*Article* **On 2-Variables Konhauser Matrix Polynomials and Their Fractional Integrals**

**Ahmed Bakhet 1,2 and Fuli He 1,\***


Received: 28 January 2020; Accepted: 4 February 2020; Published: 10 February 2020

**Abstract:** In this paper, we first introduce the 2-variables Konhauser matrix polynomials; then, we investigate some properties of these matrix polynomials such as generating matrix relations, integral representations, and finite sum formulae. Finally, we obtain the fractional integrals of the 2-variables Konhauser matrix polynomials.

**Keywords:** Konhauser matrix polynomial; generating matrix function; integral representation; fractional integral

**MSC:** 33C25; 33C45; 33D15; 65N35

#### **1. Introduction**

Special functions play a very important role in analysis, physics, and other applications, and solutions of some differential equations or integrals of some elementary functions can be expressed by special functions. In particular, the family of special polynomials is one of the most useful and applicable family of special functions. The Konhauser polynomials which were first introduced by J.D.E. Konhauser [1] include two classes of polynomials *Y<sup>α</sup> <sup>n</sup>* (*x*; *k*) and *Z<sup>α</sup> <sup>n</sup>*(*x*; *k*), where *Y<sup>α</sup> <sup>n</sup>* (*x*; *k*) are polynomials in *x* and *Z<sup>α</sup> <sup>n</sup>*(*x*; *<sup>k</sup>*) are polynomials in *<sup>x</sup>k*, *<sup>α</sup>* <sup>&</sup>gt; <sup>−</sup>1 and *<sup>k</sup>* <sup>∈</sup> <sup>Z</sup>+. Explicit expressions for the polynomials *Z<sup>α</sup> <sup>n</sup>*(*x*; *k*) are given by

$$Z\_n^a(\mathbf{x};k) = \frac{\Gamma(a+kn+1)}{n!} \sum\_{r=0}^n (-1)^r \binom{n}{r} \frac{\mathbf{x}^{kr}}{\Gamma(a+kr+1)},\tag{1}$$

where <sup>Γ</sup>(·) is the classical Gamma function and for the polynomials *<sup>Y</sup><sup>α</sup> <sup>n</sup>* (*x*; *k*), Carlitz [2] subsequently showed that

$$\mathcal{Y}\_n^{\mathfrak{a}}(x;k) = \frac{1}{n!} \sum\_{r=0}^n \frac{x^r}{r!} \sum\_{s=0}^r (-1)^s \binom{r}{s} \left(\frac{s+a+1}{k}\right)\_{n'} \tag{2}$$

where (*a*)*<sup>n</sup>* is Pochhammer's symbol of *a* as follows:

$$(a)\_n = \begin{cases} \ \newline a(a+1)(a+2)\dots(a+n-1), & n \ge 1, \\\ 1, & n = 0. \end{cases} \tag{3}$$

It is easy to verify that the polynomials *Y<sup>α</sup> <sup>n</sup>* (*x*; *k*) and *Z<sup>α</sup> <sup>n</sup>*(*x*; *k*) are biorthogonal with respect to the weight function *w*(*x*) = *xαe*−*<sup>x</sup>* over the interval (0, ∞), which means

$$\int\_0^\infty x^a e^{-x} \chi\_i^a(x;k) Z\_j^a(x;k) dx = \frac{\Gamma(kj + a + 1)}{j!} \delta\_{ij\prime} \tag{4}$$

where *<sup>α</sup>* <sup>&</sup>gt; <sup>−</sup>1, *<sup>k</sup>* <sup>∈</sup> <sup>Z</sup><sup>+</sup> and *<sup>δ</sup>ij* is the Kronecker delta.

The Laguerre polynomials <sup>L</sup>*<sup>α</sup> <sup>n</sup>*(*x*) are defined as (see, e.g., [3])

$$\mathcal{L}\_n^{\mathfrak{a}}(\mathbf{x}) = \frac{\Gamma(a+n+1)}{\Gamma(n+1)} \sum\_{r=0}^n (-1)^r \binom{n}{r} \frac{\mathbf{x}^r}{\Gamma(a+r+1)}.\tag{5}$$

For *<sup>p</sup>*, *<sup>q</sup>* <sup>∈</sup> <sup>N</sup>, we can define the general hypergeometric functions of *<sup>p</sup>*-numerator and *q*-denominator by

$${}\_{p}F\_{q}\left[\begin{array}{c} \alpha\_{1}, \alpha\_{2}, \dots, \alpha\_{p} \\ \beta\_{1}, \beta\_{2}, \dots, \beta\_{q} \end{array}; \mathbf{x} \right] = \sum\_{n=0}^{\infty} \frac{(\alpha\_{1})\_{n}(\alpha\_{2})\_{n}\dots(\alpha\_{p})\_{n}}{(\beta\_{1})\_{n}(\beta\_{2})\_{n}\dots(\beta\_{q})\_{n}} \frac{\mathbf{x}^{n}}{n!},\tag{6}$$

such that *<sup>β</sup><sup>j</sup>* <sup>=</sup> 0, <sup>−</sup>1, <sup>−</sup>2, . . . ; *<sup>j</sup>* <sup>=</sup> 1, 2, . . . , *<sup>q</sup>*. Then, according to [3], we can rewrite <sup>L</sup>*<sup>α</sup> <sup>n</sup>*(*x*) as

$$\mathcal{L}\_n^a(\mathbf{x}) = \frac{(a+1)\_n}{n!} \,\_1F\_1\left[ \begin{array}{c} -n \\ a+1 \end{array}; \mathbf{x} \right]. \tag{7}$$

For *k* = 1, we note that the Konhauser polynomials (1) and (2) reduce to the Laguerre Polynomials L*α <sup>n</sup>*(*x*) and their special cases; when *k* = 2, the case was encountered earlier by Spencer and Fano [4] in certain calculations involving the penetration of gamma rays through matter and was subsequently discussed in [5].

On the other hand, the matrix theory has become pervasive to almost every area of mathematics, especially in orthogonal polynomials and special functions. The special matrix functions appear in the literature related to statistics [6], Lie theory [7], and in connection with the matrix version of Laguerre, Hermite, and Legendre differential equations and the corresponding polynomial families (see, e.g., [8–10]). In the past few years, the extension of the classical Konhauser polynomials to the Konhauser matrix polynomials of one variable has been a subject of intensive studies [11–14]. Recently, many authors (see, e.g., [15–18]) have proposed the generating relations of Konhauser matrix polynomials of one variable from the Lie algebra method point of view and found some properties of Konhauser matrix polynomials of one variable via the Lie algebra technique; they also obtained operational identities for Laguerre–Konhauser-type matrix polynomials and their applications for the matrix framework.

Some studies have been presented on polynomials in two variables such as 2-variables Shivley's matrix polynomials [19], 2-variables Laguerre matrix polynomials [20], 2-variables Hermite generalized matrix polynomials [21–24], 2-variables Gegenbauer matrix polynomials [25], and the second kind of Chebyshev matrix polynomials of two variables [26].

The purpose of the present work is to introduce and study 2-variables Konhauser matrix polynomials and find the hypergeometric matrix function representations; we try to establish some basic properties of these polynomials which include generating matrix functions, finite sum formulae, and integral representations, and we will also discuss the fractional integrals of the 2-variables Konhauser matrix polynomials.

The rest of this paper is structured as follows. In the next section, we give basic definitions and previous results to be used in the following sections. In Section 3, we introduce the definition of 2-variables Konhauser matrix polynomials for parameter matrices *A* and *B* and some generating matrix relations involving 2-variables Konhauser matrix polynomials deriving the integral representations. Finally, we provide some results on the fractional integrals of 2-variables Konhauser matrix polynomials in Section 4.

#### **2. Preliminaries**

In this section, we give the brief introduction related to Konhauser matrix polynomials and recall some previously known results.

Let <sup>C</sup>*N*×*<sup>N</sup>* be the vector space of N-square matrices with complex entries; for any matrix *<sup>A</sup>* <sup>∈</sup> <sup>C</sup>*N*×*N*, its spectrum *σ*(*A*) is the set of all eigenvalues of *A*,

$$\alpha(A) = \max\{\operatorname{Re}(z) : z \in \sigma(A)\}, \quad \beta(A) = \min\{\operatorname{Re}(z) : z \in \sigma(A)\}.\tag{8}$$

A square matrix *<sup>A</sup>* <sup>∈</sup> <sup>C</sup>*N*×*<sup>N</sup>* is said to be positive stable if and only if *<sup>β</sup>*(*A*) <sup>&</sup>gt; 0. Furthermore, the identity matrix and the null matrix or zero matrix in C*N*×*<sup>N</sup>* will be symbolized by **I** and **0**, respectively. If Φ(*z*) and Ψ(*z*) are holomorphic functions of the complex variable *z*, which are defined as an open set <sup>Ω</sup> of the complex plane and *<sup>A</sup>* is a matrix in <sup>C</sup>*N*×*<sup>N</sup>* with *<sup>σ</sup>*(*A*) <sup>⊂</sup> <sup>Ω</sup>, then, from the properties of the matrix functional calculus [27,28], we have

$$
\Phi(A)\Psi(A) = \Psi(A)\Phi(A). \tag{9}
$$

Furthermore, if *<sup>B</sup>* <sup>∈</sup> <sup>C</sup>*N*×*<sup>N</sup>* is a matrix for which *<sup>σ</sup>*(*B*) <sup>⊂</sup> <sup>Ω</sup> and also if *AB* <sup>=</sup> *BA*, then

$$
\Phi(A)\Psi(B) = \Psi(B)\Phi(A). \tag{10}
$$

Let *A* be a positive stable matrix in C*N*×*N*. Then, Γ(*A*) is well defined as

$$
\Gamma(A) = \int\_0^\infty t^{A-1} e^{-t} dt,\tag{11}
$$

where *<sup>t</sup>A*−*<sup>I</sup>* = exp((*<sup>A</sup>* − *<sup>I</sup>*)ln *<sup>t</sup>*). Then, the matrix Pochhammer symbol (*A*)*<sup>n</sup>* of *<sup>A</sup>* is denoted as follows (see, e.g., [29–31]):

$$\Gamma(A)\_{\mathbb{H}} = \begin{cases} A(A+I)\dots(A+(n-1)I) = \Gamma^{-1}(A)\Gamma(A+nI), & n \ge 1, \\\ I, & n = 0, \end{cases} \tag{12}$$

The Laguerre matrix polynomials are defined by Jódar et al. [8]

$$\mathcal{L}\_{n}^{(A,\lambda)}(\mathbf{x}) = \sum\_{k=0}^{n} \frac{(-1)^{k}\lambda^{k}}{k!(n-k)!} (A+I)\_{n} [(A+I)\_{k}]^{-1} \mathbf{x}^{k},\tag{13}$$

where *<sup>A</sup>* <sup>∈</sup> <sup>C</sup>*N*×*<sup>N</sup>* is a matrix such that <sup>−</sup>*<sup>k</sup>* ∈ *<sup>σ</sup>*(*A*), <sup>∀</sup>*<sup>k</sup>* <sup>∈</sup> <sup>Z</sup>+, (*<sup>A</sup>* <sup>+</sup> *<sup>I</sup>*)*<sup>k</sup>* are given by Equation (12) and *λ* is a complex number with **Re**(*λ*) > 0.

For *<sup>p</sup>*, *<sup>q</sup>* <sup>∈</sup> <sup>N</sup>, 1 <sup>≤</sup> *<sup>i</sup>* <sup>≤</sup> *<sup>p</sup>*, 1 <sup>≤</sup> *<sup>j</sup>* <sup>≤</sup> *<sup>q</sup>*, if *Ai*, *Bj* <sup>∈</sup> <sup>C</sup>*N*×*<sup>N</sup>* are matrices such that *Bj* <sup>+</sup> *kI* are invertible for all integers *k* ≥ 0, the generalized hypergeometric matrix functions are defined as [32]

$${}\_{p}F\_{q}\left[\begin{array}{c} A\_{1}, A\_{2}, \dots, A\_{p} \\ B\_{1}, B\_{2}, \dots, B\_{q} \end{array}; \mathbf{x} \right] = \sum\_{n \geq 0} \frac{(A\_{1})\_{n}(A\_{2})\_{n}\dots(A\_{p})\_{n}[(B\_{1})\_{n}]^{-1}[(B\_{2})\_{n}]^{-1}\dots([B\_{p})\_{n}]^{-1}}{n!} \mathbf{x}^{\mathbf{n}}.\tag{14}$$

It follows that, for *λ* = 1 in (13), we have

$$\mathcal{L}\_n^A(\mathbf{x}) = \frac{(A+I)\_n}{n!} \,\_1F\_1\left[ \begin{array}{c} -nI, \\ A+I \end{array}; \mathbf{x} \right]. \tag{15}$$

For commuting matrices *Ai*, *Bi*, *Ci*, *Di*, *Ei* and *Fi* in C*N*×*N*, we define the Kampé de Fériet matrix series as [32]

*Mathematics* **2020**, *8*, 232

$$\begin{split} & F\_{\begin{subarray}{c} m\_1, n\_1, l\_1\\ m\_2, n\_2, l\_2 \end{subarray}}^{m\_1, m\_1, l\_1} \left[ \begin{array}{c} A, B, \mathbf{C} \\ D, E, F \end{array}; x, y \right] = \\ & \sum\_{m, n \geq 0} \prod\_{i=1}^{m\_1} (A\_i)\_{m+n} \prod\_{i=1}^{n\_1} (B\_i)\_{m} \prod\_{i=1}^{l\_1} (\mathbf{C}\_i)\_{n} \prod\_{i=1}^{m\_2} [(D\_i)\_{m+n}]^{-1} \prod\_{i=1}^{l\_2} [(E\_i)\_{m}]^{-1} \prod\_{i=1}^{l\_2} [(F\_i)\_{n}]^{-1} \frac{\mathbf{x}^m y^n}{m! n!}, \end{split} \tag{16}$$

where *A* abbreviates the sequence of matrices *A*1, ...., *Am*<sup>1</sup> , etc. and *Di* + *kI*, *Ei* + *kI* and *Fi* + *kI* are invertible for all integers *k* ≥ 0.

