On 2-variable $q$-Hermite polynomials

1.   Introduction

The quantum calculus, briefly called $q$-calculus, is infact the investigation of calculus without limits. The $q$-calculus is a generalization of ordinary calculus as for $q\to 1^{-}$, the quantum calculus reduces to the ordinary calculus. Recently, it has got attention of many researchers in the field of applied mathematics due to the fact that it provides a connection between mathematics and physics. It has been proved very helpful in the mathematical modelling of the problems arising in quantum computing. This motivates the study of certain special functions in the context of $q$-calculus. The $q$-analogues of some special functions like $q$-Gamma and $q$-Beta functions, 1-variable $q$-Hermite polynomials and $q$-Hermite polynomials-Appell polynomials etc., are introduced and studied ^{[5,6,15,17,19]}.

In this paper, we introduce the 2-variable $q$-Hermite polynomials and investigate some of its properties like series definition, recurrence relations, $q$-differential equation, summation formulas, operational definition and some integral representations.

We review briefly some definitions and notations of the quantum calculus.

The $q$-analogue of a complex number $\alpha$ is defined as ^[2,3,11,12]:

The first five few $q$-numbers are as follows:

The $q$-factorial is defined as:

which satisfies

The Gauss $q$-binomial coefficient is defined as ^[2,3,11,12]:

The raising and lowering $q$-powers are defined as ^[2,3,11,12]:

or, equivalently

where $\begin{bmatrix} n\\ k \end{bmatrix}_{q}$ is given by Eq (1.3). For $n = 1$, it is obvious that $(u\pm a)^{1}_{q} = (u\pm a)$.

In particular, for $a = 0$, Eq (1.4) or (1.5), gives

The two $q$-exponential functions, denoted by $e_{q}(u)$ and $E_{q}(u)$, are defined as ^[2,3,11,12]:

and

respectively. The relation between both $q$-exponential functions is as follows:

Next, the $q$-derivative of a function $f$ with respect to $u$, denoted by $D_{q, u} f(u)$, is defined as ^[13]:

Also, for two arbitrary functions $g(u)$ and $h(u)$, we have

In particular, we have

and

The $m^{th}$ order $q$-derivative of the $q$-exponential functions are as follows:

and

where $D^{m}_{q, u}$ denotes the $m^{th}$ order $q$-derivative with respect to $u$.

Also, the $q$-definite integral of a function $g$ is defined as ^[11,12,14]:

and

From Eqs (1.15) and (1.16), it is clear that

The $q$-definite integral of the $q$-derivative of a function $g(u)$ is given as:

We note that, for $q\to1^{-}$, all the results in the $q$-calculus reduce to the corresponding results in ordinary calculus.

The Hermite polynomials are one of the most applicable classical orthogonal special functions. They are the solutions of the differential equations, which are equivalent to the Schrödinger equation for a harmonic oscillator in quantum mechanics and thus they appear as eigen functions. Moreover, these polynomials have significant role in the study of classical boundary-value problems in parabolic regions, through the use of parabolic coordinates, or in quantum mechanics as well as in other application areas.

Now, we recall the generating function and series definition of the classical Hermite polynomials $H_{n}(u)$ and the 2-variable Hermite polynomials $H_{n}(u, v)$.

The classical Hermite polynomials $H_{n}(u)$ are defined by means of the following generating function ^[1]:

and the series definition ^[1]:

The 2-variable Hermite polynomials (2VHP) $H_{n}(u, v)$ are defined by means of the following generating function ^[4]:

and the series definition ^[4]:

The pure and the differential recurrence relations for $H_{n}(u, v)$ are defined as ^[4]:

and

The 2VHP $H_{n}(u, v)$ satisfies the following differential equation ^[4]:

From Eqs (1.19) and (1.21), it is clear that the 2VHP $H_{n}(u, v)$ is related with the classical Hermite polynomials $H_{n}(u)$ as:

The $q$-Hermite polynomials of 1-variable, which is defined in several ways, has many literatures due to its wide applications in various fields of mathematics and physics. For instance, Berg and Ismail ^[6], have shown that 1-variable $q$-Hermite polynomials can be used to build the classical $q$-orthogonal polynomials systemetically.

Recently, Nalci and Pashaev ^[17] defined the $q$-Hermite polynomials by means of the following generating function :

and series definition

in the context of $q$-analog of shock soliton solution.

In this paper, we introduce the 2-variable $q$-Hermite polynomials and study certain properties of these polynomials. An operational definition and some integral representations of these polynomials are obtained. Some examples are also considerd to show the efficacy of the proposed method. Some concluding remarks are given. At the end of this paper the graphical representations of these polynomials of certain degrees with specified values of $q$ are given.

