Keywords

1 Star Product on Polynomials

First we start by considering well-known star product, the Moyal product. Also we introduce typical star products, that is, normal product and anti-normal product. These products are mutually isomorphic with explicit isomorphisms, which are given by changing orderings in physics.

Next we define a class of star products containing the Moyal, normal, and anti-normal products as examples. The product is defined on the space of complex polynomials and is given as power series with respect to a positive parameter ħ.

Also we discuss isomorphisms between these star products. We describe that these objects naturally give rise to a bundle of star product algebras over space of complex symmetric matrices.

1.1 Moyal Product

The Moyal product is a well-known example of star product [2, 4].

For polynomials f, g of variables (u 1, …, u m, v 1, …, v m), we define a biderivation \(\left (\overleftarrow {\partial _{v}} \cdot \overrightarrow {\partial _{u}} -\overleftarrow {\partial _u} \cdot \overrightarrow {\partial _v} \right )\) by

$$\displaystyle \begin{aligned} f\left(\overleftarrow{\partial_{v}} \cdot\overrightarrow{\partial_{u}} -\overleftarrow{\partial_u} \cdot \overrightarrow{\partial_v} \right)\, g = \sum_{j=1}^m \left( \partial_{v_j} f \ \partial_{u_j}g -\partial_{u_j}f\ \partial_{v_j} g \right) \notag\end{aligned} $$

Here the overleft arrow \(\overleftarrow {\partial }\) indicates that the partial derivative is acting on the polynomial on the left and the overright arrow indicates the right.

Then the Moyal product \(f{\ast _{{ }_O}} g\) is given by the power series of the biderivation \(\left (\overleftarrow {\partial _{v}} \cdot \overrightarrow {\partial _{u}} -\overleftarrow {\partial _u} \cdot \overrightarrow {\partial _v} \right )\) such that

$$\displaystyle \begin{aligned} f{\ast_{{}_O}} g&= f \exp\tfrac{i\hbar}{2} \left(\overleftarrow{\partial_{v}}\cdot \overrightarrow{\partial_{u}} -\overleftarrow{\partial_u}\cdot \overrightarrow{\partial_{v}}\right) \ g =f \sum_{k=0}^\infty \frac{1}{k!} \left(\tfrac{i\hbar}{2}\right)^k \left( \overleftarrow{\partial_{v}} \cdot\overrightarrow{\partial_{u}} -\overleftarrow{\partial_u} \cdot \overrightarrow{\partial_v} \right)^k g \\ &=fg+\tfrac{i\hbar}{2}f\left( \overleftarrow{\partial_{v}} \cdot\overrightarrow{\partial_{u}} -\overleftarrow{\partial_u} \cdot \overrightarrow{\partial_v} \right)\, g + \tfrac{1}{2!} \left(\tfrac{i\hbar}{2} \right)^2 f \left(\overleftarrow{\partial_{v}} \cdot\overrightarrow{\partial_{u}} -\overleftarrow{\partial_u} \cdot \overrightarrow{\partial_v} \right)^2\, g \\ &\quad +\cdots + \tfrac{1}{k!} \left(\tfrac{i\hbar}{2}\right)^k f\left( \overleftarrow{\partial_{v}} \cdot\overrightarrow{\partial_{u}} -\overleftarrow{\partial_u} \cdot \overrightarrow{\partial_v} \right)^k\, g +\cdots {}\end{aligned} $$
(1.1)

where ħ is a positive number. Although the Moyal product is defined as a formal power series of bidifferential operators, this becomes a finite sum on polynomials. One can check the associativity of the product directly, hence we have

Proposition 1.1

The Moyal product is well-defined on polynomials, and associative.

Other typical star products are normal product \(\ast _{{ }_N}\), anti-normal product \(\ast _{{ }_A}\) given similarly with replacing biderivations, respectively, by

$$\displaystyle \begin{aligned} f \ast_{{}_N} g = f \exp{i\hbar} \left(\overleftarrow{\partial_{v}} \cdot \overrightarrow{\partial_{u}} \right) \ g, \quad f \ast_{{}_A} g = f \exp-{i\hbar} \left(\overleftarrow{\partial_{u}} \cdot \overrightarrow{\partial_{v}} \right) \ g \notag\end{aligned} $$

These are also well-defined on polynomials and associative.