If *<sup>A</sup>* <sup>∈</sup> <sup>C</sup>*N*×*<sup>N</sup>* is a matrix satisfying the condition

$$\mathbf{Re}(z) > -1, \quad \forall z \in \sigma(A), \tag{17}$$

and *λ* is a complex numbers with **Re**(*λ*) > 0, we recall the following explicit expression for the Konhauser matrix polynomials (see, e.g., [11])

$$Z\_{n}^{(A,\lambda)}(\mathbf{x},k) = \frac{\Gamma(A + (kn+1)I)}{n!} \sum\_{r=0}^{n} (-1)^{r} \binom{n}{r} \Gamma^{-1}(A + (kr+1)I)(\lambda \mathbf{x})^{kr},\tag{18}$$

and

$$\mathcal{Y}\_{\rm n}^{(A,\lambda)}(\mathbf{x};k) = \frac{1}{n!} \sum\_{r=0}^{n} \frac{(\lambda \mathbf{x})^r}{r!} \sum\_{s=0}^{r} (-1)^s \binom{r}{s} \left(\frac{A + (s+1)}{k}\right)\_{\mathbf{n}'} \tag{19}$$

which are biorthogonal with respect to matrix weight function *w*(*x*) = *xAe*−*λ<sup>x</sup>* over the interval (0, ∞).

#### **3. 2-Variables Konhauser Matrix Polynomials**

In this section, we first introduce the 2-variables Konhauser matrix polynomials with parameter matrices *A* and *B*; then, we get the hypergeometric matrix function representations, generating matrix functions, finite summation formulas, and related results for the 2-variables Konhauser matrix polynomials.

**Definition 1.** *Let <sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup> <sup>C</sup>*N*×*<sup>N</sup> be matrices satisfying the condition (17). Then, for <sup>k</sup>*, *<sup>l</sup>* <sup>∈</sup> <sup>Z</sup>+*, the 2-variables Konhauser matrix polynomials Z*(*A*,*B*,*λ*,*ρ*) *<sup>n</sup>* (*x*, *<sup>y</sup>*, *<sup>k</sup>*, *<sup>l</sup>*) *are defined as follows:*

$$\begin{split} Z\_{n}^{(A,B,\lambda,\rho)}(\mathbf{x},\mathbf{y},k,l) &= \frac{\Gamma(A+(kn+1)I)\Gamma(B+(ln+1)I)}{(n!)^2} \\ &\times \sum\_{r=0}^{n} \sum\_{s=0}^{n-r} \frac{(-n)\_{r+s} \ (\lambda \mathbf{x})^{ks} (\rho \mathbf{y})^{lr}}{r! \mathbf{s}!} \Gamma^{-1}(A+(ks+1)I)\Gamma^{-1}(B+(lr+1)I), \end{split} \tag{20}$$

*where λ and ρ are complex numbers with* **Re**(*λ*) > 0 *and* **Re**(*ρ*) > 0*.*

**Remark 1.** *Furthermore, we note the following special cases of the 2-variables Konhauser matrix polynomials <sup>Z</sup>*(*A*,*B*,*λ*,*ρ*) *<sup>n</sup>* (*x*, *<sup>y</sup>*, *<sup>k</sup>*, *<sup>l</sup>*) *as follows:*


$$Z\_{n}^{(A,B,\lambda,1)}(\mathbf{x},\mathbf{y},1,1) = \frac{(A+I)\_{n}(B+I)\_{n}}{(n!)^{2}} \sum\_{r=0}^{n} \sum\_{s=0}^{n-r} \frac{(-n)\_{r+s} \ (\lambda \mathbf{x})^{s} (\mathbf{y})^{r}}{r!s!} \left[ (A+I)\_{s} (B+I)\_{r} \right]^{-1};\tag{21}$$

*iii. Letting <sup>k</sup>* <sup>=</sup> *<sup>l</sup>* <sup>=</sup> <sup>1</sup>*, <sup>B</sup> <sup>=</sup>* **<sup>0</sup>** *and <sup>y</sup>* <sup>=</sup> <sup>0</sup> *in (20), we obtain the Laguerre's matrix polynomials* <sup>L</sup>(*A*,*λ*) *<sup>n</sup>* (*x*) *defined in (13);*


#### *3.1. Hypergeometric Representation*

Now, by using (16) and (20), we obtain the hypergeometric matrix function representations

$$Z\_{n}^{(A,B,\lambda,\rho)}(\mathbf{x},y,k,l) = \frac{(A+I)\_{kn}(B+I)\_{ln}}{(n!)^2} F\_{k,l}^{1} \left[ \begin{array}{c} -nI \\ \Delta(k;A+I), \Delta(l;B+I) \end{array}; (\frac{\lambda x}{k})^k, (\frac{\rho y}{l})^l \right], \tag{22}$$

where Δ(*k*; *A*) abbreviates the array of *k* parameters such that

$$\Delta(k;A) = (\frac{A}{k})(\frac{A+I}{k})(\frac{A+2I}{k})\dots(\frac{A+(k-1)I}{k}), \quad k \ge 1,\tag{23}$$

and *F*<sup>1</sup> *<sup>k</sup>*,*<sup>l</sup>* is defined in (16).

**Remark 2.** *If <sup>A</sup>* <sup>∈</sup> <sup>C</sup>*N*×*<sup>N</sup> is a matrix satisfying the condition (17), letting <sup>B</sup>* <sup>=</sup> *<sup>0</sup> and <sup>y</sup>* <sup>=</sup> <sup>0</sup> *in (22), we obtain*

$$Z\_{n}^{(A,0,\lambda)}(\mathbf{x},0;k) = \frac{(A+I)\_{kn}}{n!} \,\_1F\_{\mathbf{k}}\left[ \begin{array}{c} -nI \\ \Delta(k;A+I) \end{array}; (\frac{\lambda\mathbf{x}}{k})^{k} \right] = Z\_{n}^{(A,\lambda)}(\mathbf{x};k),\tag{24}$$

*where <sup>Z</sup>*(*A*,*λ*) *<sup>n</sup>* (*x*; *<sup>k</sup>*) *are Konhauser matrix polynomials in [11] and* <sup>1</sup>*Fk is hypergeometric matrix function of 1-numerator and k-denominator defined in (14).*

**Remark 3.** *If <sup>A</sup>* <sup>∈</sup> <sup>C</sup>*N*×*<sup>N</sup> is a matrix satisfying the condition (17), let <sup>k</sup>* <sup>=</sup> <sup>1</sup>*, <sup>B</sup> <sup>=</sup>* **<sup>0</sup>** *and <sup>y</sup>* <sup>=</sup> <sup>0</sup> *in (22), then we get*

$$Z\_{n}^{(A,\lambda)}(\mathbf{x};1) = \frac{(A+I)\_{n}}{n!} \,\_1F\_1\left[ \begin{array}{c} -nI, \\ A+I \end{array}; \mathbf{x} \right] = \mathcal{L}\_n^A(\mathbf{x}),\tag{25}$$

*where* L*<sup>A</sup> <sup>n</sup>* (*x*) *are the Laguerre's matrix polynomials defined in (15).*

#### *3.2. Generating Matrix Relations for the 2-Variables of Konhauser Matrix Polynomials*

Generating matrix relations always play an important role in the study of polynomials, first, we give some generating matrix relations for the 2-variables of Konhauser matrix polynomials as follows:

**Theorem 1.** *Letting <sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup> <sup>C</sup>*N*×*<sup>N</sup> be matrices satisfying the condition (17), we obtain the explicit formulae of matrix generating relations for the 2-variables Konhauser matrix polynomials as follows:*

$$\begin{aligned} &\sum\_{n=0}^{\infty} Z\_n^{(A,B,\lambda,\rho)}(x,y,k,l)[(A+I)\_{kn}]^{-1}[(B+I)ln]^{-1}(n!t^n) \\ &= \varepsilon^{t^\dagger} \,\_0F\_{\mathbb{R}}\left[ \begin{array}{c} - \\ \Delta(k;A+I) \end{array}; (\frac{-\lambda x}{k})^k \right] \,\_0F\_{\mathbb{T}}\left[ \begin{array}{c} - \\ \Delta(p;B+I) \end{array}; (\frac{-\rho y}{l})^l \right], \end{aligned} \tag{26}$$

*where* <sup>0</sup>*Fk and* <sup>0</sup>*Fl are hypergeometric matrix functions of* 0*-numerator and k*, *l-denominator as (14),* Δ(*k*; *A* + *I*) *and* Δ(*l*; *B* + *I*) *are defined as (23), and the short line "*−*" means that the number of parameters is zero.*

#### **Proof.** From Equation (20), we have

$$\begin{split} &\sum\_{n=0}^{\infty} Z\_n^{(A,B,\lambda,\rho)}(x,y,k,l) [(A+I)\_{kn}]^{-1} [(B+I)ln]^{-1} (n!t^n) \\ &= \sum\_{n=0}^{\infty} n! \sum\_{r=0}^n \sum\_{s=0}^{n-r} \frac{(-n)\_{r+s} (\lambda x)^{ks} (\rho y)^{lr}}{r! s! (n!)^2} [(A+I)\_{ks}]^{-1} [(B+I)ln]^{-1} t^n \\ &= \sum\_{n=0}^{\infty} \frac{t^n}{n!} \sum\_{s=0}^{\infty} \frac{(-1)^s (\lambda x)^{ks}}{s!} [(A+I)\_{ks}]^{-1} t^s \sum\_{r=0}^{\infty} \frac{(-1)^r (\rho y)^{lr}}{r!} [(B+I)\_{lr}]^{-1} t^r \end{split} \tag{27}$$

by using

$$(A)\_{km} = k^{km} \left(\frac{A}{k}\right)\_m \left(\frac{A+I}{k}\right)\_m \dots \left(\frac{A+(k-1)I}{k}\right)\_{m'}$$

we get

$$\begin{split} &\sum\_{n=0}^{\infty} Z\_n^{(A,B,\lambda,\rho)}(x,y,k,l)[(A+I)\_{kn}]^{-1}[(B+I)ln]^{-1}(n!t^n) \\ &=\sum\_{n=0}^{\infty}\frac{t^n}{n!}\sum\_{s=0}^{\infty}\frac{(-1)^s(\lambda x)^{ks}}{k^{ks}s!}\left[\prod\_{m=1}^k(\frac{A+mI}{k})\_s\right]^{-1}t^s\sum\_{r=0}^{\infty}\frac{(-1)^r(\rho y)^{lr}}{l^{lr}r!}\left[\prod\_{n=1}^l(\frac{B+nI}{l})\_r\right]^{-1}t^r \\ &=e^t\,\_0F\_k\left[\begin{array}{c} - \\ \Delta(k;A+I) \end{array};(\frac{-\lambda x}{k})^k\right]\,\_0F\_l\left[\begin{array}{c} - \\ \Delta(l;B+I) \end{array};(\frac{-\rho y}{l})^l\right]. \end{split} \tag{28}$$

This completes the proof.

For a matrix *E* in C*N*×*N*, we can easily obtain the following generating relations for the 2-variables Konhauser matrix polynomial similar to Theorem 1

$$\begin{split} &\sum\_{n=0}^{\infty} (E)\_n [(A+I)\_{kn}]^{-1} [(B+I)ln]^{-1} (n!t^n) \\ &= (1-t)^{-E} F\_{kJ}^{1} \left[ \begin{array}{c} -E \\ \Delta (k; A+I), \Delta (l; B+I) \end{array}; \frac{t}{t-1} (\frac{\lambda x}{k})^k, \frac{t}{t-1} (\frac{\rho y}{l})^l \right], \end{split} \tag{29}$$

where *F*<sup>1</sup> *<sup>k</sup>*,*<sup>p</sup>* are defined in Equation (16), Δ(*k*; *A* + *I*) and Δ(*l*; *B* + *I*) are defined as Equation (23).