In the next section, we introduce 2-variable $q$-Hermite polynomials with the help of generating function and we obtain their series definition, recurrence relations, operational differential equations of 2V$q$HP.

2.   The 2-variable $q$-Hermite polynomials

In this section, we introduce the 2-variable $q$-Hermite polynomials and obtain their series definition, recurrence relations and differential equation.

In view of Eqs (1.21), (1.29) and (1.30), we define the 2-variable $q$-Hermite polynomials (2V$q$HP) $H_{n, q}(u, v)$ by means of the following generating function:

The series on the right hand side of the above equation converges in the finite complex plane.

Expanding the left hand side of Eq (2.1) by using Eq (1.7), we have

which on using the following series rearrangement technique^[1]:

gives

Equating the coefficients of equal powers of $t$ from both sides of the above equation, we have the following series definition of 2V$q$HP $H_{n, q}(u, v)$:

where ${\lbrack\; . \; \rbrack}$ denotes the greatest integer function. Being a finite power series, the series on the right hand side of the above equation converges in the finite complex plane.

Taking $v = 0$ and $u = 0$ one by one in Eq (2.3), we have the following boundary conditions:

Further, from Eq (2.3), it can be easily verified that

where $a$ is a constant.

Now, taking $q$-partial derivative of both the sides of Eq (2.1) with respect to $u$ and then using Eq (1.13) in the resultant equation, we have

Using Eq (2.1) in the left hand side of Eq (2.6), we have

or, equivalently

Comparing the equal powers of $t$ from both the sides of the above equation, we have

Again, taking $2^{nd}$ order $q$-partial derivative of both the sides of Eq (2.1) with respect to $u$, then using Eq (1.13) for $m = 2$ and following the steps involved in the proof of Eq (2.7), we have

Similarly, taking $m^{th}$ order $q$-partial derivative of both sides of Eq (2.1) with respect to $u$, then using Eq (1.13) and following the same steps, we get the following $m^{th}$ order $q$-partial derivative of $H_{n, q}(u, v)$ with respect to $u$:

Next, taking $q$-partial derivative of both sides of Eq (2.1) with respect to $v$ and following the steps involved in the proof of Eq (2.7), we have

Similarly, taking $m^{th}$ order $q$-partial derivative of both sides of Eq (2.1) with respect to $v$ and following the same steps, we have

Again, in view of Eqs (2.7), (2.10) and (2.5), it is easy to verify that the 2V$q$HP $H_{n, q}(u, v)$ satisfy the following $q$-partial differential equation:

Next, we proceed to establish the recurrence relations for the 2V$q$HP $H_{n, q}(u, v)$. For this, we need to prove the following lemma:

Lemma 2.1. If $D_{q, t}$ denotes the $q$-partial derivative with respect to $t$, then

Proof. Using Eqs (1.7) and (1.13), we have

The series on the right hand side of the above equation converges in the finite complex plane.

Since, from Eq (1.1), we have $\frac{[2n]_{q}}{[n]_{q}}$ = ${1+q^n}$, therefore using Eq (1.2) in the right hand side of the above equation, we have

which on simplifying the right hand side, gives

Using Eq (1.7) in the right hand side of above equation, we get the assertion (2.13).

Remark 2.1 Since, for $q\to1^{-}$, we have $D_{q, t}$ $\to$ $D_{t}$ and $e_q(vt^2)$ $\to$ $e^{vt^2}$, therefore for $q\to1^{-}$, Lemma 2.2, gives the following well known result of ordinary calculus:

where $D_t$ denotes the ordinary derivative with respect to $t$.

Now, we establish the pure and $q$-differential recurrence relations for $H_{n, q}(u, v)$ in the form of following theorem:

Theorem 2.2. The 2V$q$HP $H_{n, q}(u, v)$ satisfy the following recurrence relations:

and

Proof. Taking $q$-derivative of both sides of Eq (2.1) with respect to $t$ and then using Eq (1.11) in the right hand side of the resultant equation, we have

which on using Eqs (1.12) and (1.2) in the left hand side and using Eqs (1.13) and (2.13) in the right hand side, gives

Using Eq (2.1) in both sides of the above equation and then comparing the equal powers of $t$ from both sides of the resultant equation, we get the assertion (2.16).

If $a$ is an arbitrary constant, then in view of Eq (2.5), we have

which on using Eq (2.7), gives

Again, using Eq (2.5) in the right hand side of Eq (2.21), we have

Using Eq (2.22) for $a = 1$ and $a = q$ in the right hand side of Eq (2.16), we have the assertion (2.17).