By direct calculation we see easily

Proposition 1.2

  1. (i)

    For these star products, the generators (u 1, …, u m, v 1, …, v m) satisfy the canonical commutation relations

    $$\displaystyle \begin{aligned}{}[u_k, v_l]_{\ast_{{}_L}} = -i \hbar \delta_{kl}, \, [u_k, u_l]_{\ast_{{}_L}} = [v_k, v_l]_{\ast_{{}_L}}=0, \quad (k, l=1, 2, \ldots, m) \notag \end{aligned} $$

    where \({\ast _{{ }_L}}\) stands for \({\ast _{{ }_O}}\) , \({\ast _{{ }_N}}\) , and \({\ast _{{ }_A}}\).

  2. (ii)

    Then the algebras \((\mathbb {C}[u, v], \ast _L)\) (L = O, N, A) are mutually isomorphic and isomorphic to the Weyl algebra.

The algebra isomorphisms have explicit expressions. For example, the algebra isomorphism

$$\displaystyle \begin{aligned} I_O^N : (\mathbb{C}[u, v], {\ast_{{}_O}}) \to (\mathbb{C}[u, v], \ast_N) \notag \end{aligned} $$

is given by the power series of the differential operator such as

$$\displaystyle \begin{aligned} I^O_N\ (f) = \exp \left( -\tfrac{i\hbar}{2}\partial_u\partial_v \right) \ (f) = \sum_{l=0}^\infty \tfrac{1}{l!} (\tfrac{i\hbar}{2})^l \left(\partial_u \partial_v \right)^l \ (f) \end{aligned} $$
(1.2)

Other isomorphisms are given in similar forms (cf. [12]).

Remark 1.3

We remark here that these isomorphisms are well-known as ordering problem in physics [1] .

1.2 Star Product

Using complex matrices we generalize biderivations and we define a star product on complex domain in the following way.

Let Λ be an arbitrary n × n complex matrix. We consider a biderivation

$$\displaystyle \begin{aligned} \overleftarrow{\partial_w}\Lambda \overrightarrow{\partial_w} =(\overleftarrow{\partial_{w_1}}, \cdots, \overleftarrow{\partial_{w_{n}}}) \Lambda (\overrightarrow{\partial_{w_1}}, \cdots, \overrightarrow{\partial_{w_{n}}}) = \sum_{k,l=1}^{n} \Lambda_{kl} \overleftarrow{\partial_{w_k}} \overrightarrow{\partial_{w_l}} \end{aligned} $$
(1.3)

where (w 1, ⋯ , w n) is a generator of polynomials.

Now we define a star product similar to (1.1) by

Definition 1.4

$$\displaystyle \begin{aligned} f*_{{}_\Lambda} g = f \exp \left(\tfrac{i \hbar}2 \overleftarrow{\partial_w}\Lambda \overrightarrow{\partial_w} \right) \ g \end{aligned} $$
(1.4)

Remark 1.5 ([13])

  1. (i)

    The star product \(\ast _{{ }_\Lambda }\) is a generalization of the products ∗L (L = O, N, A). Actually

    • if we put \(\Lambda =\left ( \begin {array}{cc} 0&-1_m\\ 1_m&0 \end {array} \right )\) , then we have the Moyal product

    • if \(\Lambda =2\left ( \begin {array}{cc} 0&0\\ 1_m&0 \end {array} \right )\) , then we have the normal product and

    • if \(\Lambda =2\left ( \begin {array}{cc} 0&-1_m\\ 0&0 \end {array} \right )\) , then the anti-normal product

  2. (ii)

    If Λ is a symmetric matrix, the star product \(\ast _{{ }_\Lambda }\) is commutative.

Then similarly as before we see easily

Theorem 1.6

For an arbitrary Λ, the star product \(\ast _{{ }_\Lambda }\) is well-defined on polynomials, and associative.