**Corollary 1.** *Letting <sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup> <sup>C</sup>*N*×*<sup>N</sup> be matrices satisfying the condition (17), the following generating matrix relations of the 2-variables Konhauser matrix polynomials hold:*

$$\begin{split} &\sum\_{n=0}^{\infty} Z\_{n}^{(A,B,\lambda,\rho)}(x,y,k,l) \Gamma^{-1}(A+(nl+1)I) \Gamma^{-1}(B+(nl+1)I)(n!t^{n}) \\ &= e^{t} \, \Gamma^{-1}(A+I) \, \Gamma^{-1}(B+I) \, \, \_0F\_{k} \left[ \begin{array}{c} - \\ \Delta(k;A+I) \end{array}; (\frac{-\lambda x}{k})^{k} \right] \, \, \_0F\_{l} \left[ \begin{array}{c} - \\ \Delta(l;B+I) \end{array}; (\frac{-\rho y}{l})^{l} \right], \end{split} \tag{30}$$

*where* <sup>0</sup>*Fk and* <sup>0</sup>*Fl are hypergeometric matrix functions of 0-numerator and k,l-denominator as (14).*

**Corollary 2.** *Letting A, B, and E be matrices in* C*N*×*<sup>N</sup> satisfying the condition (17), we give explicit formulae of matrix generating relations for the 2-variables Konhauser matrix polynomials as follows:*

$$\begin{split} &\sum\_{n=0}^{\infty} (E)\_n Z\_n^{(A,B,\lambda,\rho)} (x, y, k, l) \Gamma^{-1} (A + (nk + 1)I) \Gamma^{-1} (B + (nl + 1)I) (n! l^n) \\ &= (1 - t)^{-E} \Gamma^{-1} (A + I) \, \Gamma^{-1} (B + I) F\_{k,l}^1 \left[ \begin{array}{c} -E \\ \Delta (k; A + I), \Delta (l; B + I) \end{array}; \frac{t}{t - 1} (\frac{\lambda x}{k})^k, \frac{t}{t - 1} (\frac{\rho y}{l})^l \right]. \end{split} \tag{31}$$

Considering the double series,

$$\begin{split} &\sum\_{n=0}^{\infty} \sum\_{m=0}^{\infty} \frac{[(m+n)!]^2}{n! \ m!} Z\_n^{(A,B,\lambda,\rho)}(\mathbf{x}, y, k, l) [(A+I)\_{k(m+n)}]^{-1} [(B+I)l(m+n)]^{-1} \sigma^m \tau^n \\ &= \sum\_{n=0}^{\infty} n! Z\_n^{(A,B,\lambda,\rho)}(\mathbf{x}, y, k, l) \tau^n [(A+I)\_{kn}]^{-1} [(B+I)ln]^{-1} \,\_1F\_0 \left[ \begin{array}{c} -nI \\ - \end{array}; \frac{-\sigma^\prime}{\tau} \right] \\ &= \sum\_{n=0}^{\infty} n! Z\_n^{(A,B,\lambda,\rho)}(\mathbf{x}, y, k, l) [(A+I)\_{kn}]^{-1} [(B+I)ln]^{-1} (\sigma+\tau)^n . \end{split} \tag{32}$$

Now, by making use of Theorem 1, we find

$$\begin{split} &\sum\_{n=0}^{\infty} \sum\_{m=0}^{\infty} \frac{[(m+n)!]^2}{n!} Z\_n^{(A,B,\lambda,\rho)}(\mathbf{x}, y, k, l) [(A+I)\_{k(m+n)}]^{-1} [(B+I)l(m+n)]^{-1} \sigma^m \tau^n \\ &= e^{\sigma^{r+\mathsf{T}}} \,\_0F\_{\mathbf{k}} \left[ \begin{array}{c} - \\ \Delta(k; A+I) \end{array}; (\frac{-\lambda \mathbf{x}}{k})^k (\sigma + \tau) \right] \,\_0F\_{\mathbf{l}} \left[ \begin{array}{c} - \\ \Delta(l; B+I) \end{array}; (\frac{-\rho \mathbf{y}}{l})^l (\sigma + \tau) \right]. \end{split} \tag{33}$$

Here, Equation (33) may be regarded as a double generating matrix relations for (20).

**Remark 4.** *For A in* C*N*×*N, letting k* = 1*, B* = *0 and y* = 0 *in (33), we have*

$$\begin{split} &\sum\_{n=0}^{\infty} \binom{m+n}{n} [(A+I)\_{(m+n)}]^{-1} \mathcal{L}\_{m+n}^{A}(\mathbf{x}) \; t^{n} \\ &= \sum\_{n=m}^{\infty} \frac{(-1)^{n} m! [(A+I)\_{n}]^{-1} \mathbf{x}^{n}}{(n-m)! n!} \,\_1F\_1 \left[ \begin{matrix} -(n+1)I\_{\prime} \\ (n-m+1)I \end{matrix}; t \right] \\ &= \sum\_{n=m}^{\infty} \sum\_{j=0}^{\infty} \frac{(-\mathbf{x})^{n} n! t^{n-m} [(A+I)\_{n}]^{-1} (n+1)\_{j} t^{j}}{m! (n-m)! n! (n-m+1) j!} \\ &= \sum\_{n=0}^{\infty} \sum\_{j=0}^{\infty} \frac{(-\mathbf{x})^{n} (n+1)\_{j} (A+I)\_{n}]^{-1} \dot{t}^{j}}{(1)\_{j} n! j!}, \end{split} \tag{34}$$

*we find generating matrix relations of the Laguerre's matrix polynomials.*

#### *3.3. Some Properties of the 2-Variables Konhauser Matrix Polynomials*

For the finite sum property of the 2-variables Konhauser matrix polynomials *<sup>Z</sup>*(*A*,*B*,*λ*,*ρ*) *<sup>n</sup>* (*x*, *<sup>y</sup>*, *<sup>k</sup>*, *<sup>l</sup>*), we get the generating relations together as follows:

$$\begin{split} &e^{t} \, \_0F\_{\mathbf{k}} \left[ \begin{array}{c} - \\ \Delta(k; A+I) \end{array}; (\frac{-\lambda xw}{k})^{k}t \right] \, \_0F\_{\mathbf{l}} \left[ \begin{array}{c} - \\ \Delta(l; B+I) \end{array}; (\frac{-\rho yw}{l})^{l}t \right] \\ &= e^{(1-w^{k})t} \, \_0F\_{\mathbf{k}} \left[ \begin{array}{c} - \\ \Delta(k; A+I) \end{array}; (\frac{-\lambda xw}{k})^{k}t \right] \, \_0F\_{\mathbf{l}} \left[ \begin{array}{c} - \\ \Delta(l; B+I) \end{array}; (\frac{-\rho yw}{l})^{l}t \right], \end{split} \tag{35}$$

and *<sup>n</sup>*

$$\begin{split} &\sum\_{r=0}^{n} Z\_{n}^{(A,B,\lambda,\rho)}(\mathbf{x}w, yw, k, k) [(A+I)\_{kn}]^{-1} [(B+I)\_{kn}]^{-1} t^{n} n! \\ &= \left(\sum\_{n=0}^{\infty} \frac{1-w^{kn}t^{n}}{n!} \right) \left(\sum\_{n=0}^{\infty} Z\_{n}^{(A,B,\lambda,\rho)}(\mathbf{x}, y, k, k) w^{kn} [(A+I)\_{kn}]^{-1} [(B+I)\_{kn}]^{-1} t^{n} n! \right). \end{split} \tag{36}$$

By comparing the coefficients of *t <sup>n</sup>* on both sides, we have

$$\begin{split} & Z\_{n}^{(A,B,\lambda,\rho)}(\mathbf{x}w, yw, k, k) \\ &= \sum\_{r=0}^{n} \frac{r! w^{kr} (1 - w^{k})^{n-r}}{n! (n-r)!} [(A + I)\_{kr}]^{-1} [(B + I)\_{kr}]^{-1} (A + I)\_{kr} (B + I)\_{kr} Z\_{n}^{(A,B,\lambda,\rho)}(\mathbf{x}, y, k, k). \end{split} \tag{37}$$

The integral representations for the 2-variables Konhauser matrix polynomials are derived in the following theorem.

**Theorem 2.** *Letting <sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup> <sup>C</sup>*N*×*<sup>N</sup> be matrices satisfying the condition (17), and, if t λx* <sup>&</sup>lt; <sup>1</sup>*, v ρy* <sup>&</sup>lt; <sup>1</sup>*, then we have the integral representation of the 2-variables Konhauser matrix polynomials <sup>Z</sup>*(*A*,*B*,*λ*,*ρ*) *<sup>n</sup>* (*x*, *<sup>y</sup>*, *<sup>k</sup>*, *<sup>l</sup>*) *as follows:*

$$\begin{split} \mathcal{Z}\_{n}^{(A,B,\lambda,\rho)}(\mathbf{x},\mathbf{y},k,l) &= \frac{\Gamma(A+(kn+1)I)\Gamma(B+(ln+1)I)}{(n!)^2(2\pi i)^2} \\ &\times \int\_{c\_1} \int\_{c\_2} \left(t^k v^l - (\lambda \mathbf{x})^k v^l - (\rho y)^k t^l\right)^n e^{t+v} \cdot t^{-(A+(kn+1)I)} \cdot v^{-(B+(ln+1)I)} dt dv, \end{split} \tag{38}$$

*where c*1, *c*<sup>2</sup> *are the paths around the origin in the positive direction, beginning at and returning to positive infinity with respect for the branch cut along the positive real axis.*

**Proof.** The right side of the above formulae are deformed into

$$\begin{split} &\frac{\Gamma\left(A + (kn+1)I\right)\Gamma\left(B + (ln+1)I\right)}{(n!)^2} \sum\_{r=0}^{n} \sum\_{s=0}^{n-r} \frac{(-n)\_{r+s} (\lambda x)^{kr} (\rho y)^{lr}}{r!s!} \\ & \times \frac{1}{2\pi i} \int\_{c\_1} t^{-(A + (ks+1)I)} e^t dt \times \frac{1}{2\pi i} \int\_{c\_2} v^{-(B+(lr+1)I)} e^v dv, \end{split} \tag{39}$$

and using the integral representation of the reciprocal Gamma function, which are given in [34]

$$\frac{1}{\Gamma(z)} = \frac{1}{2\pi i} \int\_c e^t t^{-z} dt,\tag{40}$$

where *c* is the path around the origin in the positive direction, beginning at and returning to positive infinity with respect for the branch cut along the positive real axis. Thus, from Equation (40), we obtain the following integral matrix functional

$$
\Gamma^{-1}(A + (kn+1)I) = \frac{1}{2\pi i} \int\_{\mathcal{L}\_1} e^t t^{-(A + (kn+1)I)} dt. \tag{41}
$$

By Equation (41), we can transfer (39) to

$$\begin{split} &\frac{\Gamma(A+(kn+1)I)\Gamma(B+(ln+1)I)}{(n!)^2} \times \\ &\sum\_{r=0}^n \sum\_{s=0}^{n-r} \frac{(-n)\_{r+s} (\lambda x)^{kr} (\rho y)^{lr}}{r!s!} \Gamma^{-1}(A+(ks+1)I)\Gamma^{-1}(B+(kr+1)I) \\ &= Z\_n^{(A,B,\lambda,\rho)}(x,y,k,l). \end{split} \tag{42}$$

This completes the proof of the theorem.