Now, replacing $n$ by $n-1$ in Eq (2.17), we have

which on using Eq (2.7) in the right hand side, gives the assertion (2.18).

Next, replacing $n$ by $n-1$ in Eq (2.16), we have

In view of Eq (2.5), for any constant $a$, we have

which on using Eq (2.10), gives

Again, using Eq (2.5) in the right hand side of Eq (2.24), we have

Using Eq (2.25) for $a = 1$ and $a = q$, in the right hand side of Eq (2.23), we have the assertion (2.19).

Again, replacing $n$ by $n-1$ in Eq (2.19), we have

which on using Eq (2.10) in the right hand side of Eq (2.26), gives the assertion (2.20).

In order, to establish the differential equation of 2V$q$HP $H_{n, q}(u, v)$, we define the following operators:

Let $f(u, v)$ be a $q$-function of two variables, then we define the shift operators $L_{a, u}$ and $L_{a, v}$ as:

and

where $a$ is a constant. In view of (2.27), the shift operator $L_{a, u}$ satisfy the following properties:

In particular, for $a$ = $b$, we have

If $L^{-1}_{a, u}$ is the inverse of the operator $L_{a, u}$, that is $L^{-1}_{a, u}\; L_{a, u} = I$, where $I$ is an identity operator such that $I f(u, v) = f(u, v).$

Then, from Eq (2.27), we have

Replacing $au$ by $u$ in the above equation, we have

which on using Eq (2.27), gives

Using induction method, Eqs (2.29) and (2.30), gives

where $r$ is any integer.

Similarly, it can be shown that $L_{a, v}$ satisfies the following properties:

where $r$ is any integer.

Now, we prove the following result:

Theorem 2.3. The 2-variable $q$-Hermite polynomials $H_{n, q}(u, v)$ satisfy the following $q$-differential equation:

where $L_{., u}$ is the shift operator defined in Eq (2.27).

Proof. Replacing $n$ by ($n-1$) in Eq (2.16) and then using (2.7) in resultant equation, we have

In view of Eq (2.27), we have

Also, in view of Eqs (2.5) and (2.27), we have

Using Eqs (2.27), (2.37) and (2.38) in Eq (2.36), we have

Now, using Eqs (2.7) and (2.8) in Eq (2.39), we have

or, equivalently

which, on further simplification yields assertion (2.35).

Remark 2.2 It can be easily verified that

Using Eq (2.41), for $m = 1$ and $m = 2$, in Eq (2.40), we have the following equivalent form of differential equation:

Another, forms of the differential Eq (2.35) are as follows

and

where $L_{q, v}$ is the shift operator defined by Eq (2.28).

Remark 2.3 Since, for $q\to1^{-}$, the $q$-calculus reduces to the ordinary calculus. Therefore, for $q\to1^{-}$, Eq (2.1) gives the generating function of 2VHP $H_{n}(u, v)$, given by Eq (1.21). Also, the series definition, recurrence relations and differential equation for 2VHP $H_{n}(u, v)$, given by the Eqs (1.21), (1.22) and (1.23)-(1.28), can be obtained by taking $q\to1^{-}$ in Eqs (2.3), (2.16)-(2.20) and (2.35), respectively.

In the next section, we establish certain summation formulas involving the 2-variables $q$-Hermite polynomials $H_{n, q}(u, v)$ and its $q$-derivatives.

3.   Summation formulas

In this section, we obtain certain summation formulas for 2V$q$HP by exploiting the identities (1.7) and (1.9) and using the generating function of $H_{n, q}(u, v)$. The following summation formulas for the 2V$q$HP $H_{n, q}(u, v)$ hold:

Theorem 3.1. The following summation formulas hold in the finite complex plane:

If $n$ is even i.e. $n = 2m$ ($m\in \mathbb{N}$), then

if $n$ is odd i.e. $n = 2m+1$ ($m\in \mathbb{N}\cup\{0\}$), then

Proof. In view of Eq (1.9), we have

which on using Eqs (1.7), (1.8) and (2.1), gives

or, equivalently

Using Eq (2.2) in the left hand side of above equation, we have

which on equating the coefficients of equal powers of $t$ from both sides, gives the assertion (3.1).

Again, using Eq (1.9), we have

Using Eqs (1.7), (1.8) and (2.1) in the above equation, we have

which on using the following series rearrangement technique^[1]:

gives

Comparing the even and odd powers of $t$ from both sides of Eq (3.7), we get the assertions (3.2) and (3.3), respectively.