1.3 Equivalence and Geometric Picture of Weyl Algebra

In this section, we take Λ as a special class of matrices in order to represent Weyl algebra (cf. [5, 10]).

Let K be an arbitrary 2m × 2m complex symmetric matrix. We put a complex matrix

$$\displaystyle \begin{aligned} \Lambda = J+K \notag \end{aligned} $$

where J is a fixed matrix such that

$$\displaystyle \begin{aligned} J= \left( \begin{array}{cc} 0&-1_m\\ 1_m&0 \end{array} \right) \notag \end{aligned} $$

Since Λ is determined by the complex symmetric matrix K, we denote the star product by \(\ast _{{ }_K}\) instead of \(\ast _{{ }_\Lambda }\).

We consider polynomials of variables (w 1, ⋯, w 2m)=(u 1, ⋯, u m, v 1, ⋯, v m). By easy calculation one obtains

Proposition 1.7

  1. (i)

    For a star product \(\ast _{{ }_K}\) , the generators (u 1, …, u m, v 1, …, v m) satisfy the canonical commutation relations

    $$\displaystyle \begin{aligned}{}[u_k, v_l]_{\ast_{{}_K}} = -i \hbar \delta_{kl}, \, [u_k, u_l]_{\ast_{{}_K}} = [v_k, v_l]_{\ast_{{}_K}}=0, \quad (k, l=1, 2, \ldots, m) \notag \end{aligned} $$
  2. (ii)

    Then the algebra \((\mathbb {C}[u, v], \ast _{{ }_K})\) is isomorphic to the Weyl algebra, and the algebra is regarded as a polynomial representation of the Weyl algebra.

Equivalence

As in the case of typical star products, we have algebra isomorphisms as follows.

Proposition 1.8

For arbitrary \((\mathbb {C}[u,v], \ast _{{ }_{K_1}})\) and \((\mathbb {C}[u,v], \ast _{{ }_{K_2}})\) we have an algebra isomorphism \(I_{K_1}^{K_2}: (\mathbb {C}[u,v], \ast _{{ }_{K_1}}) \to (\mathbb {C}[u,v], \ast _{{ }_{K_2}})\) given by the power series of the differential operator ∂ w(K 2 − K 1)∂ w such that

$$\displaystyle \begin{aligned} I_{K_1}^{K_2}\ (f) = \exp \left( \tfrac{i\hbar}{4}\partial_w (K_2-K_1)\partial_w \right) \ (f) \notag \end{aligned} $$

where \(\partial _w (K_2-K_1)\partial _w =\sum _{kl} (K_2-K_1)_{kl} \partial _{w_k}\partial _{w_l}\).

By a direct calculation we have

Theorem 1.9

Isomorphisms satisfy the following chain rule:

  1. 1.

    \(I_{K_3}^{K_1}I_{K_2}^{K_3}I_{K_1}^{K_2} =Id\) ,  ∀K 1, K 2, K 3

  2. 2.

    \(\left (I_{K_1}^{K_2}\right )^{-1} =I_{K_2}^{K_1}\) ,  ∀K 1, K 2

Remark 1.10

  1. 1.

    By Proposition 1.8 we see the algebras \((\mathbb {C}[u,v], \ast _{{ }_{K}})\) are mutually isomorphic and isomorphic to the Weyl algebra. Hence we have a family of star product algebras \(\left \{ (\mathbb {C}[u,v], \ast _{{ }_{K}}) \right \}_{K}\) where each element is regarded as a polynomial representation of the Weyl algebra.

  2. 2.

    The above equivalences are also valid for star products ∗Λ and \(\ast _{\Lambda '}\) for arbitrary Λ, Λ′ with a common skew symmetric part. More precisely, let \(\tilde J\) by an arbitrary n × n skew-symmetric matrix, and for any n × n symmetric matrices K, K′ we consider \(\Lambda =\tilde J+K\) and \(\Lambda '=\tilde J+K'\) . Then \(I_{K}^{K'}\) gives an algebra isomorphism \(I_{K}^{K'}: (\mathbb C[w], *_{\Lambda }) \to (\mathbb C[w], *_{\Lambda '})\) , where w = (w 1, …, w n) is the generator of polynomials.