#### **4. Fractional Integrals of the 2-Variable Konhauser Matrix Polynomials**

In this section, we study the fractional integrals of the Konhauser matrix polynomials of one and two variables. The fractional integrals of Riemann–Liouville operators of order *μ* and *x* > 0 are given by (see [35,36])

$$(\mathbf{I}\_a^\mu f)(\mathbf{x}) = \frac{1}{\Gamma(\mu)} \int\_a^\mathbf{x} (\mathbf{x} - t)^{\mu - 1} f(t) dt, \quad \mathbf{Re}(\mu) > 0. \tag{43}$$

Recently, the authors (see, e.g., [28]) introduced the fractional integrals with matrix parameters as follows: suppose *<sup>A</sup>* <sup>∈</sup> <sup>C</sup>*N*×*<sup>N</sup>* is a positive stable matrix and *<sup>μ</sup>* <sup>∈</sup> <sup>C</sup> is a complex number satisfying the condition **Re**(*μ*) > 0. Then, the Riemann–Liouville fractional integrals with matrix parameters of order *μ* are defined by

$$\mathbf{I}^{\mu}(\mathbf{x}^{A}) = \frac{1}{\Gamma(\mu)} \int\_{0}^{\mathbf{x}} (\mathbf{x} - t)^{\mu - 1} t^{A} dt. \tag{44}$$

**Lemma 1.** *Supposing that <sup>A</sup>* <sup>∈</sup> <sup>C</sup>*N*×*<sup>N</sup> is a positive stable matrix and <sup>μ</sup>* <sup>∈</sup> <sup>C</sup> *is a complex number satisfying the condition* **Re**(*μ*) > 0*, then the Riemann–Liouville fractional integrals with matrix parameters of order μ are defined and we have (see, e.g., [28])*

$$\mathbf{I}^{\mu}(\mathbf{x}^{A-I}) = \Gamma(A)\Gamma^{-1}(A+\mu I)\mathbf{x}^{A+(\mu-1)I}.\tag{45}$$

**Theorem 3.** *If <sup>A</sup>* <sup>∈</sup> <sup>C</sup>*N*×*<sup>N</sup> is a matrix satisfying the condition (17), then the Riemann–Liouville fractional integrals of Konhauser matrix polynomials of one variable are as follows:*

$$\mathbf{I}^{\mu}\left[ (\lambda \mathbf{x})^{A} Z\_{\mathbf{n}}^{(A,\lambda)}(\mathbf{x},k) \right] = \Gamma^{-1}(A + (k\mathbf{n} + \mu + 1)I)\Gamma(A + (k\mathbf{n} + 1)I)(\lambda \mathbf{x})^{A + \mu I} Z\_{\mathbf{n}}^{(A + \mu I,\lambda)}(\mathbf{x},k), \tag{46}$$

*where <sup>λ</sup> is a complex numbers with* **Re**(*λ*) <sup>&</sup>gt; 0, *and k* <sup>∈</sup> <sup>Z</sup>+*.*

**Proof.** From Equation (44), we find

$$\begin{split} &\mathbb{I}^{\mathbb{H}}\left[\left(\lambda x\right)^{A}\mathbb{Z}\_{n}^{(A,\lambda)}(\mathbf{x},k)\right] = \int\_{0}^{\mathbf{x}} \frac{\left(\lambda(\mathbf{x}-t)\right)^{\mu-1}}{\Gamma(\mu)} t^{A} \mathbb{Z}\_{n}^{(A,\lambda)}(t,k) dt \\ &= \frac{\Gamma(A+(kn+1)I)}{\Gamma(\mu)} \sum\_{r=0}^{n} \frac{(-1)^{r}}{r!(n-r)!} \Gamma^{-1}(A+(kr+1)I) \int\_{0}^{\mathbf{x}} (\lambda \mathbf{x})^{A+krI} \left(\lambda(\mathbf{x}-t)\right)^{\mu-1} dt \\ &= \Gamma(A+(kn+1)I) \sum\_{r=0}^{n} \frac{(-1)^{r}}{r!(n-r)!} (\lambda \mathbf{x})^{A+(kr+\mu)I} \Gamma^{-1}(A+(kr+\mu+1)I) . \end{split} \tag{47}$$

and we can write

$$\mathbf{I}^{\mu}\left[\left(\lambda\mathbf{x}\right)^{A}\mathbf{Z}\_{n}^{(A,\lambda)}(\mathbf{x},k)\right] = \Gamma^{-1}(A+(kn+\mu+1)I)\Gamma(A+(kn+1)I)(\lambda\mathbf{x})^{A+\mu I}\mathbf{Z}\_{n}^{(A+\mu I,\lambda)}(\mathbf{x},k). \tag{48}$$
  $\Box$ 

The 2-variables analogue of Riemann–Liouville fractional integrals **I***ν*,*<sup>μ</sup>* may be defined as follows

**Definition 2.** *Letting <sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup> <sup>C</sup>*N*×*<sup>N</sup> be positive stable matrices, if* **Re**(*ν*) <sup>&</sup>gt; <sup>0</sup> *and* **Re**(*μ*) <sup>&</sup>gt; <sup>0</sup>*, then the 2-variables Riemann–Liouville fractional integrals of orders ν*, *μ can be defined as follows:*

$$\mathbf{I}^{\nu,\mu} \left[ \mathbf{x}^A \mathbf{y}^B \right] = \frac{1}{\Gamma(\nu)\Gamma(\mu)} \int\_0^\mathbf{x} \int\_0^y (\mathbf{x} - \mathbf{u})^{\nu - 1} (\mathbf{y} - \mathbf{v})^{\mu - 1} \mathbf{u}^A \mathbf{v}^B d\mathbf{u} d\mathbf{v}. \tag{49}$$

**Theorem 4.** *Letting <sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup> <sup>C</sup>*N*×*<sup>N</sup> be matrices satisfying the condition (17),* **Re**(*λ*) <sup>&</sup>gt; 0, **Re**(*ρ*) <sup>&</sup>gt; <sup>0</sup>*, then, for the Riemann–Liouville fractional integral of a 2-variables Konhauser matrix polynomial, we have the following:*

$$\begin{aligned} &\Gamma^{\nu,\mu}\left[ (\lambda x)^A (\rho y)^B Z\_n^{(A,B,\lambda,\rho)} (x,y,k,l) \right] \\ &= \Gamma^{-1} (A + (kn + \nu + 1)I) \Gamma^{-1} (B + (ln + \mu + 1)I) \Gamma (A + (kn + 1)I) \\ &\Gamma (B + (ln + 1)I) (\lambda x)^{A + \nu I} (\rho y)^{B + \mu I} Z\_n^{(A + \nu I, B + \mu I, \lambda, \rho)} (x,y,k,l), \end{aligned} \tag{50}$$

*where <sup>λ</sup> and <sup>ρ</sup> are complex numbers and k*, *<sup>l</sup>* <sup>∈</sup> <sup>Z</sup>+*.*

**Proof.** By using Equation (49), we obtain

$$\begin{split} \mathbf{I}^{\nu,\mu} \left[ (\lambda \mathbf{x})^A (\rho \mathbf{y})^B Z\_n^{(A,B,\lambda,\rho)} (\mathbf{x}, \mathbf{y}, k, l) \right] &= \frac{1}{\Gamma(\nu)\Gamma(\mu)} \\ \times \int\_0^\mathbf{x} \int\_0^y \left( \lambda (\mathbf{x} - \boldsymbol{u}) \right)^{\nu - 1} (\rho (\mathbf{y} - \boldsymbol{v}))^{\mu - 1} (\lambda \boldsymbol{u})^A (\rho \boldsymbol{v})^B Z\_n^{(A,B,\lambda,\rho)} (\boldsymbol{u}, \boldsymbol{v}, k, l) d\boldsymbol{u} d\boldsymbol{v}. \end{split} \tag{51}$$

By putting *u* = *xt* and *v* = *yw*, we get

$$\begin{split} \mathbf{1}^{\mathbb{V},\mu} \left[ (\lambda \mathbf{x})^{A} (\rho y)^{B} Z\_{\mathbf{n}}^{(A,B,\lambda,\rho)} (\mathbf{x}, y, k, l) \right] &= \frac{(\lambda \mathbf{x})^{A + \nu I} (\rho y)^{B + \mu I}}{\Gamma(\nu) \Gamma(\mu)} \\ &\times \int\_{0}^{1} \int\_{0}^{1} (\lambda t)^{A} (\rho w)^{B} \left( \lambda (1 - t) \right)^{\nu - 1} \left( \rho (1 - w) \right)^{\mu - 1} Z\_{\mathbf{n}}^{(A,B,\lambda,\rho)} (\mathbf{x} t, y w, k, l) dt dw, \end{split} \tag{52}$$

from definition (20), we have

$$\begin{split} &\mathbb{E}^{\mathbb{P},\mathbb{P}}\Big[ (\lambda \mathbf{x})^{A} (\rho y)^{B} Z\_{n}^{(A,B,\lambda,\rho)} (\mathbf{x}, y, \lambda, l) \Big] \\ &= \frac{\Gamma(A + (kn+1)I)\Gamma(B + (ln+1)I)(\lambda \mathbf{x})^{A + \nu I} (\rho y)^{B + \mu I}}{(n!)^{2} \Gamma(\nu)\Gamma(\mu)} \\ &\sum\_{r=0}^{n} \sum\_{s=0}^{n-r} \frac{(-n)\_{r+s} \left(\lambda \mathbf{x}\right)^{ks} (\rho y)^{lr}}{r!s!} \Gamma^{-1}(A + (ks+1)I) \Gamma^{-1}(B + (lr+1)I). \\ &\times \int\_{0}^{1} (\lambda \mathbf{t})^{A + k \text{sI}} (\lambda (1-t))^{v-1} dt \int\_{0}^{1} (\rho w)^{B + l \text{rI}} (\rho (1-w))^{v-1} dw, \end{split} \tag{53}$$

and

$$\begin{aligned} &\mathbf{I}^{\nu,\mu}\left[ (\lambda x)^{A} (\rho y)^{B} Z\_{n}^{(A,B,\lambda,\rho)} (x, y, k, l) \right] \\ &= \frac{\Gamma(A + (kn + 1)I) \Gamma(B + (ln + 1)I) (\lambda x)^{A + \nu I} (\rho y)^{B + \mu I}}{(n!)^{2} \Gamma(\nu) \Gamma(\mu)} \\ &\sum\_{r=0}^{n} \sum\_{s=0}^{n-r} \frac{(-n)\_{r + s} (\lambda x)^{ks} (\rho y)^{lr}}{r! s!} \Gamma^{-1}(A + (ks + \nu + 1)I) \Gamma^{-1}(B + (lr + \mu + 1)I). \end{aligned} \tag{54}$$

We thus arrive at

$$\begin{split} &\Gamma^{\nu,\mu}\Big[ (\lambda x)^{A} (\rho y)^{B} Z\_{n}^{(A,B,\lambda,\rho)} (x,y,k,l) \Big] \\ &= \Gamma^{-1} (A + (kn + \nu + 1)I) \Gamma^{-1} (B + (ln + \mu + 1)I) \Gamma (A + (kn + 1)I) \\ &\Gamma (B + (ln + 1)I) (\lambda x)^{A + \nu I} (\rho y)^{B + \mu l} Z\_{n}^{(A + \nu l, B + \mu l, \lambda, \rho)} (x,y,k,l) .\end{split} \tag{55}$$

This completes the proof of Theorem 4.

**Author Contributions:** All authors contributed equally and all authors have read and agreed to the published version of the manuscript.

**Funding:** This research is supported by the National Natural Science Foundation of China (11601525). **Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


c 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Fractional Supersymmetric Hermite Polynomials**

#### **Fethi Bouzeffour 1,\* and Wissem Jedidi 2,3**


Received: 12 December 2019; Accepted: 31 January 2020; Published: 5 February 2020

**Abstract:** We provide a realization of fractional supersymmetry quantum mechanics of order *r*, where the Hamiltonian and the supercharges involve the fractional Dunkl transform as a Klein type operator. We construct several classes of functions satisfying certain orthogonality relations. These functions can be expressed in terms of the associated Laguerre orthogonal polynomials and have shown that their zeros are the eigenvalues of the Hermitian supercharge. We call them the supersymmetric generalized Hermite polynomials.

**Keywords:** orthogonal polynomials; difference-differential operator; supersymmetry

#### **1. Introduction**

Supersymmetry relates bosons and fermions on the basis of Z2-graded superalgebras [1,2], where the fermionic set is realized in terms of matrices of finite dimension or in terms of Grassmann variables [3]. The supersymmetric quantum mechanics (SUSYQM), introduced by Witten [2], may be generated by three operators *Q*−, *Q*<sup>+</sup> and *H* satisfying

$$Q^2\_{\pm} = 0, \quad [Q\_{\pm}, H] = 0, \quad \{Q\_-, Q\_+\} = H. \tag{1}$$

Superalgebra (1) corresponds to the case *N* = 2 supersymmetry. The usual construction of Witten's supersymmetric quantum mechanics with the superalgebra (1) is performed by introduction of fermion degrees of freedom (realized in a matrix form, or in terms of Grassmann variables) which commute with bosonic degrees of freedom. Another realization of supersymmetric quantum mechanics, called minimally bosonized supersymmetric quantum [1,4,5], is built by taking the supercharge as the following Dunkl-type operator:

$$
\mathbb{Q} = \partial\_x \mathbb{R} + v(x),
$$

where *v*(*x*) is a superpotential.