Remark 3.1 The summation formula, given by Eq (3.3), has the following equivalent form:

From Theorem 3.1 and Remark 3.1, we deduce the following summation formulas for $q$-derivative of 2V$q$HP $H_{n, q}(u, v)$ with respect to $u$:

Corollary 3.2. The following summation formulas hold in the finite complex plane:

and

Proof. Using Eq (2.7), we have

which on simplifying, gives

Using Eqs (3.13) and (1.6) in Eq (3.1), we have the assertion (3.9).

Similarly, using Eq (2.7), we have

Using Eqs (3.14) and (1.6) in Eq (3.2), we have the assertion (3.10).

From Eq (3.14), we have

Using Eq (3.15) in Eq (3.8), we have the assertion (3.11).

Remark 3.2 Using Eq (2.10) in the right hand sides of Eqs (3.1), (3.2) and (3.8), we have the following summation formulas for $q$-derivative of 2V$q$HP $H_{n, q}(u, v)$ with respect to $v$:

Corollary 3.3. The following summation formulas hold in the finite complex plane:

and

Remark 3.3 For $q\to 1^{-}$, Eqs (3.1)-(3.3), (3.8)-(3.11), (3.16)-(3.18) give summation formulas for 2VHP $H_{n}(u, v)$ and its derivatives.

Now, we list some examples in Table 1 to show the efficacy of our results established in sections 2 and 3.

In the next section, we obtain the operational and integral representations for the 2V$q$HP $H_{n, q}(u, v)$.

4.   Operational and integral representations

It has been realized that the use of operational identities have simplified the study of special polynomials. In this section, we obtain the operational and integral representations of 2V$q$HP $H_{n, q}(u, v)$.

First, we establish the following result:

Theorem 4.1. The operational representation of 2-variable $q$-Hermite polynomials $H_{n, q}(u, v)$ is as follows:

Proof. In view of Eq (1.12), we have

Using the above equation in the right hand side of Eq (2.3), we have

which on using Eq (1.6), gives

Using Eq (1.7) in the right hand side of above equation, we have the assertion (4.1).

Remark 4.1 For ${q\to1^{-}}$, Eq (4.1) becomes ^[4]:

Now, we obtain the following integral representations for the 2V$q$HP $H_{n, q}(u, v)$:

Theorem 4.2. The definite $q$-integral of 2V$q$HP $H_{n, q}(u, v)$ with respect to $u$ is as follows:

Proof. Using Eq (2.7), we have

which on using Eq (1.18) in the right hand side, gives the assertion (4.4).

Next, we establish the following result:

Theorem 4.3. The definite $q$-integral of 2V$q$HP $H_{n, q}(u, v)$ with respect to $v$ is as follows:

Proof. Using Eq (2.10), we have

which on using Eq (1.18) in the right hand side, gives assertion (4.5).

In view of Theorems 4.2 and 4.3, we have the following result:

Corollary 4.4. The double $q$-integration of 2V$q$HP $H_{n, q}(u, v)$ is as follows:

Proof. Integrating Eq (4.4) with respect to $v$ by taking limit from $c$ to $d$ and using Eq (1.18), we have

which in view of Eq (4.5), gives

Similarly, integrating Eq (4.5) with respect to $u$ by taking limit from $a$ to $b$ and using Eq (1.18), we have

which in view of Eq (4.4), gives

In view of Eqs (4.7) and (4.9), we have the assertion (4.6).

Remark 4.2 For $q\to1^{-}$, Eqs (4.4) and (4.5) reduce to the integral representations for 2VHP $H_{n}(u, v)$ ^[4]. Also, for $q\to1^{-}$, Eq (4.6) gives:

Examples:

Now, we consider some examples to show the efficacy of the results obtained, in this section.

I. First, we take $n = 6$ in Eqs (4.1), (4.4)-(4.6) to obtain the corresponding results for $H_{6, q}(u, v)$.

For $n = 6$, Eq (4.1), the following operational representation of $H_{6, q}(u, v)$:

Also, taking $n = 6, a = 1, b = 2, c = 2, d = 3$ in Eqs (4.4)-(4.6) and then simplifying by using Eq (2.3), we have the following integral representations of $H_{6, q}(u, v)$:

and

II. Next, we take $n = 3$ in Eqs (4.1), (4.4)-(4.6) to obtain the corresponding results for $H_{3, q}(u, v)$.

For $n = 3$, Eq (4.1), the following operational representation of $H_{3, q}(u, v)$:

Also, taking $n = 3, a = 0, b = 2, c = 3, d = 4$ in Eqs (4.4)-(4.6) and then simplifying by using Eq (2.3), we have the following integral representations of $H_{3, q}(u, v)$:

and

5.   Concluding remarks

The post quantum calculus, briefly called $(p, q)$-calculus, is considered as a generalization of the $q$-calculus by some mathematicians, as for $p = 1$, the $(p, q)$-calculus reduces to the $q$-calculus. The $(p, q)$-analogues of certain ordinary special functions and $q$-special functions as well as polynomials like Beta function, Gamma function, Euler polynomials and Bernoulli polynomials have been introduced and studied ^[5,9,16,18].