According to the previous theorem, we introduce an infinite dimensional bundle and a connection over it and using parallel sections of this bundle we have a geometric picture (cf. [14]) for the family of the star product algebras \(\left \{ (\mathbb {C}[u,v], \ast _{{K}}) \right \}_{K}\).

Algebra Bundle

We set \(\mathcal {S}=\{K\}\) the space of all 2m × 2m symmetric complex matrices. We consider a trivial bundle over \(\mathcal {S}\) with fiber the star product algebras

$$\displaystyle \begin{aligned} \pi: E=\Pi_{K\in \mathcal{S}} (\mathbb{C}[u, v], \ast_{K}) \to \mathcal{S}, \quad \pi^{-1}(K) = (\mathbb{C}[u, v], \ast_{K}). \notag \end{aligned} $$

Then Proposition 1.8 shows that each fiber \((\mathbb {C}[u, v], \ast _{K})\) is isomorphic to the Weyl algebra and any fibers of the bundle are mutually isomorphic by the intertwiners \(I_{K_1}^{K_2}\).

Connection and Parallel Sections

For a curve C : K = K(t) in the base space \(\mathcal {S}\) starting from K(0) = K, we define a parallel translation of a polynomial \(f\in (\mathbb {C}[u, v], \ast _{K})\) by

$$\displaystyle \begin{aligned} f(t)=\exp \tfrac{i\hbar}{4} \partial_w(K(t)-K)\partial_w \ (f). \notag \end{aligned} $$

It is easy to see f(0) = f. By differentiating the parallell translation we have a connection of this bundle such that

$$\displaystyle \begin{aligned} \nabla_X f(K)= \tfrac{d}{dt}f(t)|{}_{t=0} (K) = \tfrac{i\hbar}{4} \partial_w X \partial_w \ f(K) |{}_{t=0}, \quad X=\dot K(t)|{}_{t=0} \notag\end{aligned} $$

where f(K) is a smooth section of the bundle E.

We set \(\mathcal {P}\) the space of all parallel sections of this bundle, namely, f is an element of \(\mathcal P\) iff \(f(K_{2})=I_{K_1}^{K_2}f(K_{1})\) for any \(K_{1}, K_{2} \in \mathcal S\). Since \(I_{K_1}^{K_2}\) are algebra isomorphisms, namely it holds for sections f, g

$$\displaystyle \begin{aligned} I_{K_1}^{K_2} (f(K_1)\ast_{{K_1}} g(K_1)) = \left(I_{K_1}^{K_2}(f(K_1)\right) \ast_{{K_2}} \left(I_{K_1}^{K_2}(g(K_1)\right), \notag\end{aligned} $$

we have a star product f ∗ g for the parallel sections \(f, g \in \mathcal {P}\) by setting

$$\displaystyle \begin{aligned} f*g\, (K)= f(K)\ast_{K} g(K) \notag\end{aligned} $$

Then we have

Theorem 1.11

  1. (i)

    The space of the parallel sections \(\mathcal {P}\) consists of the sections such that \(\nabla _X f= \tfrac {i\hbar }{4} \partial _w X \partial _w \ f=0\) , ∀X.

  2. (ii)

    The space \(\mathcal {P}\) is canonically equipped with the star product ∗, and the associative algebra \((\mathcal {P}, \ast )\) is isomorphic to the Weyl algebra.

Remark 1.12

The algebra \((\mathcal {P}, \ast )\) is regarded as a geometric realization of the Weyl algebra.

2 Extension to Functions

We consider to extend the star products ∗Λ for an arbitrary complex matrix Λ from polynomials to functions (cf. [9]).

2.1 Star Product on Certain Holomorphic Function Space

For ordinary smooth functions, the star products ∗Λ are not necessarily well-defined, e.g., convergent in general. However, we can discuss star products by restricting the product to certain class of smooth functions. Although there may be many classes for such functions, we consider the following space of certain entire functions in this note (cf. [6]).