The fractional supersymmetric quantum mechanics of order *r* (FSUYQM) are an extension of the ordinary supersymmetric quantum mechanics for which the Z2-graded superalgebras are replaced by a Z*r*-graded superalgberas [3,6,7]. The framework of the fractional supersymmetric quantum mechanics has been shown to be quite fruitful. Amongst many works, we may quote the deformed Heisenberg algebra introduced in connection with parafermionic and parabosonic systems [3,4], the *Cλ*-extended oscillator algebra developed in the framework of parasupersymmetric quantum mechanics [8], and the generalized Weyl–Heisenberg algebra *Wk* related to Z*k*-graded supersymmetric quantum mechanics [3]. Note that the construction of fractional supersymmetric quantum mechanics without employment of fermions and parafermions degrees of freedom was started in [4,9,10]. In particular, the idea of realization of fractional supersymmetry in the form as it was presented in [3,8] was initially proposed in [4] and also in [9]. In this work, we develop a fractional supersymmetric quantum of order *r* without parafermonic degrees of freedom. We essentially use a difference-differential operators generated from a special case of the well known fractional Dunkl transform. We then investigate the characteristics of the (*r*)-scheme.

The paper is organized as follows. In Section 2, we discuss some of basic properties of the fractional Dunkl transform and we define the generalized Klein operator. In Section 3, we present a realization of the fractional supersymmetric quantum mechanics and we construct a basis involving the generalized Hermite functions that diagonalize the Hamiltonian. In Section 4, we define the associated generalized Hermite polynomials and we provide its weight function and we show that the eigenvalues of the supercharge are the zeros of the associated generalized Hermite polynomials.

#### **2. Preliminaries**

Recall that the fractional Dunkl transform on the real line, introduced in [11,12], is both an extension of the fractional Hankel transform and the Fourier transform. For 0 < |*α*| < *π*, the fractional Dunkl transform is defined by:

$$\mathcal{F}\_{\boldsymbol{\nu}}^{\boldsymbol{\kappa}}f(t) = \frac{e^{i(\boldsymbol{\nu}+1/2)(\boldsymbol{\mu}\boldsymbol{\kappa}/2-\boldsymbol{a})}}{(2|\sin(\boldsymbol{a})|)^{\boldsymbol{\nu}+1/2}\Gamma(\boldsymbol{\nu}+1/2)} \int\_{\mathbb{R}} e^{-i\frac{t^{2}+\boldsymbol{x}^{2}}{2\sin\boldsymbol{a}}} \mathcal{E}\_{\boldsymbol{\nu}}(\frac{it\boldsymbol{x}}{\sin\boldsymbol{a}})f(\boldsymbol{x})\,|\boldsymbol{x}|^{2\boldsymbol{\nu}}d\boldsymbol{x}\,\boldsymbol{\kappa}$$

where

$$
\mathfrak{a} = \operatorname{sgn}(\sin(a)),
$$

and

$$\begin{aligned} \mathcal{E}\_{\boldsymbol{\nu}}(\mathbf{x}) &:= \quad \mathcal{J}\_{\boldsymbol{\nu}-1/2}(i\mathbf{x}) + \frac{\boldsymbol{\chi}}{2\boldsymbol{\nu}+1} \, \mathcal{J}\_{\boldsymbol{\nu}+1/2}(i\mathbf{x}),\\ \mathcal{J}\_{\boldsymbol{\nu}}(\mathbf{x}) &:= \quad \Gamma(\boldsymbol{\nu}+1) \, (2/\boldsymbol{\nu})^{\boldsymbol{\nu}} \, \mathcal{J}\_{\boldsymbol{\nu}}(\mathbf{x}). \end{aligned}$$

Notice that *Jν*(*x*) is the standard Bessel function ([13] Ch. 10) and Γ(*x*) is the Gamma function. It is well known that, for *ν* > 0, the function E*ν*(*λx* ) is the unique analytic solution of the following system that can be found in [14]:

$$\begin{cases} \mathcal{Y}\_{\mathcal{V}} \mathcal{E}\_{\mathcal{V}}(\lambda x) = i \,\lambda \, \mathcal{E}\_{\mathcal{V}}(\lambda x),\\ \mathcal{E}\_{\mathcal{V}}(0) = 1, \end{cases} \tag{2}$$

where *Y<sup>ν</sup>* is the Dunkl operator related to root system *A*<sup>1</sup> (see ([14] Definition 4.4.2))), which is a differential-difference operator, depending on a parameter *<sup>ν</sup>* <sup>∈</sup> <sup>R</sup> and acting on *<sup>C</sup>*∞(R) as:

$$\mathcal{Y}\_{\nu} := \frac{d}{d\mathbf{x}} + \frac{\nu}{\mathbf{x}} (\mathbf{1} - \mathcal{R})\_{\prime} \tag{3}$$

where *R* is the Klein operator :

$$(\mathcal{R}f)(\mathbf{x}) = f(-\mathbf{x}).\tag{4}$$

*Mathematics* **2020**, *8*, 193

The operator *Yν* is also related by a simple similarity transformation to the Yang–Dunkl operator used in Refs. [1,4,10]. The corresponding Dunkl harmonic oscillator and the annihilation and creation operators take the forms [15]

$$H\_{\nu} = -\frac{1}{2}Y\_{\nu}^{2} + \frac{1}{2}\mathbf{x}^{2} = -\frac{1}{2}\frac{d^{2}}{dx^{2}} - \frac{\nu}{\mathbf{x}}\frac{d}{dx} + \frac{\nu}{2\mathbf{x}^{2}}(1 - R) + \frac{1}{2}\mathbf{x}^{2},\tag{5}$$

$$A\_{-}=\frac{1}{\sqrt{2}}(\mathbb{Y}\_{\mathbb{V}}+\mathbf{x}),\quad A\_{+}=\frac{1}{\sqrt{2}}(-\mathbb{Y}\_{\mathbb{V}}+\mathbf{x}).\tag{6}$$

They satisfy the (anti)commutation relations

$$\begin{bmatrix} A\_{-\prime} \ A\_{+} \end{bmatrix} = 1 + 2\nu R, \quad R^2 = 1, \quad \{A\_{\pm}, R\} = 0, \quad [1, A\_{\pm}] = [1, R] = 0. \tag{7}$$

The generators 1, *A*±, *R*, and relations (7) give us a realization of the *R*-deformed Heisenberg algebra [1,10]. In [9,13], the authors show that the *R*-deformed algebra is intimately related to parabosons, parafermions [13] and to the *osp*(1|2) *osp*(2|2) superalgebras.

From now, we assume that *ν* > 0. The adjoint *Y*∗ *<sup>ν</sup>* of the Dunkl operators *<sup>Y</sup><sup>ν</sup>* with domain <sup>S</sup>(R) (the space <sup>S</sup>(R) being dense in *<sup>L</sup>*2(R, <sup>|</sup>*x*<sup>|</sup> <sup>2</sup>*<sup>ν</sup> dx*)) is <sup>−</sup>*Y<sup>ν</sup>* and therefore the operator *<sup>H</sup><sup>ν</sup>* is self-adjoint, its spectrum is discrete, and the wave functions corresponding to the well-known eigenvalues

$$
\lambda\_n = n + \nu + \frac{1}{2}, \quad n = 0, 1, 2, \cdots \tag{8}
$$

are given by

$$
\psi\_n^{(\upsilon)}(\mathbf{x}) = \gamma\_n^{-1/2} e^{-\mathbf{x}^2/2} H\_n^{(\upsilon)}(\mathbf{x}),
\tag{9}
$$

where

$$\gamma\_n = 2^{2n} \Gamma([\frac{n}{2}]+1) \Gamma([\frac{n+1}{2}]+\nu+\frac{1}{2}), \ n = 0, 1, 2, \dots, \dots$$

[*x*] denotes the greatest integer function and *<sup>H</sup>*(*ν*) *<sup>n</sup>* (*x*) is the generalized Hermite polynomial introduced by Szegö [15–17] and obtained from Laguerre polynomial *L*(*ν*) *<sup>n</sup>* (*x*) as follows:

$$\begin{cases} H\_{2n}^{(\nu)}(\mathbf{x}) = (-1)^n 2^{2n} n! \, L\_n^{(\nu - \frac{1}{2})}(\mathbf{x}^2), \\ H\_{2n+1}^{(\nu)}(\mathbf{x}) = (-1)^n 2^{2n+1} n! \, \mathbf{x} \, L\_n^{(\nu + \frac{1}{2})}(\mathbf{x}^2). \end{cases}$$

It is well known that for *ν* > 0, these polynomials satisfy the orthogonality relations :

$$\int\_{\mathbb{R}} H\_n^{(\nu)}(\mathbf{x}) H\_m^{(\nu)}(\mathbf{x}) |\mathbf{x}|^{2\nu} e^{-\mathbf{x}^2} d\mathbf{x} = \gamma\_n \delta\_{n\cdot m}.\tag{10}$$

We define the generalized Klein operator *<sup>K</sup>* as a special case of the fractional Dunkl transform <sup>F</sup>*<sup>α</sup> ν* corresponding to *α* = <sup>2</sup>*<sup>π</sup> <sup>r</sup>* . That is,

$$
\mathbb{K} = \mathcal{F}\_{\nu}^{\frac{2\pi}{\nu}}.\tag{11}
$$

It is well known that the wave functions *<sup>ψ</sup>*(*ν*) *<sup>n</sup>* (*x*) form an orthonormal basis of *<sup>L</sup>*2(R, <sup>|</sup>*x*<sup>|</sup> <sup>2</sup>*<sup>ν</sup> dx*) and are also eigenfunctions of the Fourier–Dunkl transform [11,12,15]. In particular, the generalized Klein operator *<sup>K</sup>* acts on the wave functions *<sup>ψ</sup>*(*ν*) *<sup>n</sup>* (*x*) as:

$$K\psi\_n(\mathbf{x}) = \varepsilon\_r^n \psi\_n^{(\nu)}(\mathbf{x}), \quad \varepsilon\_r = \varepsilon^{\frac{2in}{r}}.$$

Let us consider the <sup>Z</sup>*r*-grading structure on the space *<sup>L</sup>*2(R, <sup>|</sup>*x*<sup>|</sup> <sup>2</sup>*<sup>ν</sup> dx*) as

$$L^2(\mathbb{R}, |\mathbf{x}|^{2\nu} \, d\mathbf{x}) = \bigoplus\_{j=0}^{r-1} L\_j^2(\mathbb{R}, |\mathbf{x}|^{2\nu} \, d\mathbf{x}), \tag{12}$$

where *L*<sup>2</sup> *<sup>j</sup>*(R, <sup>|</sup>*x*<sup>|</sup> <sup>2</sup>*<sup>ν</sup> dx*) is a linear subspace of *<sup>L</sup>*2(R, <sup>|</sup>*x*<sup>|</sup> <sup>2</sup>*<sup>ν</sup> dx*) generated by the generalized wave functions

$$\{\psi\_{nr+j}^{(\nu)}(x) \; : \; n = 0, \; 1, \; 2, \cdot \cdot \cdot \} .$$

For *<sup>j</sup>* <sup>=</sup> 0, 1, ··· , *<sup>r</sup>* <sup>−</sup> 1, we denote by <sup>Π</sup>*j*, the orthogonal projection from *<sup>L</sup>*2(R, <sup>|</sup>*x*<sup>|</sup> <sup>2</sup>*<sup>ν</sup> dx*) onto its subspace *L*<sup>2</sup> *<sup>j</sup>*(R, <sup>|</sup>*x*<sup>|</sup> <sup>2</sup>*<sup>ν</sup> dx*). The action of <sup>Π</sup>*<sup>j</sup>* on *<sup>L</sup>*2(R, <sup>|</sup>*x*<sup>|</sup> <sup>2</sup>*<sup>ν</sup> dx*) can be taken to be

$$
\Pi\_k \psi\_{nr+j}^{(\nu)}(\mathfrak{x}) = \delta\_{kj} \psi\_{nr+j}^{(\nu)}(\mathfrak{x}) .
$$

It is clear that they form a system of resolution of the identity:

$$
\Pi\_0 + \Pi\_1 + \dots + \Pi\_{r-1} = 1, \quad \Pi\_i \Pi\_j = \delta\_{ij} \Pi\_i, \quad \Pi\_j^\* = \Pi\_j. \tag{13}
$$

Note that the orthogonal projection Π*<sup>j</sup>* is related to the Klein operator *K* by

$$
\Pi\_{\vec{\jmath}} = \frac{1}{r} \sum\_{l=0}^{r-1} \varepsilon\_r^{-l\vec{\jmath}} K^l \dots
$$

#### **3. Fractional Supersymmetric Dunkl Harmonic Oscillator**

In this section, we shall present a construction of the fractional supersymmetric quantum mechanics of order *r* (*r* = 2, 3, ...) by using the generalized Klein's operator defined in Equation (11). Following Khare [6,7], a fractional supersymmetric quantum mechanics model of arbitrary order *r* can be developed by generalizing the fundamental Equations (1) to the forms

$$\begin{aligned} Q\_{\pm}^{\prime} &= 0, \quad [H, Q\_{\pm}] = 0, \quad Q\_{-}^{\dagger} = Q\_{+\prime} \\ Q\_{-}^{\prime -2}H &= Q\_{-}^{\prime -1}Q\_{+} + Q\_{-}^{\prime -2}Q\_{+}Q\_{-} + \cdots + Q\_{-}Q\_{+}Q\_{-}^{\prime -2} + Q\_{+}Q\_{-}^{\prime -1}. \end{aligned}$$