We conclude this paper with the introduction and study of $(p, q)$-analogue of 2V$q$HP $H_{n, q}(u, v)$.

We recall the $(p, q)$-number $[n]_{p, q}$, which is defined as ^[8]:

for any positive integer $n\in\mathbb{N}$. The $(p, q)$-factorial is defined as ^[8]:

Also, we recall that

The $({p, q})$-exponential function, denoted by $e_{p, q}(u)$, is defined as ^[8]:

The $({p, q})$-derivative of a function $f$ with respect to $u$, denoted by D$_{p, q, u}$ $f(u)$, is defined as ^[7]:

Recently, Duran et al. defined the 1-variable $(p, q)$-Hermite polynomials as ^[10]:

In view of Eqs (2.1) and (5.5), we define the 2-variable $({p, q})$-Hermite polynomials 2V$(p, q)$HP $H_{n, p, q}(u, v)$ by means of the following generating function:

The series on the right hand side of the above equation converges in the finite complex plane.\

Simplifying and equating the coefficients of equal powers of $t$ from both sides of Eq (5.6), gives the following series definition of 2V$({p, q}$)HP $H_{n, p, q}(u, v)$:

Being a finite power series, the series on the right hand side of the above equation converges in the finite complex plane.

Similar to the Lemma 2.1, we find the following (${p, q}$)-partial derivative of the function $e_{p, q} (vt^2)$:

which is used to obtain the following $({p, q})$-pure and differential recurrence relations for 2V$(p, q)$HP $H_{n, p, q}(u, v)$:

and

Now, in order to establish the operational representation for the 2V$({p, q})$HP $H_{n, p, q}(u, v)$, we define a $({p, q})$-operator $X_{p, q, u}$ as:

which, in view of Eqs (2.31), (2.32) and (5.2), acts on a $(p, q)$-function in the following manner:

where $L^{-1}_{p, u}$ is the inverse shift operator, defined by Eq (2.31). It is easy to verify that the shift-differential operator $X_{p, q, u}$ satisfies the following:

Using the above equation in the right hand side of Eq (5.7), we have

which on using Eq (5.2), gives

Using Eq (5.3) in the right hand side of above equation, we have the following operational identity of the 2-variable $({p, q})$-Hermite polynomials $H_{n, p, q}(u, v)$:

where the $({p, q})$-operator $X^2_{p, q, u}: = p^{-1}L^{-2}_{p, u} D^2_{p, q, u}$.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Research Group Program under Grant No. RGP. 2/51/42.

Conflict of interest

The authors declare that they have no competing interest.

Supplementary

In order to plot the graphs of the 2-variable $q$-Hermite and $(p, q)$-Hermite polynomials for certain values of $p$ and $q$, we obtain the following first few $q$-Hermite and $(p, q)$-Hermite polynomials by using Eqs (2.3) and (5.7) respectively:

$H_{0, q}(u, v) = 1$,

$H_{1, q}(x, y) = u$,

$H_{2, q}(u, v) = u^2+[2]_{q}v$,

$H_{3, q}(u, v) = u^3 +[3]_{q}[2]_{q}uv$,

$H_{4, q}(u, v) = u^4 +[4]_{q}[3]_{q}u^2v+[4]_{q}[3]_{q}v^2$

and

$H_{0, p, q}(u, v) = 1$,

$H_{1, p, q}(u, v) = u$,

$H_{2, p, q}(u, v) = pu^2+[2]_{p, q}v$,

$H_{3, p, q}(u, v) = p^3 u^3 +[3]_{p, q}[2]_{p, q}uv$,

$H_{4, p, q}(u, v) = p^6 u^4 +[4]_{p, q}[3]_{p, q}p^2u^2v+[4]_{p, q}[3]_{p, q}pv^2$.

The following graphical representations of the first few 2-variable $q$-Hermite polynomials $H_{n, q}(u, v)$ and 2-variable $(p, q)$-Hermite polynomials $H_{n, p, q}(u, v)$ for certain values of $p$ and $q$, are obtained by using the above suitable expression in the software MATLAB: