operators of Hermite, Laguerre and Legendre equations
whose Cartan
sub-algebras are Hermit and Laguerre differential operators. These algebras
could be derived by similarity transformations (conjugation) of generators of
defined
in equation (79). The similarity transformation is achieved by the transforming
operator by which the original polynomial space basis transforms to the deemed
polynomial i.e., Hermite, Laguerre and Legendre polynomials as transformed
basis. These operators could be derived from Rodrigues’ formula as has been
shown in previous examples. For each algebra there exist a set of lowering and
raising operators that derives the recursion equations for related polynomials.
3.1. Associated Lie Algebra of Hermite Differential Operator
We
search for a Lie algebra
isomorphic
to
algebra
with generators to be defined based on Hermite differential operators
.
Here we apply the transformation operator
as
described in (41) for Hermite polynomials to derive similarity transformations
(conjugation) of
bases
as follows:
,
,
(80)
Equations (69) are the similarity transformations of Lie algebra
,
that
results in an algebra with basis
isomorphic
to
.
Respect to (55) this equation reduces to:
Where
as
proved in (57), denoted as Hermite differential operator.
Since
the operator
is
commutable with both
and
,
we have:
To calculate this generator, first we know from
(57) that:
Because
commutes
with
we
obtain:
Or
:
With:
(86)
Therefor
we have:
Multiplying this with itself results in:
With substitutions, equation (84) reads as:
Then
the list for generators of this representation of
is:
The Cartan subalgebra of this algebra is .
Clearly these generators span the Lie algebra
isomorphic to
, which is a representation for an isomorphism of
. The commutation relations can be checked as:
For
,
first we note:
,
and we use
instead
without
any change in commutator result. Thus, we have:
Due
to the identity:
The
equation (80) becomes:
Substitution
of
by
gives:
Replacing
operator
with
its equivalence
results
in:
This
proves the isomorphism of the Lie algebra
with
basis
with
Lowering and Raising operators of Hermite Polynomials and its Generating function
In this section we introduce the raising and
lowering operators of Hermite polynomials which act on vector space
representation of . We denote raising and lowering operators as and respectively. These operators act on the weight
vectors which are eigenfunctions of or i.e., the Hermite polynomials . As an example, for Lie algebra the following relations could be considered.
- (1)
Due to the properties of
algebra the generator
acts as a lowering operator
. This implies that:
- (2)
Consecutive action of the
and
generators on the eigenfunction
of
(i.e., the Hermite polynomial of degree
) results in lowering of polynomial degree. Respect to (81):
This means that the operator acts as a lowering (ladder) operator in the subspaces spanned by the Cartan subalgebra of .
- (3)
The raising operator can be derived from equation (85) and (86):
If we act the right side of (94) on a Hermite
polynomial of degree respect to equation (41)
Thus (94) and (95) yields:
Therefor the operator acts as raising operator in the associated vector space spanned by .
- (4)
If this method be repeated for operator, we have:
Taking into account (95) and (96) we deduce:
Clearly the operator acts as a raising operator .
The results of this section can be used to derive
recursive relations for Hermits polynomials as follows:
Any combination of operators involved in
(92),(93),(95),(96) and (97) results in a recursive relation for Hermite polynomials.
- (5)
The generating function of Hermit polynomial can be derived by a method based on theorem 3.2 as follows.
By expansion of
and acting the operator
defined in (41) on it and taking into account the
umbral property of
proved in theorem 2.3. We have
Recall that the
are eigenfunctions of the operator
Therefor we can replace
by
in equation (*)
This yields the Hermite polynomial generating function.
3.2. Associated Lie Algebra of Laguerre Differential Operator
For Laguerre polynomials the similarity transformation of the original basis of
will be obtained by operators in equation (42):
For global transformation respect to the definition, we have:
These generators construct a Lie algebra
isomorphic to both
and
. Replacing
respect to (68) we get:
For
due to equation (36), substituting
by
and
by
simply we obtain:
Because of complex structures of
and
, we calculate the raising operator by recursive relation:
We use the operator
as a lowering operator
. Substitution of
by
gives rise to:
Multiplying both side from the left by
:
Action of left side on
equals the derivative of
, then we have:
The left side should be replaced by its operator equivalent i.e.,
For
we need
:
This operator acts as raising operator. Eventually for representation of
in basis of Laguerre polynomial and related differential is an algebra
with generators:
To prove the isomorphism of
and
first, we calculate the commutation relation
:
We know: (102)
Because
, after substitution in(100) we have:
Or:
This is compatible with algebra.