We introduce the supercharges *Q*<sup>−</sup> and *Q*<sup>+</sup> as :

$$Q\_{-}=\frac{1}{\sqrt{2}}(\boldsymbol{\Upsilon}\_{\boldsymbol{\nu}}+\boldsymbol{x})(\boldsymbol{1}-\boldsymbol{\Pi}\_{0}),\qquad Q\_{+}=\frac{1}{\sqrt{2}}(\boldsymbol{1}-\boldsymbol{\Pi}\_{0})(-\boldsymbol{\Upsilon}\_{\boldsymbol{\nu}}+\boldsymbol{x})\tag{14}$$

and the fractional supersymmetric Dunkl harmonic oscillator H*<sup>ν</sup>* by

$$\mathcal{H}\_{\mathbf{v}} = -(r-1)\frac{1}{2}\mathcal{Y}\_{\mathbf{v}}^2 + (r-1)\frac{1}{2}\mathbf{x}^2 - \sum\_{k=0}^{r-1} \Theta\_k \Pi\_{r-k-1\prime} \tag{15}$$

where

$$
\Theta\_k = \frac{(r-1)(r-2k)}{2} + 2\nu \left[ \frac{2r + (-1)^k - 1}{4} \right] R, \quad k = 0, \cdots, r - 1,\tag{16}
$$

and recall that [ . ] denotes the greatest integer function. Obviously, the operators *Q*<sup>±</sup> and H*<sup>ν</sup>* with common domain <sup>S</sup>(R) are densely defined in the Hilbert space *<sup>L</sup>*2(R, <sup>|</sup>*x*<sup>|</sup> <sup>2</sup>*<sup>ν</sup> dx*) and have the Hermitian conjugation relations

$$\mathcal{H}\_{\text{V}}^{\*} = \mathcal{H}\_{\text{V}} \qquad \mathbb{Q}\_{-}^{\*} = \mathbb{Q}\_{+}.\tag{17}$$

Furthermore, they satisfy the intertwining relations valid for *s* = 0, ··· , *r* − 1:

$$
\Pi\_{\mathfrak{s}} Q\_- = Q\_- \Pi\_{\mathfrak{s}+1}, \qquad Q\_+ \Pi\_{\mathfrak{s}} = \Pi\_{\mathfrak{s}+1} Q\_+, \qquad \mathcal{H}\_{\mathfrak{v}} \Pi\_{\mathfrak{s}} = \Pi\_{\mathfrak{s}} \mathcal{H}\_{\mathfrak{v}}.\tag{18}
$$

**Proposition 1.** *The supercharges Q*<sup>±</sup> *are nilpotent operators of order r*.

**Proof.** By making use of the following relations

$$\mathbf{Y}\_{\nu}\Pi\_{\mathbf{s}}=\Pi\_{\mathbf{s}-1}\mathbf{Y}\_{\nu} \quad \text{ x } \Pi\_{\mathbf{s}}=\Pi\_{\mathbf{s}+1}\mathbf{x} \tag{19}$$

we can easily show by induction that

$$Q\_{-}^{k} = \begin{cases} A\_{-}^{k} \left( 1 - \sum\_{s=0}^{k-1} \Pi\_{s} \right), & \text{if } 1 \le k \le r - 1, \\ 0, & \text{if } k = r. \end{cases} \tag{20}$$

Since *Q*+ = *Q*∗ <sup>−</sup>, we also have *<sup>Q</sup><sup>r</sup>* <sup>+</sup> = 0.

The first main result is

**Theorem 1.** *The Hermitian operators Q*−, *Q*<sup>+</sup> *and* H*<sup>ν</sup> defined in Equations* (14) *and* (15) *satisfy the commutation relations:*

$$\begin{aligned} (i) \quad &Q\_{\pm}^{\mathsf{T}} = 0, \quad [\mathcal{H}\_{\mathsf{V}}, \mathcal{Q}\_{\pm}] = 0, \quad \mathcal{Q}\_{-}^{\dagger} = \mathcal{Q}\_{+}, \\\ (ii) \quad &Q\_{-}^{r-2} \mathcal{H}\_{\mathsf{V}} = \mathcal{Q}\_{-}^{r-1} \mathcal{Q}\_{+} + \mathcal{Q}\_{-}^{r-2} \mathcal{Q}\_{+} \mathcal{Q}\_{-} + \dots + \mathcal{Q}\_{-} \mathcal{Q}\_{+} \mathcal{Q}\_{-}^{r-2} + \mathcal{Q}\_{+} \mathcal{Q}\_{-}^{r-1}. \end{aligned}$$

**Proof.** From the commutation relation (7), we can show by induction that

$$A\_{+}A\_{-}^{k} = A\_{-}^{k}A\_{+} - \vartheta\_{k}A\_{-}^{k-1}, \quad k \ge 1,\tag{21}$$

where

$$\theta\_k = \begin{cases} k, & \text{if } k \text{ is even,} \\ k + 2\nu R, & \text{if } k \text{ is odd.} \end{cases} \tag{22}$$

Combining this with Equation (20), we obtain, for, *k* = 1, ··· ,*r* − 2:

$$\begin{array}{rcl} Q\_{+}Q\_{-}^{r-1} & = & A\_{-}^{r-2} \left( A\_{-}A\_{+} - \vartheta\_{r-1} \right) \Pi\_{r-1}, \\ Q\_{-}^{r-1}Q\_{+} & = & A\_{-}^{r-2}A\_{-}A\_{+} \Pi\_{r-2}, \\ Q\_{-}^{r-1-k}Q\_{+}Q\_{-}^{k} & = & A\_{-}^{r-2} \left( A\_{-}A\_{+} - \vartheta\_{k} \right) \left( \Pi\_{r-2} + \Pi\_{r-1} \right). \end{array}$$

Additionally, a straightforward computation shows that

$$\sum\_{k=1}^{r-1} \theta\_k = \frac{r(r-1)}{2} + 2\nu[\frac{r}{2}] \, R.$$

Thus, we get

$$\sum\_{k=0}^{r-1} Q\_{-}^{r-1-k} Q\_{+} Q\_{-}^{k} = A\_{-}^{r-2} \left[ (r-1) A\_{-} A\_{+} \left( \Pi\_{r-2} + \Pi\_{r-1} \right) - \left( \sum\_{k=1}^{r-2} \theta\_{k} \right) \Pi\_{r-2} - \left( \sum\_{k=1}^{r-1} \theta\_{k} \right) \Pi\_{r-1} \right]$$

$$= Q\_{-}^{r-2} \left[ (r-1) A\_{-} A\_{+} - \Theta\_{1} \Pi\_{r-2} - \Theta\_{0} \Pi\_{r-1} \right].\tag{23}$$

*Mathematics* **2020**, *8*, 193

From Equation (13), we easily see that

$$\left(\Pi\_{r-2} + \Pi\_{r-1}\right) \sum\_{k=2}^{r-1} \Theta\_k \Pi\_{r-k-1} = 0\_r$$

and combining with Equation (23), we get

$$\sum\_{k=0}^{r-1} Q\_{-}^{r-1-k} Q\_{+} Q\_{-}^{k} = Q\_{-}^{r-2} \mathcal{H}\_{\vee}.$$

It remains to prove that [H*ν*, *Q*−]=[H*ν*, *Q*+] = 0. Observe that for *k* = 0, ··· , *r* − 1, we have

$$r - 1 = [\frac{2r + (-1)^k - 1}{4}] + [\frac{2r + (-1)^{k+1} - 1}{4}]\_r$$

and then, for *k* = 0, ··· , *r* − 2, we have

$$
\Theta\_k - (r - 1)(1 - 2\nu \mathbb{R}) = \Theta\_{k+1} \tag{24}
$$

which leads to

$$\begin{aligned} Q\_-\mathcal{H}\_V &= \left\{ (r-1)A\_-A\_+ + 1 - 2\nu \mathcal{R} - \sum\_{k=0}^{r-2} \Theta\_k \Pi\_{r-k-2} \right\} A\_- (1 - \Pi\_0) \\ &= \left\{ (r-1)A\_-A\_+ - \sum\_{k=0}^{r-2} \Theta\_{k+1} \Pi\_{r-k-2} \right\} (1 - \Pi\_{r-1})A\_- \\ &= \mathcal{H}\_V Q\_-. \end{aligned}$$

Finally, we have obtained [H*ν*, *Q*−] = 0, and since the operator H*<sup>ν</sup>* is self-adjoint and *Q*<sup>+</sup> = *Q*<sup>∗</sup> −, we conclude that [H*ν*, *Q*+] = 0.

**Proposition 2.** *For even integer r*, *the fractional supersymmetric Dunkl harmonic oscillator* H*<sup>ν</sup> has r*/2*-fold degenerate spectrum and acts on the wave functions <sup>ψ</sup>*(*ν*) *<sup>n</sup>* (*x*) *as:*

$$\mathcal{H}\_{\nu} \psi\_{nr+s}^{(\nu)}(\mathbf{x}) = \lambda\_{nr} \psi\_{nr+s}^{(\nu)}(\mathbf{x}), \quad s = 0, 1, \dots \\ r - 1, \quad n = 0, 1, 2, \dots$$

*where*

$$
\lambda\_{nr} = (r-1)(nr+\nu+\frac{r+1}{2}) + (-1)^s \nu r, \quad s = 0, \dots, r-1.
$$

**Proof.** From ([15] [formulas (3.7.1) and (3.7.2)]), the creation and annihilation operators *A*<sup>+</sup> and *A*<sup>−</sup> act on the wave functions *ψ<sup>ν</sup> nr*+*<sup>s</sup>* as:

$$\begin{aligned} A\_- \psi\_{nr+s}^\nu &= \sqrt{nr + s + \nu(1 - (-1)^s)} \psi\_{nr+s-1}^\nu \\ A\_+ \psi\_{nr+s}^\nu &= \sqrt{nr + s + 1 + \nu(1 - (-1)^{s+1})} \psi\_{nr+s+1}^\nu \end{aligned}$$

Then, the supercharges *Q*<sup>−</sup> and *Q*<sup>+</sup> take the value

$$Q\_{-}\psi\_{nr+s}^{\nu} = \sqrt{\left(nr+s+\nu(1-(-1)^{s})\right)/2} \,\psi\_{nr+s-1^{s}}^{\nu} \text{ s} = 1, \cdot, \cdot, r-1,\tag{25}$$

$$Q\_{+} \psi\_{nr+s}^{\nu} = \sqrt{\left(nr+s+1+\nu(1-(-1)^{s+1})\right)/2} \left.\psi\_{nr+s+1}^{\nu}, s = 0, \cdots, r-2,\tag{26}$$

$$Q\_-\Psi\_{nr}^\nu = 0, \quad Q\_+\Psi\_{(n+1)r-1}^\nu = 0. \tag{27}$$

A straightforward computation shows that

$$\mathcal{H}\_{\boldsymbol{\nu}} \boldsymbol{\psi}\_{nr+s}^{\boldsymbol{\nu}} = \lambda\_{nr} \boldsymbol{\psi}\_{nr+s\boldsymbol{\nu}}^{\boldsymbol{\nu}} \quad \text{s} = 0, \cdot \cdot \cdot , r-1, \boldsymbol{\nu}$$

where *<sup>λ</sup>nr* = (*<sup>r</sup>* <sup>−</sup> <sup>1</sup>)(*nr* <sup>+</sup> *<sup>ν</sup>* <sup>+</sup> *<sup>r</sup>*+<sup>1</sup> <sup>2</sup> )+(−1)*sνr*.