Semi-norm

Let f(w) be a holomorphic function on \(\mathbb {C}^{n}\). For a positive number p, we consider a family of semi-norms {|⋅|p,s}s>0 given by

$$\displaystyle \begin{aligned} |f|{}_{p, s} = \sup_{w\in \mathbb{C}^{n}} |f(w)| \exp (-s|w|{}^p), \quad |w|=\sqrt{|w_1|{}^2+\cdots+|w_{n}|{}^2}. \notag \end{aligned} $$

Space

We put

$$\displaystyle \begin{aligned} \mathcal{E}_p= \{f :\, \text{entire}\ | \ |f|{}_{p, s}<\infty, \, \forall{s>0}\} \notag \end{aligned} $$

With the semi-norms the space \(\mathcal {E}_p\) becomes a Fréchet space.

As to the star products, we have for any matrix Λ

Theorem 2.1

  1. (i)

    For 0 < p ≤ 2, \((\mathcal {E}_p, \ast _{\Lambda })\) is a Fréchet algebra. That is, the product converges for any elements, and the product is continuous with respect to this topology.

  2. (ii)

    Moreover, for any Λ′ with the common skew symmetric part with Λ, \(I_{\Lambda }^{\Lambda '} =\exp (\tfrac {i\hbar }{4} \partial _w (\Lambda '-\Lambda ) \partial _w)\) is well-defined algebra isomorphism from \((\mathcal {E}_p, \ast _{\Lambda })\) to \((\mathcal {E}_p, \ast _{\Lambda '})\) . That is, the expansion converges for every element, and the operator is continuous with respect to this topology.

  3. (iii)

    For p > 2, the multiplication \(\ast _\Lambda : \mathcal {E}_p \times \mathcal {E}_{p'} \to \mathcal {E}_p \) is well-defined for p′ such that \(\tfrac {1}{p}+\tfrac {1}{p'}=2\) , and \((\mathcal {E}_p, \ast _\Lambda )\) is a \(\mathcal {E}_{p'}\) -bimodule.

In this topology, the parameter ħ can be taken as \(\hbar \in \mathbb C\).

3 Star exponentials

Since we have a complete topological algebra, we can consider exponential elements in the star product algebra \((\mathcal {E}_p, \ast _\Lambda )\). (cf. [13]).

3.1 Definition

For a polynomial H ∗, we want to define a star exponential \(e_*^{t\tfrac {H_*}{i\hbar }}\). However, except special cases, the expansion \(\sum _n \frac {t^n}{n!} \left (\tfrac {H_*}{i\hbar }\right )^n\) is not convergent, so we define a star exponential by means of a differential equation.

Definition 3.1

The star exponential \(e_*^{t\tfrac {H_*}{i\hbar }}\) is given as a solution of the following differential equation

$$\displaystyle \begin{aligned} \tfrac {d}{dt}F_t=H_* \ast_\Lambda F_t, \quad F_0=1. \end{aligned} $$
(3.1)

3.2 Examples

We are interested in the star exponentials of linear and quadratic polynomials. For these, we can solve the differential equation and obtain explicit form. For simplicity, we consider 2m × 2m complex matrices Λ with the skew symmetric part \(J=\begin {pmatrix} 0&-1_m\\1_m&0 \end {pmatrix}\). We write Λ = J + K where K is a complex symmetric matrix.

First we remark the following. For a linear polynomial \(l=\sum _{j=1}^{2m} a_j w_j\), we see directly an ordinary exponential function e l satisfies

$$\displaystyle \begin{aligned} e^l \notin \mathcal{E}_1, \quad \in \mathcal{E}_{1+\epsilon}, \quad \forall \epsilon>0. \notag \end{aligned} $$

Then put a Fréchet space

$$\displaystyle \begin{aligned} \mathcal{E}_{p+} = \cap_{q>p} \mathcal{E}_q \notag \end{aligned} $$

Linear Case

Proposition 3.2

For l =∑j a j w j =< a, w > , \(a_j \in \mathbb {C}\) , we have

$$\displaystyle \begin{aligned} e_*^{t({l}/{i\hbar})} =e^{t^2{\boldsymbol{a}K\boldsymbol{a}}/{4i\hbar}} e^{t({l}/{i\hbar})} \in \mathcal{E}_{1+} \notag \end{aligned} $$

Quadratic Case

(Cf. [8]).

