Hermite polynomials are among the most applicable special functions. These polynomials arise in diverse fields of probability, physics, numerical analysis, and signal processing. As an example, in quantum mechanics, eigenfunction solutions to quantum harmonic oscillator are described in terms of Hermite polynomials. Bivariate Hermite polynomials are useful in algebraic geometry and two-dimensional quantum harmonic oscillator [3,4,5]. Respect to the field of applications Hermite polynomials in one variable are divided into probabilist and physicist versions. In present paper we focus on one and two dimensional probabilist Hermite polynomials. We prove the symmetries of associated differential equations are compatible with $\mathfrak{s}\mathfrak{l}\left(2,R\right)$ algebra. By introducing isomorphic Lie algebras whose Cartan sub-algebras are the Hermite differential operators, applying Baker-Campbell-Hausdorff formula yields new relations for one variable and bivariate Hermite polynomials. Without exception, all known generating functions for Hermite polynomials contains a factorial term in denominator. We introduce a new generating function without factorial denominator.

Probabilistic Hermite polynomials, presented as: $H_{en}$ are the solutions to Hermite differential equations: where $D$ denoted as $\frac{d}{dx}$ and ${\mathbb{D}}_H$ as Hermite differential operator. This is an eigenvalue problem with positive integer eigenvalues $n$.

The equation (1) is the transformation of basis$(1,x,x^2,x^3,...)$ under the action of operator $e^{\frac{{-D}^2}{2}}$ which is compatible with Rodrigues’ formula and results in probabilistic Hermite polynomials $H_{en}$. The monomials $x^n$ expand a polynomial vector space $\mathbb{V}$. The operator$e^{\frac{{-D}^2}{2}}$ changes the basis $x^n$ into the basis $H_{en}$. Let $\mathfrak{g}\mathfrak{l}\left(\mathbb{V}\right)$ denote the linear transformation that maps vector space $\mathbb{V}$ onto itself. We present isomorphic Lie algebras to $\mathfrak{s}\mathfrak{l}(2,R)$ defined by $\mathfrak{s}\mathfrak{l}(2,R)$ module on vector space $\mathbb{V}$ which is a linear map $\varphi$ defined by $\varphi :\mathfrak{s}\mathfrak{l}\left(2,R\right)\to \mathfrak{g}\mathfrak{l}(\mathbb{V})$ that preserves the commutator relations of $\mathfrak{s}\mathfrak{l}(2,R)$ algebra [1,2].

This representation is $\mathfrak{s}\mathfrak{l}(2,R)$ module on vector space $\mathbb{V}$.

First, we review the structure of irreducible vector field representation of $\mathfrak{s}\mathfrak{l}(2,R)$. The generators of this algebra in matrix representation are as follows:

The commutation relations for this representation of $\mathfrak{s}\mathfrak{l}(2,R)$ are:

Let define a representation of $\mathfrak{s}\mathfrak{l}(2,R)$ as its module on $\mathbb{V}$ that preserves commutation relations by differential operators as its generators [1]:

With the same commutation relations

The Cartan sub-algebra $H=\mathit{h}$ produces a decomposition of representation space:

${\mathbb{V}}_j$ are the eigenspace (eigenfunction) of generator $\mathit{h}$ as Cartan sub-algebra of $\mathfrak{s}\mathfrak{l}\left(2,R\right)$ and provide the solutions to the related differential equation.

As an example, monomials $x^n$ are eigenfunctions or eigenspaces of generator $\mathit{h}$ , realized as eigenspace${\mathbb{V}}_n$. The eigenvalues $f(j)$ in most cases equals an integer $j$ or $j(j+1)$ as we observe in Hermite, Laguerre and Legendre differential equations.

We search for a Lie algebra ${\mathfrak{L}}_H$ isomorphic to $\mathfrak{s}\mathfrak{l}(2,R)$ algebra that its generators to be defined based on Hermite differential operators. Here we apply the transformation operator $e^{\frac{{-D}^2}{2}}$ as described in (1) for Hermite polynomials to derive similarity transformations (conjugation) of $\mathfrak{s}\mathfrak{l}(2,R)$ bases as follows:

Respect to a theorem in Lie algebra theory, these generators constitute an isomorphic Lie algebra to $\mathfrak{s}\mathfrak{l}(2,R)$ with similar commutation relations. We call this algebra as “ Hermite operator Lie algebra”. Due to the equation (1) that implies the change of basis $x^n$ to $H_{en}$, the operator $xD$ with eigenfunctions $x^n$ corresponds the operator ${\mathbb{D}}_H$with eigenfunctions $H_{en}$and common eigenvalues $n$ through a similarity transformation described by

Therefor we have:

Generator $X_2$ simply be calculated as ${\partial }_x=D$.

Proposition 1. For $X_3$ we have:

Proof: by Equation (11) we have the identity:

Thus:

By equation (2) we have:

Now for $X_3$ from (10) we have:

By equations (13), (14) and (15) we get and for generators of this Lie algebra, we have: where ${\mathbb{D}}_H$ denotes the Hermite differential operator i.e., $x{\partial }_x-{\partial }_x^2$. The commutation relations coincide the Lie algebra $\mathfrak{s}\mathfrak{l}(2,R)$ and are as follows:

proof: Due to a theorem for BCH formula, if $\left[X,Y\right]=sY$ for $s\in \mathbb{R}$, we have:

The BCH formula for $X_1$ and $X_2$ generators gives:

The term $-\frac{n}{2}$ in $X_1$ omitted because it has no role in commutation relation $\left[X_1,X_2\right]$.

Multiplying both side by $H_n^e(x)$

For $e^{{\partial }_x}{H}_n^e(x)$ we obtain

Thus

Substituting in equation (21) and replacing Hermite differential operator $x{\partial }_x-{\partial }_x^2$ with ${\mathbb{D}}_H$ gives

or

It is notable to compare this equation with

An ordinary definition for bivariate Hermite polynomials is as follows [3,4].

With $a,c>0$ and $ac-b^2>0.$ These polynomials satisfy the partial differential equation:

Let denote $\mathfrak{D}$ as the differential operator in equation (2)

If we denote ${\partial }_x=\frac{\partial }{\partial x}$ and ${\partial }_y=\frac{\partial }{\partial y}$, with the identities:

The equation (28) converts to

We denote the new polynomials as $u_{m,n}(x,y)$

If these polynomials are assumed as linearly independent basis, the transformation from these basis to $H_{n,m}(x,y)$ is as follows:

Therefor the corresponding differential operator with $u_{n,m}\left(x,y\right)$as its eigenfunctions could be derived by similarity transformation:

$\mathfrak{D}$ denoted as the differential operator given in eigenvalue equation (29). Thus, we have

Then due to commutativity of we have ${\partial }_x^2$ and ${\partial }_y^2$ we get

Respect to (11) and (12) this reduces to

Therefor the differential operator ${\mathfrak{D}}^{\prime }$ satisfy the differential equation:

Its eigenvalues are the same as the differential equation (24), because $\mathfrak{D}$ and ${\mathfrak{D}}^{\prime }$ related by the similarity relation (35).