#### **4. Supersymmetric Generalized Hermite Polynomials**

#### *4.1. Associated Generalized Hermite Polynomials*

Starting form the following recurrence relations for the generalized Hermite polynomials {*H*(*ν*) *<sup>n</sup>* (*x*)},

$$\begin{aligned} H\_{n+1}^{(\upsilon)}(\mathbf{x}) &= 2\mathbf{x}H\_n^{(\upsilon)}(\mathbf{x}) - 2\left(\mathbf{n} + \nu(1 - (-1)^n)\right)H\_{n-1}^{(\upsilon)}(\mathbf{x})\\ H\_0^{(\upsilon)}(\mathbf{x}) &= 1, \quad H\_1^{(\upsilon)}(\mathbf{x}) = 2\mathbf{x}, \end{aligned} \tag{28}$$

given in [15–17], one defines, for each real number *<sup>c</sup>*, the system of polynomials *<sup>H</sup>*(*ν*) *<sup>n</sup>* (*x*, *<sup>c</sup>*) by the recurrence relation:

$$H\_{n+1}^{(\nu)}(\mathbf{x}, \mathbf{c}) = 2\mathbf{x}H\_n^{(\nu)}(\mathbf{x}, \mathbf{c}) - 2\left(n + \mathbf{c} + \nu(1 - (-1)^n)\right)H\_{n-1}^{(\nu)}(\mathbf{x}, \mathbf{c}),\tag{29}$$

with initial conditions

$$H\_0^{(\nu)}(\mathbf{x}, \mathbf{c}) = 1, \quad H\_1^{(\nu)}(\mathbf{x}, \mathbf{c}) = \mathbf{2}\mathbf{x}.\tag{30}$$

Now, assume that

$$c > 0, \quad c + 2\nu > -1. \tag{31}$$

By Favard's theorem [16], it follows that the family of polynomials {*H*(*ν*) *<sup>n</sup>* (*x*, *<sup>c</sup>*)} satisfying the recurrence relation (29) and the initial condition (30), is orthogonal with respect to some positive measure on the real line. We shall refer to the polynomials {*H*(*ν*) *<sup>n</sup>* (*x*, *<sup>c</sup>*)} as the associated generalized Hermite polynomials. As shown in ([18] Theorem 5.6.1)(see also [19–21]), there are two different systems of associated Laguerre polynomials denoted by *L*(*ν*) *<sup>n</sup>* (*x*, *<sup>c</sup>*) and <sup>L</sup>(*ν*) *<sup>n</sup>* (*x*, *c*). They satisfy the recurrence relations:

$$(2\mathbf{u} + 2\mathbf{c} + \nu + 1 - \mathbf{x})L\_n^{(\nu)}(\mathbf{x}, \mathbf{c}) = (\mathbf{u} + \mathbf{c} + 1)L\_{n+1}^{(\nu)}(\mathbf{x}, \mathbf{c}) + (\mathbf{u} + \mathbf{c} + \nu)L\_{n-1}^{(\nu)}(\mathbf{x}, \mathbf{c}),\tag{32}$$

$$L\_0^{(\nu)}(\mathbf{x}, \mathbf{c}) = 1, \quad L\_1^{(\nu)}(\mathbf{x}, \mathbf{c}) = \frac{2\mathbf{c} + \nu + 1 - \mathbf{x}}{\mathbf{c} + 1} \tag{33}$$

and

$$(2n+2c+\nu+1-\mathbf{x})\mathcal{L}\_n^{(\nu)}(\mathbf{x},\mathbf{c}) = (n+c+1)\mathcal{L}\_{n+1}^{(\nu)}(\mathbf{x},\mathbf{c}) + (n+c+\nu)\mathcal{L}\_{n-1}^{(\nu)}(\mathbf{x},\mathbf{c}),\tag{34}$$

$$
\mathcal{L}\_0^{(\nu)}(\mathbf{x}, \mathbf{c}) = 1, \qquad \mathcal{L}\_1^{(\nu)}(\mathbf{x}, \mathbf{c}) = \frac{\mathbf{c} + \nu + 1 - \mathbf{x}}{\mathbf{c} + 1}. \tag{35}
$$

Recall the Tricomi function Ψ(*a*, *c*; *x*) given by

$$\Psi(a,c;\mathbf{x}) = \frac{1}{\Gamma(a)} \int\_0^\infty e^{-\mathbf{x}t} t^{a-1} (1+t)^{c-a-1} \, dt, \qquad \Re(a), \Re(\mathbf{x}) > 0.$$

By [18], the polynomials *L*(*ν*) *<sup>n</sup>* (*x*, *<sup>c</sup>*) and <sup>L</sup>(*ν*) *<sup>n</sup>* (*x*, *c*) satisfy the orthogonality relations

$$\int\_{0}^{\infty} L\_{n}^{(\nu)}(\mathbf{x}, \mathbf{c}) L\_{m}^{(\nu)}(\mathbf{x}, \mathbf{c}) \ge \mathbf{c}^{\nu} e^{-\mathbf{x}} \frac{|\Psi(\mathbf{c}, 1 - \nu; \mathbf{x} e^{-i\pi})|^{-2}}{\Gamma(\mathbf{c} + 1)\Gamma(\nu + \mathbf{c} + 1)} d\mathbf{x} \quad = \frac{(\nu + \mathbf{c} + 1)\_{\mathbf{n}}}{(\mathbf{c} + 1)\_{\mathbf{n}}} \delta\_{\mathbf{n}\mathbf{m}},\tag{36}$$
 
$$\int\_{0}^{\infty} \mathbf{c}(\mathbf{y}) \, \delta\_{\mathbf{n}\mathbf{m}}(\mathbf{y}) \, \delta\_{\mathbf{n}\mathbf{m}} \, \dots \, \mathbf{y} \, \, \exp\left[\mathbf{Y}(\mathbf{c}, -\nu; \mathbf{x} e^{-i\pi})\right]^{-2} \, \dots \qquad (\nu + \mathbf{c} + 1)\_{\mathbf{n}} \, \dots$$

$$\int\_{0}^{\infty} \mathcal{L}\_{\text{n}}^{(\nu)}(\mathbf{x}, \mathbf{c}) \mathcal{L}\_{\text{m}}^{(\nu)}(\mathbf{x}, \mathbf{c}) \ge \mathbf{x}^{\nu} e^{-\mathbf{x}} \frac{|\Psi(\mathbf{c}, -\nu; \mathbf{x} e^{-i\pi})|^{-2}}{\Gamma(\mathbf{c} + 1)\Gamma(\nu + \mathbf{c} + 1)} d\mathbf{x} \quad = \frac{(\nu + \mathbf{c} + 1)\_{\text{n}}}{(\mathbf{c} + 1)\_{\text{n}}} \delta\_{nm\nu} \tag{37}$$

when one of the following conditions is satisfied:

*ν* + *c* > −1, *c* ≥ 0 or *ν* + *c* ≥ −1, *c* ≥ −1.

The monic polynomial version of *H<sup>ν</sup> <sup>n</sup>*(*x*, *c*) is given by

$$\mathcal{H}\_n^{(\nu)}(\mathbf{x}, \mathbf{c}) = \mathbf{2}^{-n} H\_n^{(\nu)}(\mathbf{x}, \mathbf{c}), \ n = 0, 1, \cdots, \nu$$

and satisfies

$$\begin{aligned} \mathcal{H}\_{n+1}^{(\upsilon)}(\mathbf{x}, \mathbf{c}) &= \mathbf{x} \mathcal{H}\_{n}^{(\upsilon)}(\mathbf{x}, \mathbf{c}) - \frac{1}{2} \Big( \mathbf{n} + \mathbf{c} + \nu (1 - (-1)^{n}) \big) \mathcal{H}\_{n-1}^{(\upsilon)}(\mathbf{x}, \mathbf{c}), \\ \mathcal{H}\_{-1}^{(\upsilon)}(\mathbf{x}, \mathbf{c}) &= 0, \quad \mathcal{H}\_{0}^{(\upsilon)}(\mathbf{x}, \mathbf{c}) = 1. \end{aligned} \tag{38}$$

It is easy to see that the polynomial (−1)*n*H(*ν*) *<sup>n</sup>* (−*x*, *<sup>c</sup>*) also satisfies (38). Thus,

$$
\mathcal{H}\_n^{(\upsilon)}(-\mathfrak{x}, \mathfrak{c}) = (-1)^n \mathcal{H}\_n^{(\upsilon)}(\mathfrak{x}, \mathfrak{c}).
$$

Thus, by induction, we write them in the form

$$\mathcal{H}\_{2n}^{(\nu)}(\mathbf{x}, \mathbf{c}) = \mathcal{S}\_n(\mathbf{x}^2) \quad \text{and} \quad \mathcal{H}\_{2n+1}^{(\nu)}(\mathbf{x}, \mathbf{c}) = \mathbf{x} \mathcal{Q}\_n(\mathbf{x}^2), \tag{39}$$

where *Sn*(*x*), *Qn*(*x*) are monic polynomials of degree *n*.

**Theorem 2.** *Let <sup>c</sup>* <sup>&</sup>gt; <sup>0</sup> *and <sup>ν</sup>* <sup>&</sup>gt; <sup>−</sup>*c*/2. *The associated generalized Hermite polynomials <sup>H</sup>*(*ν*) *<sup>n</sup>* (*x*, *<sup>c</sup>*)*, defined in* (29)*, have the explicit form:*

$$\begin{aligned} H\_{2n}^{(\nu)}(\mathbf{x}, \mathbf{c}) &= (-1)^n 2^{2n} (1 + c/2)\_n \mathcal{L}\_n^{(\nu - 1/2)}(\mathbf{x}^2, \mathbf{c}/2), \\ H\_{2n+1}^{(\nu)}(\mathbf{x}, \mathbf{c}) &= (-1)^n 2^{2n+1} (1 + c/2)\_n \ge L\_n^{(\nu + 1/2)}(\mathbf{x}^2, \mathbf{c}/2), \end{aligned}$$

*and the orthogonality relations*

$$\int\_{\mathbb{R}} H\_n^{(\upsilon)}(\mathbf{x}, \mathbf{c}) H\_m^{(\upsilon)}(\mathbf{x}, \mathbf{c}) \, |\mathbf{x}|^{2\nu} e^{-\mathbf{x}^2} \frac{|\Psi(\mathbf{c}/2, 1/2 - \nu; \mathbf{x}^2 e^{-i\pi\tau})|^{-2}}{\Gamma(1 + \varepsilon/2)\Gamma(\nu + \varepsilon/2 + 1/2)} = \zeta\_n \, \delta\_{nm\_\tau} \tag{40}$$

*where*

$$
\zeta\_n = \begin{cases}
2^{4k} (1 + c/2)\_k (\nu + c/2 + 1/2)\_{k'} & \text{if} \quad n = 2k, \\
2^{4k+2} (1 + c/2)\_k (\nu + c/2 + 3/2)\_{k'} & \text{if} \quad n = 2k + 1.
\end{cases}
$$

**Proof.** It is directly verified that the polynomials *Sn*(*x*), *Qn*(*x*) given in (39) are orthogonal as they satisfy the recurrence relations

$$\begin{array}{rcl} \mathcal{S}\_{n+1}(\mathbf{x}) &=& (\mathbf{x} - (2\mathbf{n} + \mathbf{c} + \mathbf{v} + 1/2)) \mathcal{S}\_{\mathbf{n}}(\mathbf{x}) - (\mathbf{n} + \mathbf{c}/2), \\ &\quad \times (\mathbf{n} + \mathbf{c}/2 - 1/2 + \mathbf{v}) \mathcal{S}\_{\mathbf{n}-1}(\mathbf{x}), \\ \mathcal{S}\_{-1}(\mathbf{x}) &=& 0, \quad \mathcal{S}\_{0}(\mathbf{x}) = 1, \end{array}$$

and

$$\begin{array}{rcl} Q\_{n+1}(\mathbf{x}) &=& \left( \mathbf{x} - (2n + c + 3/2 + \nu) \right) Q\_n(\mathbf{x}) - \left( n + c/2 \right) \text{big}(\mathbf{x}) \\ &\times \left( n + (1 + c)/2 + \nu \right) Q\_{n-1}(\mathbf{x}) \\ Q\_{-1}(\mathbf{x}) &=& 0, \quad Q\_0(\mathbf{x}) = 1. \end{array}$$

From Equation (32), we see that the polynomials *Sn*(*x*) satisfy the same recurrence relation as (−1)*n*(<sup>1</sup> + *<sup>c</sup>*/2)*n*L(*ν*−1/2) *<sup>n</sup>* (*x*, *<sup>c</sup>*/2), so that

$$S\_n(\mathbf{x}) = (-1)^n (1 + c/2)\_n \mathcal{L}\_n^{(v - 1/2)}(\mathbf{x}, c/2). \tag{41}$$

A similar analysis shows that

$$Q\_{\mathbb{H}}(\mathbf{x}) = (-1)^{n} (1 + \mathfrak{c}/2)\_{\mathbb{H}} L\_{n}^{(v+1/2)}(\mathbf{x}, \mathfrak{c}/2). \tag{42}$$

In view of Equations (41) and (42), the explicit form of the associated generalized Hermite polynomials is given by

$$H\_{2n}^{(\upsilon)}(\mathbf{x}, \mathbf{c}) = (-1)^n 2^{2n} (1 + \mathbf{c}/2)\_n \mathcal{L}\_n^{(\upsilon - 1/2)}(\mathbf{x}^2, \mathbf{c}/2), \tag{43}$$

$$H\_{2n+1}^{(\nu)}(\mathbf{x}, \mathbf{c}) = (-1)^n 2^{2n+1} (1 + c/2)\_n \text{x} L\_n^{(\nu + 1/2)}(\mathbf{x}^2, \mathbf{c}/2). \tag{44}$$