Proposition 3.3

For Q ∗ = 〈w A, w〉∗ where A is a 2m × 2m complex symmetric matrix,

$$\displaystyle \begin{aligned} e_*^{t(Q_*/i\hbar)} =\frac{2^m} {\sqrt{\det(I-\kappa+e^{-2t\alpha} (I+\kappa))}} e^{\tfrac{1}{i\hbar} \langle \boldsymbol{w} \tfrac{1}{I-\kappa+e^{-2t\alpha} (I+\kappa)} (I-e^{-2t\alpha})J,\boldsymbol{w}\rangle} \notag \end{aligned} $$

where κ = KJ and α = AJ.

Remark 3.4

The star exponentials of linear functions are belonging to \(\mathcal {E}_{1+}\) then the star products are convergent and continuous. But for a quadratic polynomial Q ∗ , it is easy to see

$$\displaystyle \begin{aligned} e_*^{t(Q_*/i\hbar)} \in \mathcal{E}_{2+}, \quad \notin \mathcal{E}_{2} \notag \end{aligned} $$

and hence star exponentials \(\{e_*^{t(Q_*/i\hbar )}\}\) are difficult to treat. Some anomalous phenomena happen. (cf. [7]) .

Remark 3.5

Beside solving differential equation, we also construct star exponential of quadratic polynomial by so-called path-integral method. Namely, we divide the time interval by a positive integer N. Consider commutative exponential functions e (t∕N)(Q∕iħ) , N = 1, 2, …, and take N-multiple products e (t∕N)(Q∕iħ) ∗⋯ ∗ e (t∕N)(Q∕iħ) . Then taking a limit we have (cf. [3])

$$\displaystyle \begin{aligned} e_*^{t(Q_*/i\hbar)} = \lim_{N\to \infty} e^{(t/N) (Q/i\hbar)} *\cdots *e^{(t/N) (Q/i\hbar)} \end{aligned} $$

4 Star Functions

There are many applications of star exponential functions (cf. [11,12,13,14]). In this note we show examples using linear star exponentials.

In what follows, we consider the star product for the simple case where Λ has only one nonzero entry

$$\displaystyle \begin{aligned} \Lambda =\left( \begin{array} [c]{cc} \rho & 0\\ 0 & 0 \end{array} \right), \quad \rho \in \mathbb{C} \end{aligned}$$

Then we see easily that the star product is commutative and explicitly given by \(p_1\ast _{{ }_\Lambda } p_2 = p_1 \exp \left ( \tfrac {i \hbar \rho }{2} \overleftarrow {\partial _{w_1}} \overrightarrow {\partial _{w_1}} \right ) p_2\). This means that the algebra is essentially reduced to the space of functions of one variable w 1. Thus, we consider functions f(w), g(w) of one variable \(w\in \mathbb {C}\) and we consider a commutative star product \(\ast _{{ }_\tau }\) with complex parameter τ such that

$$\displaystyle \begin{aligned} f(w)\ast_{{}_\tau} g(w) = f(w) e^{\frac {\tau}{2} \overleftarrow \partial_w \overrightarrow \partial_w} g(w) \end{aligned}$$

4.1 Star Hermite Function

Recall the identity

$$\displaystyle \begin{aligned} \exp\left( \sqrt{2} tw - \tfrac{1}{2} t^2 \right) = \sum_{n=0}^\infty H_n(w) \tfrac{t^n}{n!} \notag \end{aligned} $$

where H n(w) is an Hermite polynomial. We remark here that

$$\displaystyle \begin{aligned} \exp\left( \sqrt{2} tw - \tfrac{1}{2} t^2 \right) = \exp_* ( {\sqrt{2} tw_*} )_{\tau=-1} \notag \end{aligned} $$

Since \(\exp _* ( {\sqrt {2} tw_*} ) = \sum _{n=0}^\infty (\sqrt {2} tw_*)^n \, \tfrac {t^n}{n!}\), we have \(H_n(w) =(\sqrt {2} tw_*)^n _{\tau =-1}\).