Respect to (101) and (103), in second term, substitution of
by
yields:
Replacing second term of (***) by this, yields:
Replacement of
with relations of
gives:
This proves isomorphism of and as expected.
Lowering and Raising operators of Laguerre Polynomials and its Generating function
Applying the method used to derive lowering and raising operators for Hermite polynomial could be repeated for Laguerre polynomials too. Respect to the properties of Lie algebra
, the generator
acts as lowering operator
and
acts as raising operator
on the weight vectors
which are the eigenfunctions of
or
:
The action of
on
is also a lowering operator:
To derive raising operator due to the equation (100) we have:
Action of both side on
gives:
Thus, the operator acts as the raising operator in weight vector space of Laguerre polynomials.
Proposition 3.1
The generating function of Laguerre polynomial is derived by projection operator method.
proof
due to umbral properties of operator
, as we proved in theorem 2.3, we have:
Substitution the series in
powers with
and the identity
gives
If denoted as extractor coefficient operator for , Then the term is equivalent to
Expansion of the right side in terms of
with some algebra results in Laguerre generating function
3.3. Associated Lie Algebra of Legendre Differential Operator
The main difference between Legendre differential operator and Hermite or Laguerre differential operator is its eigenvalues. For Hermite and Laguerre differential operators the eigenvalue are the same as the eigenvalues of original differential operator . The eigenvalues of are integers
Correspond to eigenfunctions
. The Hermite and Laguerre differential operators have the same eigenvalues and therefore we can apply the similarity transformation
to derive both operators from
. Note that operator
is defined specific for each differential operator. For Legendre differential operator the eigenvalues are
which differs from eigenvalues of operator
whose eigenvalues are integers
and eigenfunctions are
. In this case we alter the original operator
to turn the same eigenvalues
. This allows us to use similarity transformation
to construct Legendre associated Lie algebra isomorphic to
. Let to add
to
and act the result on the original basis
Therefor we choose
for similarity transformation of the form
. Now we search for a Lie algebra
isomorphic to
algebra with generators to be defined based on Legendre differential operators
. We define the following generators for Lie Algebra of Legendre Differential Operator.
The generators
,
,
are different from
,
,
defined for
in previous sections. These operators are defined to be compatible for original basis
. An isomorphic algebra to
with generators
,
,
represented as
The commutation relations of these basis are:
For
we use the identity
With these commutation relations, respect to Jacobi identity we have
This proves that generators
,
,
gives an isomorphic algebra to
. Based on these basis and conjugation them with operator
which is defined for Legendre polynomials in equation(71), we could derive its adjoint algebra with basis that are formed by Legendre differential operator. Due to (34) and common eigenvalues of and
and
(not be confused with
for Laguerre differential operator) we have
For another basis it is required to calculate
. The action of this operator on Legendre polynomial
gives
This implies that
acts as raising operator and is equivalent to
This equation and (113) gives
Or (114)
For
respect to (**) and (*) we have
Thus, the set of generators for Lie algebra of Legendre differential operator are as follows
3.4. Adjoint representation of based on Hermite differential operator
An appropriate representation of
algebra presented as 9]:
The commutation relations of these generators will be unchanged after omitting the imaginary
from
and
yields a representation of
with commutation relations of equation (79):
The adjoint representation of elements of this Lie algebra, can be derived by conjugation with any element of the group
:
The element
could be derived by exponential map of generators of
:
assume
and
, then the adjoint representation elements will read as:
Respect to equations (81) to (88):
The eigenfunctions of
as Cartan subalgebra of
are
. After conjugation with
, the adjoint representation’s Cartan subalgebra will be
with eigenfunctions or weight vectors
. The transformation of
to
, respect to (39) is given by the relation:
Therefore, the conjugation of generators of algebra by an element group , results in an isomorphic adjoint algebra that its Cartan subalgebra’s weight vectors (eigenfunctions) could be derived by action of the same group element on the eigenfunctions of the original Lie algebra i.e., .
If we choose the exponent of generator
as group element
we have:
Due to Example 2: This implies that the weight vectors of adjoint algebra should be . And can be verified by the action of on .
For
we get:
And
Thus, the eigenfunctions of this operator would be .
3.5. Representation of and Hermite differential operator
Let introduce the basis
,
,
of
given by
With commutation relations
These commutation relations coincide the complexified algebra of that is the same as complexified .
Comparing these basis with the generators of
presented in (78) reveals the relations
Conjugation of these basis with an element of the group
gives the adjoint representation of
. Let use the operator introduced in (41) to derive Hermite polynomials from monomials
. The similarity transformations
Substituting the basis
by (78) and (79) gives