From Equations (36) and (37), we deduce that the system <sup>H</sup>*<sup>ν</sup> <sup>n</sup>*(*x*, *c*) satisfies the orthogonality relations

$$\int\_{\mathbb{R}} H\_n^{(\upsilon)}(\mathbf{x}, \mathbf{c}) H\_m^{(\upsilon)}(\mathbf{x}, \mathbf{c}) \, |\mathbf{x}|^{2\upsilon} e^{-\mathbf{x}^2} \frac{|\Psi(\mathbf{c}/2, 1/2 - \upsilon; \mathbf{x}^2 e^{-i\pi\tau})|^{-2}}{\Gamma(1 + \upsilon/2)\Gamma(\upsilon + \upsilon/2 + 1/2)} = \zeta\_n \, \delta\_{nm\prime} \tag{45}$$

with

$$\zeta\_{\mathbb{M}} = \begin{cases} 2^{4k} (1 + c/2)\_k (\nu + c/2 + 1/2)\_{k\prime} & \text{if} \quad n = 2k, \\ 2^{4k+2} (1 + c/2)\_k (\nu + c/2 + 3/2)\_{k\prime} & \text{if} \quad n = 2k+1. \end{cases}$$

#### *4.2. Supersymmetric Generalized Hermite Polynomials*

In the sequel, we assume that *r* is an even integer and we consider the Hermitian supercharge operator *<sup>Q</sup>*, defined on <sup>S</sup>(R), by

$$Q = \frac{1}{\sqrt{2}} \mathcal{Y}\_{\boldsymbol{\nu}} (\boldsymbol{\Pi}\_{\boldsymbol{r}-1} - \boldsymbol{\Pi}\_0) + \frac{\boldsymbol{\varkappa}}{\sqrt{2}} (2 - \boldsymbol{\Pi}\_0 - \boldsymbol{\Pi}\_{\boldsymbol{r}-1}).$$

From Equation (14), we have

$$Q = \frac{1}{\sqrt{2}}(Q\_- + Q\_+)\_{\prime}$$

so it has a self-adjoint extension on *<sup>L</sup>*2(R, <sup>|</sup>*x*<sup>|</sup> <sup>2</sup>*νdx*). Furthermore, it acts on the basis *ψ<sup>ν</sup> <sup>n</sup>* as

$$\begin{aligned} Q\psi\_{nr+s}^{\nu} &= a\_s^{(n)}\psi\_{nr+s-1}^{\nu} + a\_{s+1}^{(n)}\psi\_{nr+s+1}^{\nu} \text{ s} = 1, \dots, r-1, \\ Q\psi\_{nr}^{\nu} &= a\_1^{(n)}\psi\_{nr+1}^{\nu} \quad Q\psi\_{(n+1)r-1}^{\nu} = a\_{r-1}^{(n)}\psi\_{(n+1)r-1}^{\nu} \end{aligned} \tag{46}$$

where

$$a\_s^{(n)} := \sqrt{(nr + s + \nu(1 - (-1)^s))/2}, \qquad s = 1, \dots, r - 1.$$

On the other hand, by (46), we see that the operator *Q* leaves invariant the finite dimensional subspace of *<sup>L</sup>*2(R, <sup>|</sup>*x*<sup>|</sup> <sup>2</sup>*νdx*) generated by *ψ<sup>ν</sup> nr*+*s*, *s* = 0, 1, ··· , *r* − 1. Hence, *Q* can be represented in this basis by the following *<sup>r</sup>* <sup>×</sup> *<sup>r</sup>* tridiagonal Jacobi matrix *<sup>A</sup>*(*n*) *<sup>r</sup>*

$$A\_r^{(n)} = \begin{pmatrix} 0 & a\_1^{(n)} & 0 & & & \\ a\_1^{(n)} & 0 & a\_2^{(n)} & 0 & & \\ 0 & a\_2^{(n)} & 0 & a\_3^{(n)} & \ddots & & \\ & \ddots & \ddots & \ddots & \ddots & 0 \\ & & \ddots & a\_{r-2}^{(n)} & 0 & a\_{r-1}^{(n)} \\ & & & 0 & a\_{r-1}^{(n)} & 0 \end{pmatrix}$$

It is well known that, if the coefficients of the subdiagonal of some Jacobi Matrix are different from zero, then all the eigenvalues of this matrix are real and nondegenerate [16]. We introduce the normalized eigenvectors *φ<sup>s</sup>* of the supercharge *Q*

$$Q\phi\_s = \mathbf{x}\_s \phi\_s, \qquad s = 0, \ \cdots, r - 1 \tag{47}$$

.

that can be expanded in the basis *ψnr*+*k*, *k* = 0, 1, ··· , *r* − 1, as

$$\Phi\_{\mathfrak{k}} = \sum\_{k=0}^{r-1} \sqrt{w\_{\mathfrak{s}}} p\_k(\mathbf{x}\_{\mathfrak{s}}) \psi\_{nr+k\mathfrak{s}} \tag{48}$$

where the coefficients *pk* obey the three-term recurrence relation [22]

$$\begin{aligned} a\_k^{(n)} p\_{k-1}(\mathbf{x}) + a\_{k+1}^{(n)} p\_{k+1}(\mathbf{x}) &= \mathbf{x} p\_k(\mathbf{x}), \\ p\_{-1}(\mathbf{x}) = 0, \quad p\_0(\mathbf{x}\_s) &= 1, \end{aligned}$$

Hence, they become orthogonal polynomials. We denote by *Pk*(*x*), the monic orthogonal polynomial related to *pk*(*x*) by

$$P\_k(\mathbf{x}) = h\_k p\_k(\mathbf{x}),\tag{49}$$

where

$$h\_k = a\_k^{(n)} \cdot \cdots \cdot a\_1^{(n)} \tag{50}$$

and satisfying

$$\begin{aligned} \mathbf{x}P\_k(\mathbf{x}) &= P\_{k+1}(\mathbf{x}) + \frac{1}{2} (k + nr + \nu(1 - (-1)^k)) P\_{k-1}(\mathbf{x}), \ \ k = 0, \cdots, r - 1, \\\ P\_{-1}(\mathbf{x}) &= 0, \quad P\_0(\mathbf{x}) = 1. \end{aligned} \tag{51}$$

From the three terms recurrence relations (51), the polynomials *Pk*(*x*) can be identified with the associated generalized Hermite polynomial <sup>H</sup>(*ν*) *<sup>k</sup>* (*x*, *c*), namely,

$$P\_k(\mathbf{x}) = \mathcal{H}\_k^{(\upsilon)}(\mathbf{x}, m\mathbf{r}).$$

It is well known from the theory of orthogonal polynomials that the eigenvalues of the Jacobi matrix *<sup>A</sup>*(*n*) *<sup>r</sup>* coincide with the roots of the characteristic polynomial <sup>H</sup>(*ν*) *<sup>r</sup>* (*x*, *nr*) [16,22]. The weights *ws* defined in (56) are given by the following formula

$$w\_{\mathfrak{s}} = \frac{h\_r^2}{\mathcal{H}\_{r-1}^{(\upsilon)}(\mathbf{x}\_{\mathfrak{s}}, nr) (\mathcal{H}\_r^{(\upsilon)})'(\mathbf{x}\_{\mathfrak{s}}, nr)},\tag{52}$$

where (H(*ν*) *<sup>r</sup>* ) (*x*, *nr*) denotes the derivative of <sup>H</sup>(*ν*) *<sup>r</sup>* (*x*, *nr*), *hr* is defined in Equation (50) and *xnr*,1 <sup>&</sup>gt; ··· <sup>&</sup>gt; *xnr*,*<sup>r</sup>* are the zeros of <sup>H</sup>(*ν*) *<sup>r</sup>* (*x*, *nr*). For more detail, we refer to [16]. Then, it turns out that

$$\Phi\_{\sf s} = \sum\_{k=0}^{r-1} \mu\_{\sf ks}^{(n)} \Psi\_{nr+k\sf \prime} \tag{53}$$

where

$$u\_{ks}^{(n)} = \frac{h\_r}{h\_k} \frac{\mathcal{H}\_k^{(\upsilon)}(\mathbf{x}\_s, nr)}{(\mathcal{H}\_{r-1}^{(\upsilon)}(\mathbf{x}\_s, nr)(\mathcal{H}\_r^{(\upsilon)})'(\mathbf{x}\_s, nr))^{1/2}}, \quad 0 \le s, k \le r - 1. \tag{54}$$

Since both bases {*ψnr*+*k*, *k* = 0, ··· *r* − 1} and {*φs*, *s* = 0, ··· *r* − 1} are orthonormal and all the coefficients are real, then the matrix (*u*(*n*) *ks* ) is orthogonal and hence the system {H(*ν*) *<sup>k</sup>* (*x*)} becomes orthogonal polynomials:

$$\sum\_{s=0}^{r-1} w\_s \mathcal{H}\_k^{(\upsilon)}(\mathbf{x}\_s) \mathcal{H}\_{k'}^{(\upsilon)}(\mathbf{x}\_s) = \delta\_{kk'} / h\_k^2. \tag{55}$$

We call supersymmetric generalized Hermite polynomials the orthogonal polynomials, denoted by H(*r*,*ν*) *<sup>N</sup>* (*x*), extracted form the orthogonal function *φs*:

$$\mathbb{H}\_N^{(r,\nu)}(\mathbf{x}) = \sum\_{k=0}^{r-1} H\_k^{(\nu)}(\mathbf{x}\_{\nu}, nr) H\_{nr+k}^{(\nu)}(\mathbf{x}), \ N = nr + \mathbf{s}, \tag{56}$$

and we obtain the following:

**Theorem 3.** *The supersymmetric generalized Hermite polynomials* H(*r*,*ν*) *<sup>N</sup>* (*x*) *satisfy the orthogonality relations*

$$\int\_{-\infty}^{\infty} \mathbb{H}\_N^{(r,\nu)}(\mathbf{x}) \mathbb{H}\_{N'}^{(r,\nu)}(\mathbf{x}) |\mathbf{x}|^{2\nu} e^{-\mathbf{x}^2} d\mathbf{x} = \varrho\_N \delta\_{NN'\nu} \tag{57}$$

*where <sup>N</sup>* = *γnr*/*ws for s* = 0, ··· *r* − 1 *and N* = *nr* + *s.*

**Proof.** From Equations (10) and (56), we obtain

$$\begin{aligned} \int\_{-\infty}^{\infty} \mathbb{H}\_{nr+s}^{(r,\nu)}(\mathbf{x}) \mathbb{H}\_{n'r+s'}^{(r,\nu)}(\mathbf{x}) \, |\, \mathbf{x}|^{2\nu} e^{-\mathbf{x}^2} \, d\mathbf{x} &= \delta\_{nn'} \sum\_{k=0}^{r-1} H\_k^{(\nu)}(\mathbf{x}\_{s}, nr) H\_k^{(\nu)}(\mathbf{x}\_{s'}, nr) \gamma\_{nr+k} \\ &= \delta\_{nn'} \gamma\_{nr} \sum\_{k=0}^{r-1} h\_k^2 \mathcal{H}\_k^{(\nu)}(\mathbf{x}\_{s\_{\nu}} nr) \mathcal{H}\_k^{(\nu)}(\mathbf{x}\_{s'}, nr) \end{aligned}$$

and, from ([18] Theorem 2.11.2), we obtain the dual orthogonality relation for {H(*ν*) *<sup>k</sup>* (*x*)}:

$$\sum\_{k=0}^{r-1} \mathcal{H}\_k^{(\upsilon)}(x\_s) \mathcal{H}\_k^{(\upsilon)}(x\_{s'}) \frac{[\frac{nr+k}{2}]! \, \Gamma(\frac{nr+k+1}{2} + \nu + \frac{1}{2})}{[\frac{nr}{2}]! \, \Gamma(\frac{nr+1}{2} + \nu + \frac{1}{2})} = \delta\_{ss'} / w\_s \tag{58}$$

and, finally,

$$\int\_{-\infty}^{\infty} \mathbb{H}\_{nr+s}^{(r,\nu)}(\mathfrak{x}) \mathbb{H}\_{n'r+s'}^{(r,\nu)}(\mathfrak{x}) \, |\, \mathfrak{x}|^{2\nu} \, \mathfrak{e}^{-\mathfrak{x}^2} \, d\mathfrak{x} = \delta\_{nn'} \delta\_{ss'} \gamma\_{nr} / w\_{\mathfrak{s}} .$$

**Author Contributions:** Formal analysis, F.B. and W.J.; Methodology, F.B. and W.J.; Writing—original draft, F.B. and W.J.; Writing—review and editing, F.B. and W.J. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saudi University for funding this Research Group No. (RG-1437-020).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


c 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
