1. Introduction

As a kind of special function, the Hermite polynomial has been widely used in quantum mechanics and mathematical physics.^[1–3] The one-variable Hermite polynomials are just eigenfunctions of the Hamiltonian of the quantum harmonic oscillator, while the two-variable Hermite polynomials can represent transition amplitude from the number state |n〉 to |m〉 in time evolution of a forced harmonic oscillator.^[4] The Hermite–Gaussian modes exist in optical propagation in quadratic Gradient Index (GRIN) lenses.^[5] In Ref. [6], the optical field’s quadrature excitation state is introduced which turns out to be a Hermite polynomial excited state. Mathematically, the one-variable Hermite polynomial H_n (x) can be introduced by its generating function

or

In Ref. [7] new generating function formulae of even- and odd-Hermite polynomials are obtained, they are

and

On the other hand, associated with the construction of bipartite entangled state representation^[8,9]

we have introduced the generating function of two-variable Hermite polynomials H_n,m (x,y)

or

In this equation when x^n−ly^m−l is replaced by H_n−l (ix) H_m−l (iy) we can introduce a kind of special function^[10]

for which we have found its generating function

or

Such a function really appears in some calculations of quantum optics theory, for example, the following anti-normally ordered product of two single-variable Hermite-polynomial operators is equal to

where

is used. Then the overlap of two Hermite-polynomial excited coherent states is equal to

An interesting question thus challenges us: if we replace the power-series function x^n−l or y^m−l in Eq. (8) by suitable one-variable Hermite polynomial, like: x^n−l → H_n−l (fx) or y^m−l→H_m−l (y), e.g.,

then how is such a new function Is it a meaningful special function? If it is true, what is its generating function and under what situation does it appear? We shall employ the operator Hermite polynomial method (OHPM)^[7] to tackle these problems. In Section 2 we briefly review OHPM. In Section 3 we show how we get the form of in quantum optics calculations, noting its expansion involving both power-series and Hermite polynomials. In Section 4 we obtain the generating function of .

2. Brief review of OHPM

The essential point of OHPM is replacing classical Hermite polynomial H_n (x) by H_n (X), where X is the coordinate operator in quantum mechanics

and then using the fundamental operator commutative relations among normal, anti-normal or Weyl ordering, to obtain many operator identities about X, finally let X return back to x, we can derive some new relations about special functions. For instance, from Eq. (1) we have

where :  : denotes normal ordering, then we have the useful and concise operator identity

Similarly, for the two-mode case, after introducing the second-mode coordinate operator

and noting [a + b^†,b + a^†] = 0, from Eq. (7) we have

which indicates

this is another concise operator identity. Hence, the OHPM is powerful in deriving new operator identities.

3. The appearance of

We now demonstrate in what physical circumstances will the form of new special function appear? For example, when we calculate photon number distribution of the Hermite-polynomial-excited state , i.e., we consider

where

is a number state, due to the integration formula

and using the integration within normal product of operators, we have

where is the completeness relation of the coordinate eigenstate. When

then

Since

so equation (21) becomes

Thus, we see the appearance of the form a^lH_n (fX), its Hermite conjugate is in the form H_n (fX) a^†l. Now using

we consider

where

so the normally ordered expansion of H_n (fX) a^†l is

Comparing with the definition of in Eq. (14), we can see

which indicates that when one calculates photon number distribution of the Hermite-polynomial-excited state, one encounters the new function .

Similarly, we can derive

Then using the anti-normally ordered form of H_n (X)

and comparing with Eq. (14) we have

4. The generating function of

Now we explore the generating function of . According to the idea of OHPM we consider

Combining with Eq. (35) leads to

Both sides of Eq. (37) are in anti-normally ordering, so letting , we obtain the generating function of

In summary, by virtue of the operator Hermite polynomial method and the technique of integration within the ordered product of operators,^[11,12] we have proposed a new kind of special functions , whose expansion involves both power-series and Hermite polynomials. Its application in calculating physical quantity in quantum optics theory is presented.