We define ∗-Hermite function by

$$\displaystyle \begin{aligned} H_n(w, \tau) =(\sqrt{2} tw_*)^n, \quad (n=0, 1, 2, \cdots) \notag \end{aligned} $$

with respect to ∗τ product. Then we have

$$\displaystyle \begin{aligned} \exp_*( {\sqrt{2} tw_*}) = \sum_{n=0}^\infty H_{n}(w, \tau ) \, \tfrac{t^n}{n!} \notag \end{aligned} $$

Trivial identity \(\tfrac {d}{dt} \exp _*( {\sqrt {2} tw_*}) = \sqrt {2} w* \exp _*( {\sqrt {2} tw_*})\) yields at every \(\tau \in \mathbb {C}\) the identity

$$\displaystyle \begin{aligned} \tfrac{\tau}{\sqrt{2}} H_n^{\prime}(w, \tau) +\sqrt{2} w H_n(w, \tau) = H_{n+1}(w, \tau), \quad (n=0,1,2,\cdots). \notag \end{aligned} $$

The exponential law \(\exp _*( {\sqrt {2} sw_*}) * \exp _*( {\sqrt {2} tw_*}) = \exp _*( {\sqrt {2} (s+t) w_*})\) yields at every \(\tau \in \mathbb {C}\) the identity

$$\displaystyle \begin{aligned} \sum_{k+l=n} \tfrac{n! }{k!l!} H_k(w,\tau)\ast_\tau H_l(w, \tau) = H_n(w,\tau). \notag \end{aligned} $$

4.2 Star Theta Function

In this note we consider the Jacobi’s theta functions by using star exponentials as an application.

A direct calculation gives

$$\displaystyle \begin{aligned} \exp_{*_{{}_\tau}}i\ t w =\exp (i\ t w-(\tau/4) t^2) \notag \end{aligned} $$

Hence for \(\Re \tau >0\), the star exponential \(\exp _{*_{{ }_\tau }}n i\ w =\exp (n i\ w-(\tau /4) n^2)\) is rapidly decreasing with respect to integer n and then we can consider summations for τ satisfying \(\Re \tau >0\)

$$\displaystyle \begin{aligned} \sum_{n=-\infty}^\infty &\exp_{\ast_{{}_\tau}}2n i\ w =\sum_{n=-\infty}^\infty \exp\left(2n i\ w-\tau\,n^2\right) =\sum_{n=-\infty}^\infty q^{n^2} e^{2n i\ w}, \quad (q=e^{-\tau}) \notag \end{aligned} $$

This is Jacobi’s theta function θ 3(w, τ). Then we have expression of theta functions as

$$\displaystyle \begin{aligned} \theta_{1\ast_{{}_\tau}}(w) =\tfrac{1}{i}\sum_{n=-\infty}^\infty (-1)^n\exp_{\ast_{{}_\tau}}(2n+1)i\ w, \quad \theta_{2\ast_{{}_\tau}}(w) =\sum_{n=-\infty}^\infty \exp_{\ast_{{}_\tau}}(2n+1)i\ w \end{aligned}$$
$$\displaystyle \begin{aligned} \theta_{3\ast_{{}_\tau}}(w) =\sum_{n=-\infty}^\infty \exp_{\ast_{{}_\tau}}2n i\ w, \quad \theta_{4\ast_{{}_\tau}}(w) =\sum_{n=-\infty}^\infty (-1)^n\exp_{\ast_{{}_\tau}}2n i\ w \end{aligned}$$

Remark that \( \theta _{k\ast _{{ }_\tau }}(w)\) is the Jacobi’s theta function θ k(w, τ), k = 1, 2, 3, 4, respectively. It is obvious by the exponential law

$$\displaystyle \begin{aligned} &\exp_{\ast_{{}_\tau}}2i\ w\ast_{{}_\tau} \theta_{k\ast_{{}_\tau}}(w) =\theta_{k\ast_{{}_\tau}}(w) \quad (k=2, 3) \\ &\exp_{\ast_{{}_\tau}}2 i\ w\ast_{{}_\tau} \theta_{k\ast_{{}_\tau}}(w) =-\theta_{k\ast_{{}_\tau}}(w) \quad (k=1, 4) \end{aligned} $$

Then using \(\exp _{\ast _{{ }_\tau }}2 i\ w = e^{-\tau } e^{2 i\ w}\) and the product formula directly we have

$$\displaystyle \begin{aligned} &e^{2 i\ w-\tau}\theta_{k\ast_{{}_\tau}} (w+i\ \tau) =\theta_{k\ast_{{}_\tau}}(w) \quad (k=2,3) \\ &e^{2 i\ w-\tau}\theta_{k\ast_{{}_\tau}} (w+i\ \tau) =-\theta_{k\ast_{{}_\tau}}(w) \quad (k=1,4) \end{aligned} $$

4.3 ∗-Delta Functions

Since the ∗τ-exponential \(\exp _*(itw_*) = \exp (itw-\tfrac {\tau }{4}t^2)\) is rapidly decreasing with respect to t when \(\Re \tau >0\), then the integral of ∗τ-exponential

$$\displaystyle \begin{aligned} \int_{-\infty}^\infty \exp_*(it(w-a)_*) \, dt = \int_{-\infty}^\infty \exp_*(it(w-a)_*) dt = \int_{-\infty}^\infty \exp (it(w-a)-\tfrac{\tau}{4} t^2) dt \notag \end{aligned} $$

converges for any \(a\in \mathbb {C}\). We put a star δ-function

$$\displaystyle \begin{aligned} \delta_*(w-a) = \int_{-\infty}^\infty \exp_*(it(w-a)_*) dt \notag \end{aligned} $$

which has a meaning at τ with \(\Re \tau >0\). It is easy to see for any element \(p_*(w) \in \mathcal {P}_*(\mathbb {C})\),

$$\displaystyle \begin{aligned} p_*(w)*\delta_*(w-a) =p(a) \delta_*(w-a), \, w_* * \delta_*(w)=0. \notag \end{aligned} $$

Using the Fourier transform we have

Proposition 4.1

$$\displaystyle \begin{aligned} \theta_{1*}(w) &=\tfrac{1}{2} \sum_{n=-\infty}^\infty (-1)^n \delta_* (w+\tfrac{\pi}{2}+n\pi) \notag\\ \theta_{2*}(w) &=\tfrac{1}{2} \sum_{n=-\infty}^\infty (-1)^n \delta_* (w+n\pi) \notag\\ \theta_{3*}(w) &=\tfrac{1}{2} \sum_{n=-\infty}^\infty \delta_* (w+n\pi) \notag\\ \theta_{4*}(w) &= \tfrac{1}{2} \sum_{n=-\infty}^\infty \delta_* (w+\tfrac{\pi}{2}+n\pi) \notag. \end{aligned} $$

Now, we consider the τ with the condition \(\Re \tau > 0\). Then we calculate the integral and obtain \(\delta _*(w-a) = \tfrac {2\sqrt {\pi }}{\sqrt {\tau }} \exp \left ( - \tfrac {1}{\tau } (w-a)^2\right )\). Then we have

$$\displaystyle \begin{aligned} \theta_3(w, \tau) & = \tfrac{1}{2} \sum_{n=-\infty}^\infty \delta_* (w+n\pi) = \sum_{n=-\infty}^\infty \tfrac{\sqrt{\pi}}{\sqrt{\tau}} \exp\left( - \tfrac {1}{\tau} (w+n\pi)^2\right) \notag\\ &= \tfrac{\sqrt{\pi}}{\sqrt{\tau}} \exp\left(- \tfrac {1}{\tau}\right) \sum_{n=-\infty}^\infty \exp\left( -2n \tfrac {1}{\tau} w - \tfrac {1}{\tau} n^2 \tau ^2 \right)\\ &= \tfrac{\sqrt{\pi}}{\sqrt{\tau}} \exp\left(- \tfrac {1}{\tau}\right) \theta_{3*} (\tfrac{2\pi w}{i\tau}, \tfrac{\pi^2}{\tau}). \notag \end{aligned} $$

We also have similar identities for other ∗-theta functions by the similar way.